INVISIBLE INK !
T HE A PPLIED M ATH S PELLBOOK
P IERSON G UTHREY
Version as of September
8, 2014
Pierson Guthrey
pguthrey@iastate.edu
CONTENTS
Contents
I
Elementary Logic and Proofs
8
1 Logic
1.1 Logical Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Truths, Implications, and Equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
II
Algebra
8
8
8
9
10
2 Structures and Properties
10
2.1 Algebraic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Morphisms
10
4 Algebraic Structures
4.1 Groups . . . . . . . . . . . . . . .
4.2 Ring . . . . . . . . . . . . . . . . .
4.3 Fields . . . . . . . . . . . . . . . .
4.3.1 Archimedian Property . . .
4.4 Fundamental Theorem of Algebra
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Analysis
11
11
11
11
11
11
12
5 Sets
5.1 Subsets . . . . . . . . . .
5.2 Collection of Sets . . . . .
5.3 Union and Intersect . . .
5.3.1 de Morgan’s Laws
5.4 Sequences (xn ) . . . . .
5.4.1 Subsequences . .
5.4.2 Nested Sequences
5.4.3 Limits lim pn = p
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12
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13
13
5.4.4 Cauchy Condition . . . . . . . . .
5.4.5 Series . . . . . . . . . . . . . . . .
5.5 Open and Closed Sets . . . . . . . . . . .
5.5.1 Limit Set . . . . . . . . . . . . . . .
5.5.2 Neighborhood . . . . . . . . . . .
5.5.3 Inheritance principle . . . . . . . .
5.5.4 Closed Sets . . . . . . . . . . . . .
5.5.5 Open Sets . . . . . . . . . . . . .
5.5.6 Neither closed nor open set . . . .
5.5.7 Clopen sets . . . . . . . . . . . . .
5.6 Closure, Interior, and Boundaries of Sets
5.6.1 Closure S, cl(S) . . . . . . . . . .
5.6.2 Interior int(S) . . . . . . . . . . . .
5.6.3 Boundary ∂S . . . . . . . . . . . .
5.7 Unit Pieces . . . . . . . . . . . . . . . . .
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13
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16
n→∞
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1
Pierson Guthrey
pguthrey@iastate.edu
CONTENTS
5.8 Set Properties . . .
5.8.1 Convexity . .
5.8.2 Set Diameter
5.9 Zero Set . . . . . . .
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16
16
16
16
6 Topology
6.0.1 Topological Invariance . . . . . . . . . . .
6.0.2 Topological Equivalence . . . . . . . . . .
6.1 Intervals . . . . . . . . . . . . . . . . . . . . . . .
6.2 - Principle . . . . . . . . . . . . . . . . . . . . .
6.2.0.1 Form 1: . . . . . . . . . . . . . .
6.2.0.2 Form 2: . . . . . . . . . . . . . .
6.2.1 Preimage f pre (V ) . . . . . . . . . . . . .
6.3 Cardinality . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Finite Set . . . . . . . . . . . . . . . . . .
6.3.2 Infinite Set . . . . . . . . . . . . . . . . .
6.3.3 Denumerable Sets . . . . . . . . . . . . .
6.3.4 Countable Sets . . . . . . . . . . . . . . .
6.3.5 Uncountable Sets . . . . . . . . . . . . .
6.4 Boundedness . . . . . . . . . . . . . . . . . . . .
6.4.1 Upper Bounds and Suprenums . . . . . .
6.4.2 Lower Bounds and Infimums . . . . . . .
6.5 Clustering and Condensing . . . . . . . . . . . .
6.5.1 Cluster Points S 0 . . . . . . . . . . . . . .
6.5.2 Condensation Points . . . . . . . . . . . .
6.6 Compactness . . . . . . . . . . . . . . . . . . . .
6.6.1 Precompact . . . . . . . . . . . . . . . . .
6.6.2 Nested Compacts . . . . . . . . . . . . .
6.6.3 Continuous Transformation on Compacts
6.6.4 Embeddedness . . . . . . . . . . . . . . .
6.7 Connectedness . . . . . . . . . . . . . . . . . . .
6.7.1 Totally Disconnected . . . . . . . . . . . .
6.7.2 Point Connected . . . . . . . . . . . . . .
6.7.3 Path Connected . . . . . . . . . . . . . .
6.8 Perfect Sets . . . . . . . . . . . . . . . . . . . . .
6.9 Dense Sets . . . . . . . . . . . . . . . . . . . . .
6.10 Coverings . . . . . . . . . . . . . . . . . . . . . .
6.10.0.1 Lebesgue Number of a Covering
6.11 Topological Objects . . . . . . . . . . . . . . . .
6.11.1 Topologist’s Sine Curve . . . . . . . . . .
6.11.2 The Cantor Sets . . . . . . . . . . . . . .
6.11.3 Explicit Representations . . . . . . . . . .
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17
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7 Real Analysis
7.1 Expansions . . . . . . . . . . . . .
7.1.1 Binomial Expansion . . . .
7.2 Metric Spaces (X, d) . . . . . . . .
7.2.1 Definition . . . . . . . . . .
7.2.2 Common metric spaces . .
7.2.3 Common metric spaces???
7.2.4 Common Metrics . . . . . .
7.2.5 Metric Subspaces . . . . .
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25
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.
Pierson Guthrey
pguthrey@iastate.edu
CONTENTS
7.3
7.4
7.5
7.6
7.7
7.8
7.2.6 Product Metrics . . . . . . . . . . . . . . . . . . .
7.2.6.1 Common Product Metrics . . . . . . . .
Equivalent Metrics . . . . . . . . . . . . . . . . . . . . .
7.3.1 Completeness . . . . . . . . . . . . . . . . . . .
7.3.1.1 Completions . . . . . . . . . . . . . . .
Linear Spaces (or Vector Space) . . . . . . . . . . . . .
Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5.1 Seminorms . . . . . . . . . . . . . . . . . . . . .
7.5.1.1 Seminorm Topology and Convergence
7.5.1.2 Hausdorff Topologies . . . . . . . . . .
7.5.1.3 Frechet space . . . . . . . . . . . . . .
7.5.2 Inner Product Spaces . . . . . . . . . . . . . . .
7.5.2.1 Normed Linear Spaces . . . . . . . . .
Banach Spaces . . . . . . . . . . . . . . . . . . . . . . .
7.6.1 Schauder Basis . . . . . . . . . . . . . . . . . . .
7.6.1.1 Isomorphisms . . . . . . . . . . . . . .
Bounded Linear Maps . . . . . . . . . . . . . . . . . . .
Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . .
7.8.1 Orthogonality . . . . . . . . . . . . . . . . . . . .
7.8.1.1 Projections . . . . . . . . . . . . . . . .
7.8.1.2 Direct Sum . . . . . . . . . . . . . . . .
7.8.1.3 Decomposition . . . . . . . . . . . . . .
7.8.2 Hilbert Bases . . . . . . . . . . . . . . . . . . . .
7.8.2.1 Gram Schmidt . . . . . . . . . . . . . .
7.8.2.2 Reisz Fischer Critertion . . . . . . . . .
7.8.2.3 Riemann Lebesgue Lemma . . . . . .
7.8.2.4 Version 1 . . . . . . . . . . . . . . . . .
7.8.2.5 Version 2 . . . . . . . . . . . . . . . . .
7.8.2.6 Version 3 . . . . . . . . . . . . . . . . .
7.8.2.7 Version 4 . . . . . . . . . . . . . . . . .
7.8.2.8 Version 5 . . . . . . . . . . . . . . . . .
7.8.2.9 Version 6 . . . . . . . . . . . . . . . . .
7.8.2.10 Bessel’s Inequality . . . . . . . . . . . .
7.8.2.11 Weierstrass Approximation Theorem .
7.8.3 Functionals on H . . . . . . . . . . . . . . . . . .
7.8.4 The Dual Space of H . . . . . . . . . . . . . . .
7.8.4.1 Riesz Representation Theorem . . . .
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26
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35
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35
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8 Functions
8.0.4.2 Graph . . . . . . . . . . . . . . . . . .
8.0.4.3 Linearity . . . . . . . . . . . . . . . .
8.0.4.4 Affine . . . . . . . . . . . . . . . . . .
8.0.4.5 Homogeneity . . . . . . . . . . . . . .
8.0.4.6 Injective function (1-1 or one-to-one) .
8.0.4.7 Surjective function (onto) . . . . . . .
8.0.4.8 Bijective . . . . . . . . . . . . . . . .
8.0.4.9 Convex . . . . . . . . . . . . . . . . .
8.0.4.10 Concave . . . . . . . . . . . . . . . .
8.0.4.11 Composition . . . . . . . . . . . . . .
8.0.4.12 Preimage . . . . . . . . . . . . . . . .
8.1 Function Sequences . . . . . . . . . . . . . . . . . . .
8.1.0.13 Uniform Norm . . . . . . . . . . . . .
3
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.
Pierson Guthrey
pguthrey@iastate.edu
CONTENTS
8.2 Essential Suprenum, Essential Infimum . . . . . . . . . . . . . . . . .
8.3 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3.1 Oscillation oscx (f ) . . . . . . . . . . . . . . . . . . . . . . . . .
8.3.2 Discontinuity set of f . . . . . . . . . . . . . . . . . . . . . . . .
8.3.3 Uniformly Continuous . . . . . . . . . . . . . . . . . . . . . . .
8.3.4 Lipschitz Continuous . . . . . . . . . . . . . . . . . . . . . . . .
8.3.5 Absolutely Continuous . . . . . . . . . . . . . . . . . . . . . . .
8.3.5.1 Analysis Definiton . . . . . . . . . . . . . . . . . . . .
8.3.5.2 Calculus Definiton . . . . . . . . . . . . . . . . . . . .
8.3.5.3 Fundamental Theorem of Lebesgue Integral Calculus
8.3.5.4 Properties of AC(I) . . . . . . . . . . . . . . . . . . .
8.3.6 Equicontinuous Families . . . . . . . . . . . . . . . . . . . . . .
8.3.7 Homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . .
8.3.8 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3.9 Fundamental Theorem of Continuous Functions . . . . . . . .
8.3.10 Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . .
8.4 Properties of Functions . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4.1 Support supp(f ) . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4.2 Other Properties . . . . . . . . . . . . . . . . . . . . . . . . . .
8.5 Special Types of Functions . . . . . . . . . . . . . . . . . . . . . . . .
8.6 Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.6.0.1 Set Properties of Function Spaces . . . . . . . . . . .
8.7 Function Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.7.1 Continuity Classes C(Ω) . . . . . . . . . . . . . . . . . . . . . .
8.7.2 Bounded Function Classes B(Ω) . . . . . . . . . . . . . . . . .
8.7.3 Differentiability Classes C m (Ω) . . . . . . . . . . . . . . . . . .
8.7.4 Integrability Classes R,Lp . . . . . . . . . . . . . . . . . . . . .
9 Calculus
9.1 Differentiation . . . . . . . . . . . . . . . . . . . . .
9.1.1 Mean Value Theorem (MVT) . . . . . . . .
9.1.2 Ratio Mean Value Theorem (RMVT) . . . .
9.1.3 L’Hospital’s Rule . . . . . . . . . . . . . . .
9.1.4 Intermediate Value Theorem for Derivatives
9.1.5 Inverse Function Theorem . . . . . . . . . .
9.2 Higher Derivatives and More Dimensions . . . . .
9.2.1 Leibniz Product Rule . . . . . . . . . . . . .
9.3 Analytic Functions C ω . . . . . . . . . . . . . . . .
9.3.1 Taylor Approximation Theorem . . . . . . .
9.4 Differentiability Classes C m (Ω) . . . . . . . . . . .
9.5 Integration . . . . . . . . . . . . . . . . . . . . . . .
9.5.1 Partitions . . . . . . . . . . . . . . . . . . .
9.5.2 Riemann Integration . . . . . . . . . . . . .
9.5.3 Darboux Integration . . . . . . . . . . . . .
9.5.4 Riemann-Lebesgue Theorem: . . . . . . .
9.5.5 Fundamental Theorem of Calculus . . . . .
9.5.6 Antiderivative . . . . . . . . . . . . . . . . .
9.5.7 Integration Techniques . . . . . . . . . . . .
9.5.7.1 Integration by Substitution . . . .
9.5.7.2 Integration by Parts . . . . . . . .
9.5.8 Improper Integrals . . . . . . . . . . . . . .
9.6 Higher Dimensional Calculus . . . . . . . . . . . .
4
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37
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48
48
48
49
49
49
49
49
Pierson Guthrey
pguthrey@iastate.edu
CONTENTS
9.6.1 Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
9.6.2 Green’s Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
9.6.3 Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
IV
Differential Equations
51
10 Solutions
51
10.1 General Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
10.1.1 Well-Posed Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
11 Ordinary Differential Equations
11.1 Senses of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.1.0.1 Classical Solution . . . . . . . . . . . . . . . . . . . . . . . .
11.1.0.2 Weak Solution . . . . . . . . . . . . . . . . . . . . . . . . . .
11.1.0.3 Distributional Solution . . . . . . . . . . . . . . . . . . . . . .
11.1.0.4 Regularity of Solutions . . . . . . . . . . . . . . . . . . . . .
11.1.1 Initial Value Problems (IVP) . . . . . . . . . . . . . . . . . . . . . . . .
11.1.2 Boundary Value Problems (BVP) . . . . . . . . . . . . . . . . . . . . .
11.1.3 Systems of ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2.1 General solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2.1.1 Linear Systems of ODEs . . . . . . . . . . . . . . . . . . . .
11.3 Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3.1 Exactly Solvable Cases . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3.2 Relation between Euler Equations and Constant Coefficient Equations
11.4 Nonlinear ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.4.1 Integraton Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.4.1.1 Integrating Factor . . . . . . . . . . . . . . . . . . . . . . . .
11.4.1.2 Variation of Parameters . . . . . . . . . . . . . . . . . . . . .
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51
52
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56
56
12 Integral Equations
12.0.2 Solutions . . . . . . . . . . . . . . . . . .
12.1 Fredholm IE of the First Kind; λ = 0 . . . . . . .
12.2 Fredholm IE of the Second Kind; λ ≤ 0 . . . . .
12.2.0.1 Solutions via Fixed Point Method
12.3 Special Kernels . . . . . . . . . . . . . . . . . . .
12.4 Volterra Equations . . . . . . . . . . . . . . . . .
12.4.1 Volterra IE of the First Kind (λ = 0) . . . .
12.4.2 Volterra IE of the Second Kind (λ 6= 0) . .
12.5 Famous IEs . . . . . . . . . . . . . . . . . . . . .
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57
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59
13 Partial Differential Equations
13.1 Multiindex notation . . . . . . . . . . . .
13.2 Senses of Solution . . . . . . . . . . . .
13.2.0.1 Classical Solution . . .
13.2.0.2 Weak Solution . . . . .
13.2.0.3 Distributional Solution .
13.3 Distributional Solutions . . . . . . . . . .
13.3.1 Fundamental Solutions . . . . .
13.3.1.1 Translational Invariance
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.
Pierson Guthrey
pguthrey@iastate.edu
CONTENTS
13.4
13.5
13.6
13.7
13.3.1.1.1 Convergence . . . . . . . . . . . .
13.3.2 Green’s Function . . . . . . . . . . . . . . . . . . . .
13.3.2.1 Example . . . . . . . . . . . . . . . . . . .
13.3.2.2 Example: . . . . . . . . . . . . . . . . . . .
13.3.2.3 Example: . . . . . . . . . . . . . . . . . . .
13.3.3 Symbol of Operator L . . . . . . . . . . . . . . . . .
13.3.3.1 Example: . . . . . . . . . . . . . . . . . . .
13.3.3.2 Example: . . . . . . . . . . . . . . . . . . .
13.3.4 Duhamel’s Principle . . . . . . . . . . . . . . . . . .
13.3.4.1 Regularity of Solutions . . . . . . . . . . .
Characteristic Curves . . . . . . . . . . . . . . . . . . . . .
Types of PDEs . . . . . . . . . . . . . . . . . . . . . . . . .
13.5.1 Linear PDEs . . . . . . . . . . . . . . . . . . . . . .
13.5.2 Semilinear PDEs . . . . . . . . . . . . . . . . . . . .
13.5.3 Quasilinear PDEs . . . . . . . . . . . . . . . . . . .
13.5.4 Nonlinear PDEs . . . . . . . . . . . . . . . . . . . .
First Order Equations (m = 1) . . . . . . . . . . . . . . . . .
13.6.1 Linear Equations . . . . . . . . . . . . . . . . . . . .
13.6.1.1 Constant Coefficients . . . . . . . . . . . .
13.6.1.2 Nonconstant Coefficients . . . . . . . . . .
13.6.1.3 Cauchy Problem . . . . . . . . . . . . . . .
13.6.2 Semilinear Equations . . . . . . . . . . . . . . . . .
Second Order Equations (m = 2) . . . . . . . . . . . . . . .
13.7.1 Types of Equations . . . . . . . . . . . . . . . . . . .
13.7.1.1 Parabolic Equations b2 − ac = 0 . . . . . .
13.7.1.2 Hyperbolic Equations b2 − ac > 0 . . . . .
13.7.1.3 Elliptic Equations b2 − ac < 0 . . . . . . . .
13.7.2 Separation of Variables . . . . . . . . . . . . . . . .
13.7.3 Boundary Conditions . . . . . . . . . . . . . . . . . .
13.7.3.1 Dirichlet Conditions (First Type) . . . . . .
13.7.3.2 Neumann (Second Type) . . . . . . . . . .
13.7.3.3 Robin (Third Type) . . . . . . . . . . . . . .
13.7.4 Common Well-Posed Problems . . . . . . . . . . . .
13.7.4.1 Poisson’s Equation . . . . . . . . . . . . .
13.7.4.1.1 Using Distributions . . . . . . . . .
13.7.4.2 Fundamental Solution . . . . . . . . . . . .
13.7.4.3 Wave Equation . . . . . . . . . . . . . . . .
13.7.4.4 Fundamental Solution . . . . . . . . . . . .
13.7.4.4.1 Uniqueness . . . . . . . . . . . .
13.7.4.4.2 Four Point Property . . . . . . . .
13.7.4.4.3 Using Distributions . . . . . . . . .
13.7.4.5 Heat Equation . . . . . . . . . . . . . . . .
13.7.4.5.1 Uniqueness . . . . . . . . . . . .
13.7.4.6 Fundamental Solution . . . . . . . . . . . .
13.7.4.7 Reason: . . . . . . . . . . . . . . . . . . .
13.7.4.7.1 Using FT . . . . . . . . . . . . . .
13.7.4.7.2 Using FT and Duhamel’s Principle
13.7.4.8 Maximum Principle . . . . . . . . . . . . .
13.7.4.9 Shrodinger Equation . . . . . . . . . . . . .
13.7.4.10Fundamental Solution . . . . . . . . . . . .
13.7.4.11Helmholtz Equation . . . . . . . . . . . . .
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71
Pierson Guthrey
pguthrey@iastate.edu
CONTENTS
13.7.4.12Fundamental Solution .
13.7.4.13Euler-Tricomi Equation
13.7.4.14Biharmonic Oscillator .
13.7.4.15Fundamental Solution .
13.7.4.16Klein Gordon Equation
13.7.4.17Fundamental Solution .
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71
72
72
72
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72
14 Distribution Theory in DE
72
14.1 ODEs in D0 sense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
7
Pierson Guthrey
pguthrey@iastate.edu
Part I
Elementary Logic and Proofs
1
Logic
1.1
Logical Operations
Given propositions p and q,
• The negation of p is ¬p
p
T
F
¬p
F
T
• The disjunction of p and q is p ∨ q and is read ”p or q”
p
T
T
F
F
q
T
F
T
F
p∨q
T
T
T
F
• The conjunction of p and q is p ∧ q and is read ”p and q”
p
T
T
F
F
q
T
F
T
F
p∧q
T
F
F
F
• The conditional of p and q is p → q and is read ”if p then q”. p is the hypothesis and q is the
conclusion. Whenever the hypothesis is false, p → q is said to be vacuously true
p
T
T
F
F
1.2
q
T
F
T
F
p∧q
T
F
T (vacuously)
T (vacuously)
Truths, Implications, and Equivalences
• q =⇒ p is the converse of p → q
• ¬q =⇒ ¬p is the contrapositive of p → q
• A proposition is a tautology if it is true for all p,q
• A proposition is a contradiction if is is false for all p,q
• A proposition is a contingency if it is true for at least one pair p,q and false for at least one pair p,q
• A compound proposition is one that combines p,q with logical operations such as P = p ∨ q
8
1.3
Pierson Guthrey
pguthrey@iastate.edu
Proofs
• Given possibly compound propositions P and Q, if Q is true whenever P is true, P =⇒ Q (”P implies
Q”) is a logical implication
• If both P =⇒ Q and Q =⇒ P then we say P and Q are a logically equivalence
• Any conditional proposition is logically equivalent to its contrapositive
1.3
Proofs
the following are equivalent:
A =⇒ B
,
¬A ∨ B
,
¬B =⇒ ¬A
The following proofs are equivalent.
Direct Proof: assume A, prove B
Note: When proving ”iff” ( ⇐⇒ ) statements, show A =⇒ B THEN show B =⇒ A
Contrapositive: assume ¬B, prove ¬A
Contradiction: assume A and ¬B, arrive at a contradiction
9
Pierson Guthrey
pguthrey@iastate.edu
Part II
Algebra
2
Structures and Properties
Algebraic structures refer to a carrier set and at least one finitary operation defined on that set, which obey
certain algebraic properties.
2.1
Algebraic Properties
Closure
a + b ∈ R, ab ∈ R
(1)
a + b = b + a, ab = ba
(2)
(a + b) + c = a + (b + c), (ab)c = a(bc)
(3)
(a + b)c = ac + bc
(4)
a + 0 = 0 + a = a, 1a = a1 = a
(5)
a + (−a) = 0, a(1/a) = 1
(6)
a+x=a+y ⇒x=y
(7)
a0 = 0a = 0
(8)
−(−a) = a, (−a)b = a(−b) = −(ab), (−a)(−b) = ab
(9)
Commutativity
Associativity
Distributivity
Identity
Invertbility
Cancelation
Zero-Factor
Negation
3
Morphisms
Morphisms preserve the group structure.
Homeomorphism: A homeomorphism f : S → T from a set S to a set T is a continuous bijecton with a
continuous inverse.
A homeomorphism bijects the collection of open sets of M and the collection of open sets in N.
Isomorphism: An isomorphism is a homeomorphism in both directions.
10
Pierson Guthrey
pguthrey@iastate.edu
4
Algebraic Structures
4.1
Groups
Sets A where you take (a, b) ∈ A and perform an opertation to make a third element c ∈ A
Axioms: Closure, Associativity, Identity, Invertibility
Commutative groups are known as Abelian groups.
4.2
Ring
A ring is an Abelian group that defines the behavior of two operations.
Axioms: include group axioms, distributivity, and
Bilinearity
[ax + by, z] = a [x, z] + b [y, z]
[z, ax + by] = a [z, x] + b [z, y]
4.3
Fields
A field F is a commutative ring consisting of a set of elements and two operations (addition and multiplication) both with the following axioms:
Group properties (Closure, Associativity, Identity, Invertibility), Ring Properties (Commutativity, Distributivity), Cancelation, Zero-Factor, Negation
They also have the following properties
• Transitivity: x < y < z ⇒ x < z
• Trichotomy: Either x < y, x > y, or x = y, exclusively
• Translation: x < y ⇒ x + z < y + z
The Standard Fields are R, C, Q
4.3.1
Archimedian Property
• Definition: The property of having no infinitely large or infinitely small elements
• Infinitesimals have been used to express the idea of objects so small that there is no way to see them
or to measure them. The insight with exploiting infinitesimals was that objects could still retain certain
specific properties, such as angle or slope, even though these objects were quantitatively small
• An algebraic structure in which any two non-zero elements are comparable, in the sense that neither
of them is infinitesimal with respect to the other, is said to be Archimedean. A structure which has
a pair of non-zero elements, one of which is infinitesimal with respect to the other, is said to be nonArchimedean.
4.4
Fundamental Theorem of Algebra
Every polynomial pn (x) over C of degree n has exactly n roots (allowing for multiples).
If pn (x) is real, then complex roots come in complex cojugate pairs.
11
Pierson Guthrey
pguthrey@iastate.edu
Part III
Analysis
Equivalence Relations
An equivalence relation ∼ defines clusters of elements of A. For each a ∈ A, and equivalence of class a
w.r.t to R
[a]R = b ∈ A : aRb
An equivalence relation between A, B,and C satisfies three properties
• Identity: A ∼ A
• Symmetry: A ∼ B =⇒ B ∼ A
• Transitivity: A ∼ B and B ∼ C =⇒ A ∼ C
5
Sets
5.1
Subsets
For a set A, B is a subset of A denoted B ⊆ A if ∀ b ∈ B it is also true that b ∈ A
Superset: For a set B, A is a superset of B denoted A ⊇ A if ∀ b ∈ b it is also true that b ∈ A
Proper subset: A proper subset A0 of A, denoted A0 ⊂ A, is a subset of A but is missing at least one
element of A
Equivalent sets: For any two sets A and B, A = B iff A ⊆ B and B ⊆ A
Power Set: The power set P(S) of S is the collection of all subsets of S.
5.2
Collection of Sets
A collection of sets is denoted {Eα }α∈A where A is an indexing set and Eα is a set in the collection.
5.3
Union and Intersect
A union of two sets is
A ∪ B = {x : x ∈ A ∨ x ∈ B}
The intersection of two sets is
A ∩ B = {x : x ∈ A and x ∈ B}
If A ∩ B = ∅, A and B are disjoint Similiarly , A union of n sets is
n
[
{Ai } = {x : ∃ i s.t. x ∈ Ai }
i=1
an intersection of n sets is
n
\
{Ai } = {x : x ∈ Ai ∀ i}
i=1
12
5.4
5.3.1
Pierson Guthrey
pguthrey@iastate.edu
Sequences (xn )
de Morgan’s Laws
(E ∪ F )c = E c ∩ F c
(E ∩ F )c = E c ∪ F c
5.4
Sequences (xn )
Sequences are functions f : N → M with pn = f (n) denoted (xn ) or (xn )∞
n=1
Repetition is allowed.
Sequences aren’t in general surjections (Not everything in M needs to be used.)
5.4.1
Subsequences
Subsequences (qk ) of a sequence (pn ) are simply qk = pnk , where n1 < n2 < ... Every subsequence of a
convergent sequence converges and it converges to the same limit as does the mother sequence.
There can be further sub-subsequences. (rl ) s.t rl = qkl = pnkl = pnk(l)
5.4.2
Nested Sequences
(An ) is a nested sequence if A1 ⊃ A2 ⊃ ... ⊃ An ⊃ ... and
\
An = {p : ∀ n, p ∈ An }
5.4.3
Limits lim pn = p
n→∞
The sequence (an ) converges to the limit p in M if
∀ > 0 ∃ N ∈ N s.t. n ∈ N and n ≥ N =⇒ d(pn , p) < Otherwise it is said to diverge.
Limits of sequences are unique.
5.4.4
Cauchy Condition
A sequence is Cauchy iff ∀ > 0 ∃ N ∈ N s.t. n, m ≥ N ⇒ d(an , am ) < Every convergent sequence obeys a Cauchy Condition.
Cauchy Convergence Criterion: A sequence (an ) in a complete metric space converges ⇐⇒ it obeys
the Cauchy condition.
5.4.5
Series
n
P
Given a sequence (sn ) such that each is a partial sum sn =
xk , if sn converges to a limit s, we say the
k=1
P
series
xk converges to s.
n
P
It is said to be absolutely convergent if sn =
|xk | converges.
k=1
13
5.5
Pierson Guthrey
pguthrey@iastate.edu
Open and Closed Sets
5.5
5.5.1
Open and Closed Sets
Limit Set
A point p is a limit point (or just limit) of S if there exists a sequence (pn ) in S that converges to it.
The limit set of a set S is the set
limS = {p ∈ M : p is a limit of S}
limS = S
5.5.2
Neighborhood
A neighborhood of a point p is any open set that contains p.
The r-neighbordhood of p is the set
Mr p = {q ∈ M : d(p, q) < r}
The following are equivalent:
a. p is a limit of S
b. ∀ r > 0, Mr (p) ∩ S 6= 0
Openness is dual to closedness: The complement of an open set is closed and the complement of a
close set is open.
5.5.3
Inheritance principle
If K ⊂ N ⊂ M where M is a metric space and N is a metric space then K is closed in N if and only if there
is some subset L or M st L is closed in M and K = L ∩ N . It is said that N inherits its closeds from M.
Dually, a metric subspace inherits opens.
Assume that N is a metric subspace of M and also is a closed subset of M. A set K ∈ N is closed in N
if and only if it is closed in M. Dually,
Assume that N is a metric subspace of M and also is an open subset of M. A set U ∈ N is open in N if and
only if it is closed in M.
5.5.4
Closed Sets
A set K is closed if it contains all of its limit points.
• A closed set K has an open complement K c .
• For the metric space R, [a, b] is closed for a, b ∈ R.
T
• The intersection of any number of closed sets K is closed.
• A finite union of closed sets
n
S
K is closed.
• K is closed if and only if K 0 ⊂ K
• The lub and glb of a nonempty bounded closed set K are in K.
• A countable collection of closed sets is known as a Fσ set
14
5.6
Pierson Guthrey
pguthrey@iastate.edu
Closure, Interior, and Boundaries of Sets
5.5.5
Open Sets
A set is open if
∀ p ∈ S ∃ r > 0 s.t. d(p, q) < r =⇒ q ∈ S
or
∀ p ∈ S ∃ r > 0 s.t. {q} = {q : d(p, q) < r} =⇒ {q} ⊂ S
• An open set is a set whose complement is closed.
• For the metric space R, (a, b) is open.
• The union of opens sets is open.
• A countable collection of open sets is known as a Gδ set
• Every open set can be uniquely expressed as a countable union of disjoint open intervals.
• Open subspaces of R are in one of the following forms: (−∞, b), (a, b), (a, ∞), and R = (−∞, ∞)
5.5.6
Neither closed nor open set
For the metric space R, (a, b] is neither closed nor open.
5.5.7
Clopen sets
A clopen set is both close and open. For the metric space R, the empty set ∅ and the entire set of reals R
are both clopen.
5.6
Closure, Interior, and Boundaries of Sets
5.6.1
Closure S, cl(S)
T
The closure S = K where K ranges through the collection of all closed sets that contains S.
S = {x ∈ M : if K is closed and S ⊂ K then x ∈ K}
The closure is closed (intersect of closeds)
S is the smallest set that contains S.
S = limS
When it is unclear, use clM (S) to denote closure in M
5.6.2
Interior int(S)
S
The interior int(S) = U where U ranges through the collection of all open sets contained in S.
int(S) = {x ∈ M : for some open U ⊂ S, x ∈ U }
When it is unclear, use intM (S) to denote boundary of S with respect to M
15
5.7
Pierson Guthrey
pguthrey@iastate.edu
Unit Pieces
5.6.3
Boundary ∂S
The boundary ∂S = S ∩ S c
or equivalently S \ int(S)
The bounday is closed (intersect of closeds)
When it is unclear, use ∂M (S) to denote boundary within with respect to M
5.7
Unit Pieces
Unit Ball: B m = {x ∈ Rm : |x| ≤ 1}
Unit Sphere: S m−1 = {x ∈ Rm : |x| = 1}
Unit Cube: [0, 1]m = [0, 1] × ... × [0, 1]
5.8
5.8.1
Set Properties
Convexity
A set E ⊂ Rm is convex if ∀ x, y ∈ E, the straight line segment between x and y is also in E. That is, E is
convex iff
∀ x, y ∈ E, sx + (1 − s)y ∈ E for 0 ≤ s ≤ 1
Convex Combinations: Linear combinations of the form z = sx + (1 − s)y for 0 ≤ s ≤ 1 are called convex combinations. The line segment formed from all of the convex combinations of x and y are denoted [x, y]
5.8.2
Set Diameter
of a nonempty set S ⊂ M is sup {d(x, y)}, for x, y ∈ S
5.9
Zero Set
N
A set E is a zero set if for all > 0 there exists a countable collection of open intervals {In = (an , bn )}n=1
N
N
S
P
such that E ⊂
In and
(bn − an ) < for N ≤ ∞
n=1
n=1
For a zero set E,
• E has Lebesgue measure zero
• Almost Everywhere (a.e.): Properties of a set that hold except possibly for points in a zero set are
said to hold a.e.
The following are zero sets
• A countable union of zero sets
• A subset of a zero set
• A countable or finite set of points
• Q
• The Cantor Set
16
Pierson Guthrey
pguthrey@iastate.edu
6
Topology
Any property of a metric space or of a maping between metric spaces that can be described solely in terms
of open sets is a topological property.
Namely: Countability, Connectedness, Compactness, Metrizability...and more
The topology of a set M is the collection of all open subsets of M. A topology is closed under union
(unions of opens are open), finite intersection (finite intersections of open sets are open), and contains ∅
and M (both are open sets).
6.0.1
Topological Invariance
A property is topologically invariant if it is invariant under homeomorphisms. That is, if X has a topologically
invariant property and X and Y are homeomorphic, then Y has that topological property.
6.0.2
Topological Equivalence
A homeomorphism f : M → N bijects topologies of M and N.
6.1
Intervals
For any open interval I = (a, b)
1. I Has no smallest element, either irrational or rational
2. There are strictly more irrational numbers in the interval than rational numbers.
6.2
- Principle
6.2.0.1
Form 1: If a, b ∈ R s.t. a ≤ b + ∀ > 0, then a ≤ b
6.2.0.2
Form 2: If a, b ∈ R s.t. |a − b| < ∀ > 0, then a = b
6.2.1
Preimage f pre (V )
Let f : M → N be given. The pre-image of a set V ⊂ N is
f pre (V ) = {p ∈ M : f (p) ∈ V }
6.3
Cardinality
If there exists a bijection between A and B, they are said to have equal cardinality.
Denoted A ∼ B
A ∼ A, A ∼ B =⇒ B ∼ A, A ∼ B ∼ C =⇒ A ∼ C
6.3.1
Finite Set
A set A is finite if A 1, 2, ..., n for some n ∈ N.
17
6.4
Pierson Guthrey
pguthrey@iastate.edu
Boundedness
6.3.2
Infinite Set
A set is infinite if it is not finite.
An infinite set B is denumerable if ∃ f : A → B s.t. f is a surjection from a denumerable set A.
n
Ex. Q+ is denumerable. ∃ f : N × N → Q+ , namely f (n, m) = m
6.3.3
Denumerable Sets
A set is denumerable if A ∼ N
Compositions of denumerable sets are denumerable (Ex. N × N)
Infinite subsets of denumerable sets are deumerable
If f : N → B is a surjection and B is infinite, then B is denumerable
Denumerable unions of denumerable sets are denumerable
Q is denumerable because ∃ f : N × N → Q and f is surjective
6.3.4
Countable Sets
• A Set is countable if it is denumerable or finite.
k
• Any countable union of countable sets is countable. Ex. ∪∞
k=1 N is countable
• Nk is countable
• Nk can be embedded into N∞ as such: (a1 , a2 , ...ak ) → (a1 , a2 , ..., ak , 0, 0, ...)
6.3.5
Uncountable Sets
A set is uncountable if it is not countable.
• Every non-empty, perfect, complete metric space is uncountable.
6.4
Boundedness
A subset of a metric space M is bounded if for some p ∈ M and some r > 0, S ⊂ Mr (p)
Otherwise, it is said to be unbounded.
• A bounded function f : M → N is one with a bounded range. f (M ) ⊂ Nr (q) for q ∈ N
• Graphs of bounded functons can be unbounded: x → sin(X) is bounded, (x, y) ∈ R2 : y = sin(x)
is not.
• Bolzano Weierstrass Theorem Any bounded sequence in R has a convergent subsequence.
Further, a space A ⊂ M is totally bounded if for each > 0 there exists a finite covering of A by neigborhoods.
18
6.5
Pierson Guthrey
pguthrey@iastate.edu
Clustering and Condensing
6.4.1
Upper Bounds and Suprenums
Upper Bound: M ∈ R is an upper bound for S ⊂ R if ∃ s ∈ S, s ≤ M . Also, M is the Least Upper Bound or
l.u.b. for S if M is less than all other Upper Bounds for S.
Least Upper Bound Property: If S is a non-empty supset of R and is bounded above then in R there
exists a least upper bound for S.
Suprenum: The sup of S is it’s least upper bound when it is bounded above and +∞ otherwise.
6.4.2
Lower Bounds and Infimums
Lower Bound: m ∈ R is a Lower Bound for S ⊂ R if ∃ s ∈ S, s ≥ m. Also, m is the Greatest Lower Bound
for S if m is greater than all other Lower Bounds for S.
Greatest Lower Bound Property: If S is a non-empty supset of R and is bounded below then in R there
exists a Greatest Lower Bound for S.
Infimum: The inf of S is it’s greatest lower bound it is bounded below and −∞ otherwise.
6.5
6.5.1
Clustering and Condensing
Cluster Points S 0
A set S clusters at p (and p is a cluster point of S, not necessarily in S) if each Mr (p) contains infinitely
many points of S
Fact: S ∪ S 0 = (S)
The following are equivalent conditions to S clustering at p:
1. There is a sequence of distinct points in S that converges to p
2. Each neighborhood of p contains infinitely many points of S
3. Each neighborhood of p contains at least two points of S
4. Each neighborhod of p contains at least one point of S other than p
6.5.2
Condensation Points
A set S condenses at p (and p is a condensation point of S) if each Mr (p) contains uncountably many points
of S
6.6
Compactness
A space M is covering compact if every open cover has a finite subcover.
A space is sequentially compact if every sequence (xn ) has a convergent subsequence (xnk ).
A space is sequentially compact if and only if it is covering compact.
Every compact set is closed and bounded.
The following are compact:
• A closed subset of a compact set
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Compactness
• A box of closed intervals [a1 , b1 ] × ... × [am , bm ] ∈ Rm
• The Cartesian product of finitely many compact sets
• Heine-Borel Theorem Every closed and bounded subset of Rm is compact
• Generalized Heine-Borel Theorem Every closed and totally bounded subset of a complete metric
space is compact
• A metric space is compact if and only if it is complete and totally bounded
• Any finite subset of a metric space
• The empty set
• The union of infinitely many compact sets
• The intersection of arbitarily many compact sets
• The boundary of a compact set (such as the unit 2-sphere in R3 )
• The Cantor Set
• A nested sequence of non-empty compacts is compact and nonempty
6.6.1
Precompact
A subset A of a metric space X is pecompact if its closure in X is compact.
A set is compact if and only if it is closed and precompact.
• A bounded subset of Rn is precompact
• A function family F ⊂ C(Ω) is precompact if is uniformly bounded and equicontinuous.
6.6.2
Nested Compacts
A nested sequence of non-empty compacts is compact and nonempty
T
A nested nonempty compact sequence with a diameter that tends to 0 as n → ∞, then A = An is a single
point
6.6.3
Continuous Transformation on Compacts
If f : M → N is continuous and A is a compact subset of M then f (A) is a compact subset of N
The continuous image of a compact is compact.
Every continuous function defined on a compact is uniformly continuous.
A continuous real valued function defined on a compact set is bounded; it assumes minimum and maximum
values.
If M is homeomorphic to N, then N is compact.
If M is compact then a continuous bijection f : M → N is a homeomorphism, its inverse f −1 : M → N is
continuous.
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Connectedness
6.6.4
Embeddedness
h : M → N embeds M into N if h is a homeomorphism from M onto h(M )
Any property that holds for every embedded copy of M is an absolute or intrinsic property.
A compact is absolutely closed and absolutely bounded.
6.7
Connectedness
If M has a proper clopen subset A, M is disconnected
If M is disconnected, the separation of M is M = A t Ac
Otherwise, M is connected
• The continuous image of a connected is connected. If M is connected, f : M → N is continuous, and
f is onto, then N is connected.
• If M is connected and M is homeomorphic to N then N is connected.
• Generalized Intermediate Value Thm Every continuous real valued function defined on a connected
domain has the intermediate value property.
• R is connected.
• Intermediate Value Thm for R Every continuous function R → R has the intermediate value property.
• The following metric spaces are connected: (a, b), [a, b], the circle
• If S ⊂ T ⊂ S and S is connected, then T is connected.
• The closure of a connected set is connected.
6.7.1
Totally Disconnected
A metric space M is totally disconnected if each point p ∈ M has arbitrarily small clopen neighborhoods.
Given and p ∈ M , there exists a clopen set U
p ∈ U ⊂ M (p)
• A discrete space is totally disconnected
• Q and N is totally disconnected
6.7.2
Point Connected
The union of connected sets sharing at least one common point is connected.
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Perfect Sets
6.7.3
Path Connected
A path joining p to q in a metric space M is a continuous function f : [a, b] → M such that f (a) = p and
f (b) = q. If each pair of points in M can be joined by a path M then M is path connected.
• Path connectedness implies connectedness, but the converse need not be true.
• All connected subsets of R are path connected
• Every open connected subset of R is path connected
• The Topologist’s Sine Curve is connected yet not path connected
6.8
Perfect Sets
The following are equivalent: A set M is perfect
• if it has no isolated points. Given any point and a neighborhood of that point, there is another point
within that neighborhood.
• if M 0 = M : Each p ∈ M is a cluster point of M
Otherwise, a set is said to be imperfect. The imperfection of an imperfect set is the set of isolated points.
• R is a connected perfect space
• The Cantor set is perfect
• Every open subset of a perfect set is perfect
• The imperfection of N is itself
• Every non-empty, perfect, complete metric space is uncountable
6.9
Dense Sets
The following definitions are equivalent:
A subset A of a set X is dense in X...
• every point x ∈ X either belongs to A or is a limit point of A
• if any neighborhood of every point x ∈ X contains at least one point in A
• if every open cover of A is also an open cover of X.
• if and only if the only closed subset of X containing A is itself.
• if the closure of A is X: A = X
• if the interior of the complement of A in X is empty
The density of a set X is the least cardinality of all dense subsets of X. Density is topologically invariant.
• Q is dense in R with countable density
• The Irrationals are dense in R yet disjoint.
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Coverings
• Every space without isolated points is dense in itself
• Every metric space is dense in its completion
• Transitivity: If A ⊆ B ⊆ C and if A is dense in B and B is dense in C, then A is dense in C
• The image of a dense subset under a continuous surjections is dense.
• A space X with a subset A which is connected and dense in A must be connected
A subset is somewhere dense if
• There exists an open non-empty set U ⊂ X such that X ∩ U ⊃ U
• intX 6= ∅
A subset is nowhere dense if there is no neighborhood in X in which A is dense.
• A subset is nowhere dense if it is not somewhere dense
• A subset is nowhere dense if and only if the interior of its closure is empty
• The interior of the complement of a nowhere dense set is always dense
• The complement of a closed set that is nowhere dense is a dense open set
Related Notions
• A subset A of X is meagre if it can be expressed as the union of countably many nowhere dense
subsets of X. Q are meagre as a subset of R
• A space X with a countable subset that is dense in X is called separable
Lp (Ω), 1 ≤ p < ∞ is separable but not for p = ∞
• A space is resolvable if it is the union of two disjoint subsets and k-resolvable if it contains k pairwise
disjoint dense sets
• An embedding of a space as a dense subset of a compact space is a called a compactification of X
• A linear operator between vector spaces X and Y is densely defined if its domain is a dense subset
of X and if its range is contained within Y
6.10
Coverings
A collection U of subsets of M covers A ⊂ M if A is containted in the union of of the sets belonging to U,
and we say U is a covering or cover of A.
• The members of U are called scraps and are not themselves covers.
• The elements of U should be distinct but may overlap.
• A whole space forms an open cover of any set in it
• For covers that are intervals {(ai , bi )},
N
P
(bn − an ) is the total length of the covering
n=1
Open covering : A open cover U is one whose sets (or scraps) are all open.
Subcoverings : V is a subcover of U if U and V both cover A and if V ⊂ U ( ∀ V ∈ V, V ∈ U). We say
U reduces to V.
Covering Compact: A is covering compact if every open covering of A reduces to a finite subcovering.
A set is covering compact if and only if it is sequentially compact.
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Topological Objects
6.10.0.1 Lebesgue Number of a Covering The Lebesgue number λ ∈ R, λ ≥ 0 of a covering U of A is
the number such that ∀ a ∈ A ∃ U ∈ U s.t. Mλ (a) ⊂ U , with λ independent of a.
That is, we know each point a ∈ A is contained in some U ∈ U
• Noncompact sets may have open covers with no positive Lebesgue number
• Lebesgue Number Lemma: Every open covering of a sequentially compact set has a Lebesgue
number λ > 0
6.11
Topological Objects
6.11.1
Topologist’s Sine Curve
The Topologist’s Sine Curve is the set M = G ∪ Y where
1
1
G = (x, y) ∈ R2 : y = sin
and 0 < x ≤
x
π
2
Y = (0, y) ∈ R : −1 ≤ y ≤ 1
• M is a closed set
6.11.2
The Cantor Sets
The Cantor Ternary Set is constructed by removing the open middle portion if a set of line segments, beginning with [0, 1] and continuing ad infinitum. The standard Cantor set is based on removing the middle
thirds.
The Cantor Ternary Set is constructed by removing the open middle third if a set of line segments, beginning with [0, 1] and continuing ad infinitum. Recursively,
Cn−1
2 Cn−1
Cn =
∪
+
3
3
3
so
C0 ⊃ C1 ⊃ C2 ⊃ ... ⊃ Cn
The Cantor set contains all points not deleted at any step of the infinite process.
6.11.3
Explicit Representations
The Cantor set is
C=
∞
\
m=1
3m−1
\−1 ∞
\
Cn =
0,
m=1
k=0
3k + 1
3k + 2
∪
,
1
3m
3m
or alternatively
C = [0, 1] \
∞
[
m=1
3m−1
[−1 k=0
3k + 1 3k + 2
,
3m
3m
• The Cantor set is compact, nonempty, perfect, and totally disconnected.
• The Cantor set is uncountable.
• The Cantor set contains no interval
• The Cantor set is nowhere dense
• Anything with a 1 in its base 3 expansion if removed.
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7
Real Analysis
7.1
7.1.1
Expansions
Binomial Expansion
For a ∈ [0, 1], (a1 , ..., an ) are chosen in {0, 1} s.t.
1.a1 = 1 if a ≥ 21
n
P
ak
2. Take 0 ≤ x = 2n+1 (a −
) < 1. an+1 = 1 if x ≥
2k
k=1
1
2
This expansion is not unique because any numbr of the form
lently as (a1 , a2 , ..., an = 1, 0, 0...) or (a1 , a2 , ..., an = 0, 1, 1...).
7.2
m
2k
for m, k ∈ N can be expressed equiva-
Metric Spaces (X, d)
A metrix space is a set X equipped with a metric d
A metric d : X × X → R+ on a nonempty set X takes two elements of some set X as input and outputs a
non-negative real number with the properties that for every x, y, z in our set:
7.2.1
Definition
1. d(x, y) ≥ 0 ∀ x, y ∈ X, (distances are nonnegative)
and d(x, y) = 0 ⇐⇒ x = y (the only point of distance zero from x is x itself)
2. d(x, y) = d(y, x) (symmetry)
3. d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality)
7.2.2
Common metric spaces
• R is a metric space with metric |x − y|
Triangle Inequality for R: For all x, y ∈ R, |x + y| ≤ |x| + |y|
• C is a metric space with metric |z − w|
• T, the unit circle, is a metric space with any metric on R2
n
n
• (X, dp ) is a metric space where X ⊂ R or X ⊂ C and dp (x, y) =
n
P
|xi − yi |
p
p1
for p s.t. 1 ≤ p ≤
i=1
∞
Triangle inequality proved using Holder Inequality
• (C([a, b]), dmax ) is a metric space where f is continuous on [a, b] and dmax (x, y) = max |f (x) − g(x)|
a≤x≤b
• C m (K), m < ∞ is a metric space with metric d(f, g) =
7.2.3
P
max |Dα (f − g)|
|α|≤m x∈X
Common metric spaces???
• If (A, d) is a metric space and B ⊂ A, then (B, d) is a metric space for nonempty A, B
• d is an ultrametric if
d(x, y) ≤ max {d(x, z), d(z, y)}
25
∀ x, y, z ∈ X
7.3
Pierson Guthrey
pguthrey@iastate.edu
Equivalent Metrics
• d is a pseudometric if two distinct points can be zero. That is, d(x, y) = 0 does not imply x = y.
Points governed by a pseudometric need not be distinguishable.
7.2.4
Common Metrics
• Absolute Value Metric: d1 (~x, ~y) = |x1 − y1 | + ... + |xm − ym |
(
0 if ~x = ~y
• Discrete Metric: d0 (~x, ~y) =
1 if ~x 6= ~y
p
• Euclidean Metric: d2 (~x, ~y) = dE (~x, ~y) = (x1 − y1 )2 + ... + (xm − ym )2
• Maximum Metric: dmax (~x, ~y) = d∞ (~x, ~y) = max {|xi − yi |}
i
7.2.5
Metric Subspaces
A metric (Y, d|Y ) is a metric subspace of X if X ⊂ Y and we restrict the metric of (X, d) to Y .
(R, k·k ) is a metric subspace of (C, k·k ) and (Q, k·k ) is a metric subspace of (R, k·k )
7.2.6
Product Metrics
A product metric is a metric on a cartesian product M = M1 × M2 × ... × Mm of m metric spaces with metric
di (pi1 , pi2 ) on each Mi .
• The default product metric on R2 is dX×Y ((x1 , y1 ), (x2 , y2 )) = dX (x1 , x2 ) + dY (y1 , y2 )
7.2.6.1 Common Product Metrics For pi1 , pi2 ∈ Mi , and let p1 = (p11 , p21 , ..., pm1 ) ∈ M , and p2 =
(p12 , p22 , ..., pm2 ) ∈ M , the most common product metrics are:
dsum (p1 , p2 ) =d1 (p11 , p12 ) + d2 (p21 , p22 ) + ... + dm (pm1 , pm2 )
p
dE (p1 , p2 ) = d1 (p11 , p12 )2 + d2 (p21 , p22 )2 + ... + dm (pm1 , pm2 )2
dmax (p1 , p2 ) =max {d1 (p11 , p12 ), d2 (p21 , p22 ), ..., dm (pm1 , pm2 )}
7.3
Equivalent Metrics
Given two metric spaces (X, d1 ) and (X, d2 ) , d1 and d2 are equivalent if there exist nonzero constants c, C
such that
cd1 (x1 , x2 ) ≤ d2 (x1 , x2 ) ≤ Cd1 (x1 , x2 )
∀ x1 , x2 ∈ X
• Equivalent metrics give rise to the same open sets hlCheck this
• Equivalent metrics form an equivalence relation?
The following are equivalent
For a sequence pn = (p1n , p2n , ..., pmn ) in M = M1 × M2 × ... × Mm
1. (pn ) converges with respect to the metric dmax
2. (pn ) converges with respect to the metric dE
3. (pn ) converges with respect to the metric dsum
4. Each (pin ) converges in it’s respective Mi
Fact: dmax ≤ dE ≤ dsum ≤ mdmax
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Linear Spaces (or Vector Space)
7.3.1
Completeness
A metric space M is complete if each Cauchy sequence in M converges to a limit in M.
Rm is complete
Every closed subset of a complete metric space is a complete metric space.
Every closed subset of Euclidean Space is a complete metric space.
7.3.1.1 Completions
˜ is called a completion of (X, d) if the following conditions are satisfied
A metric space (X̃, d)
˜ and (X, d) are isomorphic (there exists an isometry ı : X → X̃)
1. (X̃, d)
2. The image space ı(X) is dense in X̃
˜ is complete
3. The space (X̃, d)
• Every metric space has a completion
• Completions are unique up to isomorphism
7.4
Linear Spaces (or Vector Space)
A linear space X over the scalar field C is a set of points or vectors on which are defined operations of
vector addition or scalar multiplication with the following properties.
• X is commutative with respect to vector addition
• There is a zero vector
• Each vector has a unique additive inverse
• Vector spaces are closed under linear combinations
7.5
Norms
A Norm on a vector space V is any function k k : V → R with the properties for any given v, w, ∈ V and
λ∈R
• Nonnegative: kvk ≥ 0
• Positive Definite: kvk = 0 ⇐⇒ v = 0
• Homogenous: kλvk = |λ| kvk
• Triangle Inequality: kv + wk ≤ kvk + kwk
Two norms k·ka and k·kb on a linear space X are equivalent if there are constants c and C such that
ck·ka ≤ k·kb ≤ Ck·ka ∀ x ∈ X
Equivalent norms generate the same topologies (same collection of open sets)
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Norms
7.5.1
Seminorms
A function X → R is a seminorm on X if:
• p(x) ≥ 0 for all x ∈ X
• p(x + y) ≤ p(x) + p(y) for all x, y ∈ X
• p(λx) = |λ| p(x) for every x ∈ X and λ ∈ C
• However, p(x) = 0 need not imply x = 0
7.5.1.1 Seminorm Topology and Convergence
A countable or uncountable subset of seminorms {pα }α∈A indexed by a set A defined on a linear space X
yields open neighborhoods defined in the following way:
N = {Nα1 ,...,αn ; (x) : x ∈ X, α1 , ..., αn ∈ A, and > 0}
Nα1 ,...,αn ; (x) = {y ∈ X : pαi (x − y) < for i = 1, ..., n}
A sequence (xn ) converges x ∈ X in this topology if and only if pα (x − xn ) → 0 as n → ∞ for each α ∈ A
7.5.1.2 Hausdorff Topologies
A family of seminorms {pα }α∈A separates points if pα (x) = 0 for every α implies x = 0.
A topological linear space space whose topology may be derived from a family of seminorms that separates
points is called a locally convex space
If the family of seminorms is finite and separates points, then
kxk = p1 (x) + ... + pn (x)
defines a norm on X
7.5.1.3 Frechet space A locally convex space with X whose topology is generated by a countably infinite family of seminorms {pn : n ∈ N} has a metrizeable topology with metric
d(x, y) =
X 1 pn (x − y)
2n 1 + pn (x − y)
n∈N
7.5.2
Inner Product Spaces
An Inner Product Space is a vector space X equipped with an Inner Product.
An inner product on a complex linear space X is a continuous map
h·, ·i : X × X → C
with the following properties for x, z, y ∈ X, α, β ∈ C
1. Linear in the First Argument: hαx + βy, zi = α hx, zi + β hy, zi
2. Hermitian symmetric: hx, yi = hy, xi
3. Nonnegative: hx, xi ≥ 0
4. Positive Definite: hx, xi = 0 if and only if x = 0
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Banach Spaces
• Properties 1 and 2 imply h·, ·i is antilinear (or conjugate linear) in the second argument
hx, αy + βzi = α hx, yi + β hx, zi
(sesquilinear)
• If X ∈ R then h·, ·i is billinear. For a, b, c, d ∈ R and x, y, z, w ∈ X,
hax + by, cz + dwi = ac hx, zi + bc hy, zi + ad hx, wi + bd hy, wi
(bilinear)
• A linear space with an inner product is an inner product space or pre-Hilbert space
• Cauchy-Schwarz Inequality: If (X, h·, ·i) is an IPS, then
|hx, yi| ≤ kxk kyk
∀ x, y ∈ X
• The Cartesian product X × Y = {(x, y) : x ∈ X, y ∈ Y } of two inner product spaces (X, h·, ·iX ) and
(Y, h·, ·iY ) have an inner product and norm
h(x1 , x2 ), (y1 , y2 )iX×Y = hx1 , x2 iX + hy1 , y2 iY
k(x, y)k =
q
2
2
kxk + kyk
Examples
• X = Rn , hx, yi =
n
P
xj yj
j=1
• X = Cn , hx, yi =
n
P
xj yj
j=1
• X = L2 (Ω), Ω ⊂ Rn , hu, vi =
R
Ω
u(x)v(x)dx
7.5.2.1 Normed Linear Spaces
If X is a vector/linear space with an inner product h·, ·i, then it is a normed linear space with norm and
metric
p
kxk = hx, xi
d(x, y) = kx − yk
p
• A normed linear space X is an inner product space with norm kxk = hx, xi if and only if
2
2
2
2
kx + yk + kx − yk = 2kxk + 2kyk
(Parallelogram Law)
• If a normed linear space that satisfies the paralleogram law, then there is at most one inner product
that can exist because of the polarization formula:
hx, yi =
7.6
1
2
2
2
2
kx + yk − kx − yk − ikx + iyk + ikx − iyk
4
Banach Spaces
A Banach Space is normed linear space that is complete with respect to the metric d(x, y) = kx − yk
The following are Banach Spaces
• C(Ω) = {f : f is continuous} when equipped with the sup norm
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Bounded Linear Maps
• C(K) = {f : f is continuous on the compact set K} when equipped with the sup norm
• C k (Ω) when equipped with the C k norm kf kC k = kf k∞ + kf 0 k∞ + ... + f (k) ∞
• Lp (Ω) for 1 ≤ p < ∞
• Lp (Ω) for p = ∞ when f ∈ L∞ (Ω) is essentially bounded (bounded on a subset of Ω whose
complement has measure 0). The norm on f is the essential suprenum
kf k∞ = inf {M : |f (x)| ≤ M a.e. in [a,b]}
• For linear maps, boundedness is equivalent to continuity. A linear map is bounded if and only if it is
continous.
7.6.1
Schauder Basis
∞
A sequence {xn }n=1
∈ X is a Schauder Basis of X if for evey x ∈ X there is a unqiue sequence of scalars
P∞
(cn ) such that x = n=1 cn xn
It is necessary but insufficient that X is separable (since linear combinations of xn with rational cofficients
is dense and countable)
7.6.1.1
Isomorphisms For a linear map T : X → Y
• X and Y are linearly isomorphic if there is a one-to-one and onto (bijective) linear map T .
• If X and Y are normed linear spaces with both T and T −1 bounded then X and Y are topologically
isomorphic.
• If T preserves norms such that kT xk = kxk ∀ x ∈ X then X and Y are isometrically isomorphic.
7.7
Bounded Linear Maps
A linear map or linear operator T between spaces X and Y is a function T : X → Y such that
∀ α, β ∈ R and x, y ∈ X
T (αx + βy) = αT x + βT y
• Also known as a linear transformation of X or linear operator on X. We denote the set of all linear
maps T : X → Y by L(X, Y ), or L(X) when X = Y
• If T is one-to-one and onto (bijection) then it is nonsingular or invertible and we can define a linear
map T −1 : Y → X with the properties T −1 T = T T −1 = I and T −1 y = x ⇐⇒ y = T x
• A linear map is bounded if there is a constant M such that kT xk ≤ M kxk ∀ x ∈ X. Otherwise, T is
unbounded
• The operator norm or uniform norm kT k of T by
kT k = inf {M : kT xk ≤ M kxk ∀ x ∈ X}
We denote the set of all bounded linear maps T : X → Y by B(X, Y ) or by B(X) when X = Y
• If T is a contraction, then T n → ∞ for n → ∞
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Hilbert Spaces
7.8
Hilbert Spaces
A Hilbert Space H is a complete inner product space with respect to the standard inner product space
norm.
p
kxk = hx, xi
• Every Hilbert Space is a Banach space
• Rn , Cn , L2 (Ω), `2 are all Hilbert Spaces
• The standard inner product on Cn is
n
X
hx, yi =
xj yj
j=1
where x = (x1 , x2 , ..., xn ) and y = (y1 , y2 , ..., yn ) with xj , yj ∈ C.
This space is complete and therefore a finite dimensional Hilbert Space
• If H is real, the inner product is called the dot product and has a defined angle between vectors
hx, yi
= cos(θ)
kxk kyk
• The inner product on C(Ω) is
Z
hf, gi =
f (x)g(x)dx
Ω
and the completion with respect to the norm
21
|f (x)| dx
Z
2
kf k =
Ω
2
is L (Ω)
• The bi-infinite complex sequences `2 is a Hilbert space
)
(
∞
X
2
2
∞
|zn | < ∞
` (Z) = (z)n=−∞ :
n=−∞
with inner product
∞
X
hx, yi =
xn yn
n=−∞
• Solobev Spaces Hk (Ω) = W k,2 (Ω) are Hilbert Spaces that are the completions of C k (Ω) with norm

 21
k Z 2
X
kf k = 
f (j) (x) dx
j=0
7.8.1
Ω
Orthogonality
Vectors x, y ∈ H are orthogonal, written x ⊥ y if hx, yi = 0
• Pythagorean Theorem If x ⊥ y, then x2 + y 2 = x2 + y 2 • Subsets A and B are orthogonal A ⊥ B if x ⊥ y for all x ∈ A, y ∈ B
• The orthogonal complement of a subset A, A⊥ , is the set of vectors orthogonal to A
A⊥ = {x ∈ H : x ⊥ y ∀ y ∈ A}
The orthogonal complement of a subset of a Hilbert space is always a closed linear subspace.
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7.8
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Hilbert Spaces
7.8.1.1
Projections Denoted proj x or PM x
M
For each M ⊂ H,
• There is single valued function that gives a unique closest point y ∈ M such that
kx − yk = min kx − zk
y = argmin kx − zk
z∈M
z∈M
• The point y ∈ M closest to x ∈ H is the unique element of M with the property that (x − y) ⊥ M
• If y ∈ M , z ∈ M ⊥ and x = y + z then y = proj x and z = proj x
M⊥
M
• Uniqueness. Let x = y1 + z1 = y2 + z2 with y1 , y2 ∈ M and z1 , z2 ∈ M ⊥ . This implies y1 − y2 =
z2 − z1 = w which implies ∃ w ∈ M ∩ M ⊥ , but then w = 0 (the only element in that intersection), so
y2 = y1 and z2 = z1
• Existence. Pick um ∈ M such that kx − un k = dn → d = inf kx − yk . Use parallelogram law and
y∈M
reduce to y = proj x and z = proj x for z ∈ M ⊥ .
M⊥
M
• If M = L {e1 , e2 , ..., en }, then PM (x) =
n
P
j=1
hx,ej i
hej ,ej i ej
7.8.1.2 Direct Sum
Given M ⊂ H and N ⊂ H are orthogonal closed linear subspaces (M ∩ N = 0), the orthogonal direct
sum or direct sum is defined by
M ⊕ N = {y + z : y ∈ M and z ∈ N }
7.8.1.3 Decomposition
If M is a closed subspace of a Hilbert space H, then H = M ⊕ M⊥
If M is not closed, then H = M ⊕ M⊥
7.8.2
Hilbert Bases
To check if a given sequence If {xn } is a basis, we check the following
1. {xn } is closed if the set of all finite linear combinations of them is dense in H
2. {xn } is complete if and only if there is no nonzero vector orthogonal to all xn , that is
hx, xn i = 0 ∀ n ⇐⇒ x = 0
3. If {xn } are orthonormal they are maximal orthonormal if {xn } is not contianed in any strictly larger
orthonormal set.
∞
Let {en }n=1 be orthonormal, the following are equivalent
1. {en } is maximal orthonormal
2. {en } is closed
3. {en } is complete
32
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Hilbert Spaces
4. {en } is a basis of H
5. Bessel’s Inequality holds for all x ∈ H
6.
∞
P
hx, en i hen , yi = hx, yi ∀ x, y ∈ H
n=1
So we know a complete orthonormal set forms a basis, with hx, en i known as the nth Generalized Fourier
Coefficient of x with respect to {en }
A basis is only possible in a separable Hilbert space, though nonseparable Hilbert Spaces do exist.
The Standard Basis for a Hilbert Space is en = (0, 0, ..., 1, ...) where the 1 appears in the nth dimension.
7.8.2.1 Gram Schmidt
If M = L {e1 , e2 , ..., en } is linearly independent but not orthonormal, then create a new orthonormal basis
M = L {f1 , f2 , ..., fn } with
n
X
f0
fn0 = en −
hen , fi i fi
fn = 0n
kfn k
i=1
7.8.2.2 Reisz Fischer Critertion For an infinite dimensional basis,
∞
∞
P
P
2
cn en converges in H if and only if
|cn | < ∞
n=1
This implies
∞
P
n=1
7.8.2.3
n=1
∞
cn en converges if and only if {cn }i=1 ∈ `2
Riemann Lebesgue Lemma
7.8.2.4 Version 1
∞
P
∞
2
If given {en }n=1 is ON and x ∈ H then
|hx, en i| < ∞ implies lim hx, en i = 0
n→∞
n=1
7.8.2.5 Version 2
Choose H = L2 (0, 2π) and en =
einx
√
2π
. For any f ∈ L2 (0, 2π),
√1
2π
R 2π
0
f (x)e−inx dx → 0 as n → ±∞
7.8.2.6 Version 3
Replace n by λn in version 2. Same conclusion, yet different proof needed. (Use IBP, and the fact that f,f’
are bounded)
7.8.2.7 Version 4
Use the fact that C 1 (0, 2π) is dense in L1 (0, 2π). If f ∈ L1 (0, 2π) pick > 0 and g ∈ C 1 (0, 2π) with
kf − gkL1 < (by density). Then
Z
Z
Z
f (x)e−iλx dx = (f (x) − g(x))e−iλx dx + g(x)e−iλx dx
T
So
T
T
Z
f (x)e−iλx dx ≤ kf (x) − g(x)k 1 ≤ L
T
So
Z
lim sup
|λ|→∞
2π
f (x)e−iλx dx = 0
0
Smooth enough functions are dense in C 1
33
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Pierson Guthrey
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Hilbert Spaces
7.8.2.8 Version 5
Same conculsion works if limits of integration in version 4 are replaced by a, b or if we integrate over R
7.8.2.9 Version 6
If f is real valued,
Z
f (x)e−iλx dx =
Z
Z
f (x) cos(λx)dx + i
f (x) sin(λx)dx
Both terms go to zero.
One can replace f by h + ig to get same conclusion for complex valued f
7.8.2.10 Bessel’s Inequality
L {e1 , e2 , ..., en }, which implies
For an infinite dimensional basis, SN =
∞
X
2
|hx, en i| ≤ kxk
P
hx, en i en = PMN x where MN =
2
n=1
7.8.2.11 Weierstrass Approximation Theorem The set of polynomials is dense in C([a, b]), so for a
given > 0 and f ∈ C([a, b]) there exists a polynomial P (x) such that
kf (x) − P (x)kL∞ (a,b) = sup |f (x) − P (x)| < a≤x≤b
Proof. Weierstrass, Landau
Lemma: If t > −1 then (1 + t)n ≥ (1 + nt). (Proven by induction) Lemma: If every f ∈ C [0, 1] can be
approximated by polynomials, then so can every g ∈ C [a, b] since for the polynomial
ϕ = a +(b − a)x, and
a given g ∈ [a, b], g ◦ φ = f ∈ C [0, 1] so if ∃ P s.t. kf − P k∞ < then f ◦ ϕ−1 − P ◦ ϕ−1 ∞ < since
compositions of polynomials are polynomials.
7.8.3
Functionals on H
A mapping ` : H → R or C is a function on H. If it is linear then
`(c1 x1 + c2 x2 ) = c1 `(x1 ) + c2 `(x2 )
The most common linear functional is
`(x) = hx, yi for some y ∈ H
A linear functional is bounded if
∃ c < ∞ s.t. sup k`k = sup
7.8.4
x6=0
x6=0
x∈H
x∈H
|`(x)|
=c
kxk
The Dual Space of H
The set of bounded linear functionals is exactly B(H, C) = H∗ which is the dual space of H
34
Pierson Guthrey
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7.8.4.1 Riesz Representation Theorem Given a Hilbert Space H, let H∗ denote its dual space consisting of allocntinous linear functionals ` : H → R or ` : H → C. If x ∈ H, then the function
`x (y) = hx, yi
∀y∈H
is an element of H∗ . Define Φ : H → H∗ as Φ(x) = `x
• Every element in H∗ can be written in this form
• If the target space is R, Φ is an isometric isomorphism. If the target space is C, Φ is an isometric
anti-isomorphism.
– Φ is bijective
– kxk = kΦ(x)k
If ` ∈ H∗ then ∃ ! y ∈ H such that `(x) = hx, yi
The map R : ` → y, taking H∗ onto H is also one-to-one (uniqueness), in a linear fasion, and it is an
isometry.
kyk = kRxk = kxk
Explain MORE?
• Every Banach Space has a dual space.
• Every Hilbert space is isometric to its dual space.
8
Functions
f : X → Y is a function if ∀ x ∈ X ∃ y ∈ Y s.t. y = f (x)
8.0.4.2 Graph
The graph of f is the set of all ordered pairs (x, y) such that x ∈ X, y ∈ Y and y = f (x).
8.0.4.3 Linearity
A function f : R → R is linear if f (x + y) = f (x) + f (y) and f (ax) = af (x)
• This implies f (x) = a1 x1 + ... + an xn
• f (0) = 0
8.0.4.4 Affine
A function f : R → R is affine if
f (x) + f (y) = f (x + y) − f (0)
That is, it describes a linear transformation followed by translation.
• This implies f (x) = a1 x1 + ... + an xn − b
• The coefficients {ai } can be scalars or dense or sparse matrices. The constant term is either a scalar
or a column vector.
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Pierson Guthrey
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8.0.4.5 Homogeneity
A function f is homogenous of order κ if f (ax) = aκ f (x)
8.0.4.6 Injective function (1-1 or one-to-one)
A function f : R → R is injective if f (x) = f (y) implies that x = y. In other words, it passes both the
horizontal and vertical line tests. Examples: x3 , exp(x). f (x) = x2 is not injective because although
f (−2) = f (2), −2 6= 2. If f is injective and g is injective, f ◦ g is injective.
If ∃ an injection f : A → B and ∃ an injection g : B → A, then A ∼ B
8.0.4.7 Surjective function (onto)
A function f : R → R is injective if for every element b ∈ B there exist at least one element a ∈ A s.t.
f (a) = b. In other words, for every element of the range there is an element in the domain s.t. f maps
the element in the domain to the element in the range. Examples: x3 and 2x are surjective but x2 is not
surjective because you can pick a negative range element.
8.0.4.8 Bijective
A function is bijective if it is both injective and surjective.
We write A ∼ B to mean that there exists a bijection between A and B.
For a bijection f : A → B, the cardinality of each set is equal, that is |A| = |B|
8.0.4.9 Convex
A function is convex if no secant line lies below the function. That is, ∀ x1 , x2 ∈ X and t ∈ [0, 1],
f (x1 )(1 − t) + f (x2 )(t) ≥ f (x2 (t − 1) + x2 (t))
8.0.4.10 Concave
A function is concave if no secant line lies above the function. That is, ∀ x1 , x2 ∈ X and t ∈ [0, 1],
(1 − t)f (x1 ) + (t)f (x2 ) ≤ f ((t − 1)x2 + (t)x2 )
• Schroeder-Bernstein Theorem If A,B, are sets and f : A → B, g : B → A are injections then
∃ h : A → B which is a bijection.
8.0.4.11
Composition : If f : A → B and g : B → C then the composite is the map g ◦ f : A → C
• Compositions of injective functions are injective.
• Compositions of surjections are surjections.
• Compositions of bijections are bijections.
8.0.4.12 Preimage : The preimage of function f : X → Y is defined to be f −1 (A) = {x ∈ X|f (x) ∈ A}.
It is all the elements which are mapped to A by f . Example: If f (x) = x2 then f −1 (4) = ±2. A can be either
a single element or a set. Note that inverses are functions which sometimes exist, but preimages are sets
which always exist (they may be empty).
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Pierson Guthrey
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Function Sequences
8.1
Function Sequences
A sequence of real valued functions (fn ) with fn : X → R converges to f in one of two incompatible ways
• Pointwise convergence: fn → f if fn (x) → f (x) ∀ x ∈ X (fails to preserve continuity)
• Metric convergence: fn → f if d(fn (x), f (x)) → 0
8.1.0.13
Uniform Norm The default norm for metric convergence is kf ksup = kf kL∞ (Ω) = sup |f (x)|
x∈X
• The norm is bounded if and ony if f is bounded.
• A sequence of bounded real valued functions (fn ) converges uniformly to to a function f on a metric
space X if
lim kfn − f k∞ = 0
(uniform convergence)
n→∞
• Uniform convergence implies pointwise convergence (but the converse need not be true)
• If fn is continuous ∀ n, and fn → f uniformly, then f is continuous
This concept also applies to other function metric spaces
• For fk ∈ C m (K), d(fk , f ) → 0 ⇐⇒ kfk − f kCk → 0 as k → ∞
R
p
• For fk ∈ Lp (Ω), d(fk , f ) → 0 ⇐⇒ Ω |fk − f | dx → 0 as k → ∞
p = 2 case known as root mean square convergence
8.2
Essential Suprenum, Essential Infimum
Given X and f ,
Essential suprenum: The ess sup X = a s.t. {x ∈ S : f (x) > a} is contained in a zero set. That is, a is
the essential suprenum for X if f (x) ≤ a for almost all x ∈ X
(
inf {a ∈ R : meas ({x : f (x) > a}) = 0} {x : f (x) > a} =
6 ∅
ess sup f =
∞
{x : f (x) > a} = ∅
Essential infimum: The essential lower bound
(
sup {a ∈ R : meas ({x : f (x) < a}) = 0} {x : f (x) < a} =
6 ∅
ess inf f =
−∞
{x : f (x) < a} = ∅
8.3
Continuity
The following definitions are equivalent.
Topological description for open sets: A function f : X → Y is continuous if, for every open set V ⊆ Y
the preimage f −1 (V ) is an open set in X.
Topological description for closed sets: A function f : X → Y is continuous if, for every closed set
V ⊆ Y the preimage f −1 (V ) is a closed set in X.
, δ description :
∀ > 0 and ∀ p ∈ M ∃ δ > 0 s.t.
q ∈ M and d(p, q) < δ =⇒ d(f (p), f (q)) < 37
8.3
Pierson Guthrey
pguthrey@iastate.edu
Continuity
Convergent Sequence description:
f : M → N is continuous if and only if it sends convergent sequences to convergent sequences, limits
being sent to limits.
Facts of continuous functions f, g
• Compositions (f ◦ g) are continuous.
• Multiplying functions (f × g) is continuous.
• Division of functions (f /g) is continuous if g 6= 0 everywhere
• Addition/Subtraction (f ± g) is continuous.
8.3.1
Oscillation oscx (f )
The oscillation of f at x is
oscx (f ) = lim sup f (t) − lim inf f (t)
t→x
t→x
or equivalently
oscx (f ) = lim diam (f ([x − t, x + t]))
t→0
f is continuous at x if and only if oscx (f ) = 0
8.3.2
Discontinuity set of f
The set D of discontinuity points is the set
D = {x ∈ [a, b] : f is discontinuous at the point x}
And the points at which f is continuous is DC . This can be defined as
D=
∞ [
k=1
1
x ∈ [a, b] : oscx (f ) ≥
k
• D(f ± g) ⊂ D(f ) ∪ D(g)
• D(f ∗ g) ⊂ D(f ) ∪ D(g)
• D fg = D(f ) ∪ D g1 = D(f ) ∪ D(g) ∪ (supp g)C
8.3.3
Uniformly Continuous
Given (X, d1 ) and (Y, d2 ), a function f : X → Y is uniformly continuous if it is continuous for all x ∈ X. More
precisely, if
∀ > 0 ∃ δ s.t. d1 (x1 , x2 ) < δ =⇒ d2 (f (x1 ), f (x2 )) < ∀ x1 , x2 ∈ X
• Uniform continuity implies pointwise continuity
• Every continuous function defined on a compact set is uniformly continuous
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Pierson Guthrey
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Continuity
8.3.4
Lipschitz Continuous
Given (X, d), a function f : X → R is Lipschitz continuous or Lipschitz thre exists a constant C such that
|f (x1 ) − f (x2 )| ≤ Cd(x1 , x2 )
∀ x1 , x2 ∈ X
The Lipschitz constant C = Lip(f ) is defined as the maximum ’slope’:
Lip(f ) = sup
x6=y
|f (x) − f (y)|
d(x, y)
or equivalently as the smallest possible C:
Lip(f ) = inf {C : |f (x) − f (y)| ≤ Cd(x, y) ∀ x, y ∈ X}
• Lipschitz continuity implies uniform continuity
• A continuously differentiable function with bounded partial derivatives is Lipschitz.
• A family of continuously differentiable functions with uniformly bounded partial derivatives is equicontinuous. If it is bounded, it is also precompact.
8.3.5
Absolutely Continuous
Absolute continuity is a smoothness property of functions that is stronger than continuity and uniform
continuity.
8.3.5.1 Analysis Definiton Given an interval on the reals I = [a, b], a function f : I → R is absolutely
continuous if for all > 0 there exists δ > 0 such that for all finite sequences of subintervals (xk , yk ) on I,
X
X
|yk − xk | < δ =⇒
|f (yk ) − f (xk )| < k
k
We denote this f ∈ AC(I)
8.3.5.2
Calculus Definiton f ∈ AC(I) if f has a derivative f 0 a.e. which is Lebesgue Integrable and
Z x
f (x) = f (a) +
f 0 (t)dt
∀ x ∈ I = [a, b]
a
8.3.5.3 Fundamental Theorem of Lebesgue Integral Calculus
If f there exists a Lebesgue Integrable function g and
Z x
f (x) = f (a) +
g(t)dt
∀ x ∈ I = [a, b]
a
Then g = f 0 a.e., f ∈ AC(I)
8.3.5.4
Properties of AC(I) Given f, g ∈ AC(I),
• f ± g ∈ AC(I). f ∗ g ∈ AC(I) if I is closed.
f
g
∈ AC(I) if I is closed and g is nowhere zero on I.
• Absolute conintuity is stronger than uniform and Lipschitz continuity.
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Pierson Guthrey
pguthrey@iastate.edu
Continuity
8.3.6
Equicontinuous Families
A function family F from a metric space (X, dX ) to a metric space (Y, dY ) is equicontinuous if
∀ x ∈ X, > 0 ∃ δ > 0 s.t. d(x1 , x2 ) < δ =⇒ d(f (x1 ), f (x2 )) < ∀ f ∈ F
• We require δ to not depend on f , but it may depend on x
• If δ does not depend on x, F is said to be uniformly equicontinuous
• Equicontinuous families on compact spaces are uniformly equicontinuous
8.3.7
Homeomorphisms
If f : M → N is a continuous bijection and the inverse f −1 : N → M is also a continuous bijection, the f is
a homeomorphism and M and N are said to be homeomorphic, denoted M ∼
= N.
Geometrically, a homeomorphism bends, twists, stretches, and wrinkles the space M to make it coincide
with N.
Homeomorphisms map the following
• ∂M → ∂N
• intM → intN
• Mc → Nc
8.3.8
Isometries
An isometry or isometric embedding is a map ı : X → Y that satisfies
dY (ı(x1 ), ı(x2 )) = dX (x1 , x2 )
∀ x1 , x2 ∈ X
• ı identifies a point x ∈ X with its image ı(x) ∈ Y so that ı(X) is a copy of X embedded in Y
• Two metric spaces X and Y are isomorphic if there exists an isomorphism ı : X → Y . Isomorphic
spaces are indistinguishable as metric spaces.
• Isometries are one-to-one and continuous
• An isometry that is onto is called a metric space isomorphism
• ı : C → R2 defined by ı(x + iy) = (x, y) is a metric space isomorphism between (C, |· · ·|) and
(R2 , k· · ·k ). Since ı is linear, these are isomorphic as real normed linear spaces
8.3.9
Fundamental Theorem of Continuous Functions
Every continuous real valued functionof a real variable x ∈ [a, b] is bounded, achieves maximum, intermediate, and maximum values, and is uniformly continuous.
8.3.10
Mean Value Theorem
For a continuous f : [a, b] → R which is differentiable on (a, b) there exists a c s.t.
f 0 (c) =
f (b) − f (a)
b−a
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8.4
Pierson Guthrey
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Properties of Functions
8.4
8.4.1
Properties of Functions
Support supp(f )
The support of a function is the set of points where the function is not zero-valued, or the closure of that
set. In other words, the support Y of f : X → C is
Y = {y ∈ X : f (y) 6= 0}
• A function supported in Y will vanish in X \ Y .
• A function with domain X has finite support if Y is finite.
• A function has compact support if Y is a compact subset of X.
• Equivalently, A function f : Rn → R has compact support if ∃ R > 0 such that f (x) = 0 ∀ x with
kxk > R. That is, lim f (x) = 0
kxk →+∞
8.4.2
Other Properties
• A fixed point of a function f : X → X is a point such that f (x) = x
• The diagonal of X × X is the set of all pairs (x, x) ∈ X × X
• The argmax f (x) is the x ∈ X where there maximum of f (x) is achieved
x∈X
8.5
Special Types of Functions
• Step Functions are constant except at a finite number of points where it is discontinuous.
• Characteristic function or Indicator Function χE of a set E ⊂ R is defined as
(
1 x∈E
χE (x) =
0 x∈
/E
• Rational Ruler Function is defined as
(
f (x) =
1
q
0
x = pq for some p,q in lowest terms
x∈
/Q
• The Sign Funtion is defined as


1
sign (x) = 0


−1
8.6
x>0
x=0
x<0
Function Spaces
Function spaces are infinite-dimensional linear normed spaces.
The domain of a function Ω is the set such that f : Ω → R.
We say x ∈ Ω if ∀ r > 0 Br (x) ∩ Ω 6= ∅
We say x ∈
/ Ω if ∀ r > 0 Br (x) ∩ Ωc 6= ∅
Ω can be
41
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Pierson Guthrey
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Function Classes
• Open if it is an open set and closed if it is a closed set. Similiarly it can be clopen or neither closed
nor open.
• Bounded if is contained in some Br (0) with r finite
• Compact if it is closed and bounded
8.6.0.1
Set Properties of Function Spaces
• If all Cauchy sequences of functions are convergent, then the metric space is complete. Otherwise it
is incomplete, and may be completed
8.7
8.7.1
Function Classes
Continuity Classes C(Ω)
The following function classes real linear spaces and are closed under pointwise combination. That is
If f, g ∈ C(Ω), α, β ∈ R, then αf + βg ∈ C(Ω)
• C(Ω) = {f : f is continuous on Ω}
• C(K) = {f : f is continuous on the compact set K}
This is a complete normed linear space when equipped with the uniform norm so it is a Banach
Space
Arzela-Ascoli Theorem : A subset of C(K) is compact if and only if it is closed, bounded, and
equicontinuous.
• Cb (Ω) = {f : f is continuous and bounded on Ω} (countinous functions that are bounded)
This is a complete normed linear space when equipped with the uniform norm so it is a Banach Space
• Ck (Ω) = {f : f is continuous and has compact supportΩ}
Note that Ck (Ω) ⊂ Cb (Ω), but it need not be open, so it need not be a Banach Space.
C k norm: recalling the sup norm k·k∞ , Ck (Ω) is complete with respect to the norm:
kf kC k = kf k∞ + kf 0 k∞ + ... + f k ∞
if f ∈ C0∞
• C0 (Ω) = Cc in Cb is closed and is therefore a Banach space.
Note that
C(Ω) ⊃ Cb (Ω) ⊃ C0 (Ω) ⊃ Ck (Ω)
And if Ω is compact then these are equal.
8.7.2
Bounded Function Classes B(Ω)
The following function classes real linear spaces and are closed under pointwise combination. That is
If f, g ∈ B(Ω), α, β ∈ R, then αf + βg ∈ B(Ω)
• B(Ω) = {f : f is bounded on Ω}
• CB(Ω) = {f : f is continuous and bounded on Ω}
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Pierson Guthrey
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8.7.3
Differentiability Classes C m (Ω)
• C m (Ω) = {f : Dα is continuous on Ω ∀ |α| ≤ m}
• C ∞ (Ω) = {f : Dα is continuous on Ω ∀ |α|} (Smooth Functions)
• C ω (Ω) = {f : f ∈ C ∞ and f equals the Taylor series expansion in a neighborhood around every point in Ω}
(Analytic Functions)
Note: C m (Ω) ⊂ C ω (Ω) ⊂ C ∞ (Ω)
8.7.4
Integrability Classes R,Lp
• R(Ω) = {f : f is Riemann Integrable on Ω}, Riemann Integrable functions Note R(Ω) ⊂ B(Ω)
n
o
1
R
p
• Lp (Ω) = f : Ω |f | dx p < ∞ , Lebesgue Integrable functions
9
Calculus
9.1
Differentiation
The function f : (a, b) → R is differentiable at x if there is a real number L such that
lim
t→x
and
∆f
f (t) − f (x)
= lim
=L
∆x→0 ∆x
t−x
f (t) − f (x)
− L < ∀ > 0 ∃ δ > 0 s.t. |t − x| < δ =⇒ t−x
We denote the derivative of f at x f 0 (x).
• Differentiability of f implies continuity of f
• Note that if f is differentiable at x,
∆f
∆x
= f 0 (x) + σ(δx) for some σ(δx)
• If a function is Lipschitz with constant M, then |f (x)| ≤ M
If f, g are differentiable, then the following are diferetiable with the indicated derivative
• (f + g)0 (x) = f 0 (x) + g 0 (x)
• (f × g)0 (x) = f 0 (x)g(x) + f (x)g 0 (x)
• If f = c for some constant c, then f 0 = 0
0 0
(x)g 0 (x)
• If g(x) 6= 0, fg = f (x)g(x)−f
g(x)2
• (f ◦ g)0 (x) = g 0 (f (x)) × f 0 (x)
9.1.1
Mean Value Theorem (MVT)
A continuous function f : [a, b] → R that is differentiable on the interval (a, b) has the mean value property
there exists a point θ ∈ (a, b) such that
f (b) − f (a) = f 0 (θ)(b − a)
43
9.2
Pierson Guthrey
pguthrey@iastate.edu
Higher Derivatives and More Dimensions
9.1.2
Ratio Mean Value Theorem (RMVT)
Continuous functions f, g : [a, b] → R that are differentiable on the interval (a, b) has the mean value property
there exists a point θ ∈ (a, b) such that
(f (b) − f (a)) g 0 (θ) = (g(b) − g(a)) f 0 (θ)
9.1.3
L’Hospital’s Rule
Gven continuous functions f, g : [a, b] → R that are differentiable on the interval (a, b), if f, g tend to 0 at b,
0
and if fg0 tends to a finite limit L at b then fg also tends to L at b. (assuming g(x) 6= 0, g 0 (x) 6= 0)
9.1.4
Intermediate Value Theorem for Derivatives
Give a continuous function f : [a, b] → R that is differentiable on the interval (a, b), its derivative f 0 (x) has
the intermediate value theorem.
The derivative of a differentiable function never has a jump discontinuity, since that would violate IVT for
Derivatives.
9.1.5
Inverse Function Theorem
If f : (a, b) → (c, d) is a differetiable surjection and f 0 (x) 6= 0 ∀ x ∈ (a, b),then f is a homeomorphism with a
differentiable inverse with derivative:
f −1 (y) =
1
f 0 (x)
for y = f (x)
Further, if both f : (a, b) → (c, d) is a homeomorphism of class C r for 1 ≤ r ≤ ∞ and f 0 (x) 6= 0 ∀ x, then f
is a C r diffeomorphism
9.2
Higher Derivatives and More Dimensions
• (f 0 )0 (x) = f 00 (x) = lim
t→x
f 0 (t)−f 0 (x)
t−x
• Higher derivatives are denoted f (r) = (f (r−1) )0
• If f (r) exists f is rt h order differentiable, with f (0) = f (x)
If f is rt h order differentiable, then f, f 0 , ..., f (r−1) are continuous
• If f (r) exists for all r, f is infinitely differentiable also known as smooth
If f is smooth then f (r) is continuous and smooth for all r
9.2.1
Leibniz Product Rule
Given rt h order (r ≥ 1) differentiable functions f, g : (a, b) → R,
(f × g)(r) (x) =
r X
r
f (k) (x)g (r−k) (x)
k
k=0
For multiple dimensions,
∂ α (f g) =
X
β+γ=α
α!
(∂ β f )(∂ γ g)
β!γ!
44
Pierson Guthrey
pguthrey@iastate.edu
Analytic Functions C ω
9.3
9.3
Analytic Functions C ω
A function f : (a, b) → ∞ is analytic if it can be expressed locally as as a convergent power series. That is,
for each x, there exists a δ > 0 such that if |h| < δ then the series
f (x + h) =
∞
X
ar hr
r=0
converges.
• Note that this implies
f (r) (x) = r!ar
Which gives rise to the uniquness of a power series expansion.
• C ω ⊂ C ∞ , and is a proper subset since smooth functions need not be analytic.
9.3.1
Taylor Approximation Theorem
The rth order Taylor polynomial of an rt h order differentiable function f at x is
r
f (r) (x) r X f (k) (x) k
f 00 (x) 2
h + ... +
h =
h
P (h) = f (x) + f (x)h +
2
r!
k!
0
k=0
Note that
P 0 (0) = f 0 (x)
P (0) = f (x)
P 00 (0) = f 00 (x)
P (r) (0) = f (r) (x)
• P approximates f to order r at x in the sense that the remainder
R(h) = f (x + h) − P (h)
is rth order flat at h = 0:
R(h)
hr
→ 0 as h → 0
• The Taylor polynomial is the only polynomial of degree ≤ r with this approximation property
• In addition, if f is (r + 1)th order differentiable on (a, b) then for some θ between x and x + h,
R(h) =
9.4
f (r+1) (θ) r+1
h
(r + 1)!
Differentiability Classes C m (Ω)
• C m (Ω) = {f : Dα is continuous on Ω ∀ |α| ≤ m}
• C ∞ (Ω) = {f : Dα is continuous on Ω ∀ |α|} (Smooth Functions)
• C ω (Ω) = {f : f ∈ C ∞ and f equals the Taylor series expansion in a neighborhood around every point in Ω}
(Analytic Functions)
• C 1 ⊂ C 2 ⊂ ... ⊂ C ω ⊂ C ∞
with proper subsets
45
9.5
Pierson Guthrey
pguthrey@iastate.edu
Integration
9.5
Integration
The integral of f : [a, b] → R for f (x) > 0 is
Z
b
f (x)dx = area of U
a
Where U is the undergraph of f :
U = {(x, y) : a ≤ x ≤ b and 0 ≤ y ≤ f (x)}
9.5.1
Partitions
A partition pair P, T ⊂ [a, b] is of the form P = {x0 , ..., xn } and T = {t1 , ..., tn } interlaced as
a = x0 ≤ t1 ≤ x1 ≤ t2 ≤ . . . ≤ xn−1 ≤ tn ≤ xn = b
And if we let ∆xi = xi − xi−1 , they define a Riemmann sum
R(f, T, P ) =
n
X
f (ti )∆xi = f (t1 )∆x1 + . . . + f (tn )∆xn
i=1
which is the area o the rectangles which approximate the area uder the graph of f .
• Each ti are known as sample points of f
• The mesh of P is max {∆xi }
i
• A coarse partition is one with a large mesh, and a fine partition is one with a small mesh
• A partition P 0 refines P if P 0 ⊃ P
9.5.2
Riemann Integration
I ∈ R is the Riemann integral of f over [a, b] if it satisfies:
∀ > 0 ∃ δ > 0 s.t. meshP < δ =⇒ |R − I| < If it exists, I is unique, and we say f is Riemann integrable. We denote I:
Z
I=
lim
R(f, P, T ) =
meshP →0
b
f (x)dx
a
• If f is Riemann integrable then it is bounded
• R is the set of all functions that are Riemann Integrable over [a, b]
Rb
• R is a vector space and f → a f (x)dx is a linear map R → R
R(αf + βg, P, T ) = αR(f, P, T ) + βR(g, P, T )
and so
Z
b
Z
(αf (x) + βg(x))dx = α
a
b
Z
f (x)dx + β
a
b
g(x)dx
a
• The constant function h(x) ∈ R where h(x) = k for some k, and its integral is k(b − a)
46
9.5
Pierson Guthrey
pguthrey@iastate.edu
Integration
• Any polynomial P (x) ∈ R
• The Rational Ruler function is in R
• Zeno’s Staircase is in R
• χQ ∈
/R
• Monotonicity of the Integral (Theorem): If f, g ∈ R, and f (x) ≤ g(x) ∀ x, then
Z
b
Z
f (x)dx ≤
b
g(x)dx
a
a
R
b
Corollary: If |f (x)| ≤ M ∀ x, then a f (x)dx ≤ (b − a)M
9.5.3
Darboux Integration
The upper sum and lower sum of f : [a, b] → [−M, M ] with repsect to a partition P of [a, b] is
U (f, P ) =
n
X
Mi ∆xi
and
L(f, P ) =
i=1
n
X
mi ∆xi
i=1
where
Mi = inf {f (t) : xi−1 ≤ t ≤ xi }
and
mi = sup {f (t) : xi−1 ≤ t ≤ xi }
which lead to the upper integral and lower integral over all partitions P
I = inf U (f, P )
P
and
I = sup L(f, P )
P
And f is assumed to be bounded so that mi , Mi ∈ R ∀ i. We see for all f, P, T ,
L(f, P ) ≤ R(f, P, T ) ≤ U (f, P )
And f is Darboux Integrable if I = I = I. That is, a bounded function is Darboux integrable if and only if
∀ > 0 ∃ P s.t. U (f, P ) − L(f, P ) < • Darboux Integrability and Riemann Integrability are equivalent, which gives rise to
Riemann’s Integrability Criterion: f is Riemann Integrable if and only if
∀ > 0 ∃ P s.t. U (f, P ) − L(f, P ) < • Refinement Principle Refining a partition causes the lower sum to increase and the upper sum to
decrease
• For a refinement P 0 of P, let the Common Refinement be P ∗ = P ∪ P 0 , then refinement principle says
L(f, P ) ≤ L(f, P ∗ ) ≤ U (f, P ∗ ) ≤ U (f, P )
47
9.5
Pierson Guthrey
pguthrey@iastate.edu
Integration
9.5.4
Riemann-Lebesgue Theorem:
Let f be a bounded function defined on [a, b] then f is Riemann integrable if and only if the set of point at
which f is discontinuous is a zero set.
Corollaries: The following are Riemann integrable
• Every continuous function and every piecewise continuous function
• The characteristic function of S ⊂ [a, b] if and only if ∂S is a zero set
• Every monotone funtion
• The product of Riemann integrable functions
• For f : [a, b] → [c, d] s.t. f ∈ R, and g : [c, d] → R s.t. g ∈ C 1 ([c, d]), then (g ◦ f )(x) = g(f (x)) ∈ R
• For f : [a, b] → R s.t. f ∈ R, and g : [c, d] → [a, b] is a bijecton whose inverse is Lipschitz then
(f ◦ g)(x) = f (g(x)) ∈ R
• For f : [a, b] → R s.t. f ∈ R, and g : [c, d] → [a, b] is a C 1 diffeomorphism then (f ◦g)(x) = f (g(x)) ∈ R
• If f ∈ R then |f | ∈ R
• If f : [a, b] ∈ R and a < c < b, then its restrictions to [a, c] and [c, b] are Riemann integrable and
Z b
Z c
Z b
f (x)dx =
f (x)dx +
f (x)dx
a
a
c
• If f : [a, b] → [0, M ] s.t. f ∈ R for some M > 0 and I = 0, then f (x) = 0 at every continuity point x of
f . This implies f (x) = 0 a.e.
9.5.5
Fundamental Theorem of Calculus
For a continuous real valued fuction f : [a, b] → R s.t. f ∈ R, let its indefinite integral F be defined on
[a, b] by
Z x
f (t)dt
F (x) =
a
F is continuous on [a, b], differentiable on (a, b), and for all x ∈ (a, b) such that f (x) is continuous,
F 0 (x) = f (x)
Theorem The values of a continuous function defined on an interval [a, b] form a bounded subset of R.
There exist m, M ∈ R s.t. ∀ x ∈ [a, b], m ≤ f (x) ≤ M
• The derivative of an indefinite integral exists a.e. and equals the integrand a.e.
9.5.6
Antiderivative
If f = g 0 , then g is the Antiderivative of f When G is an antiderivatve of g : [a, b] → R, we have
G0 (x) = g(x)
∀ x ∈ [a, b]
Notice that this is for all x, not merely almost all x
• Every continuous function has an antiderivative
• Antiderivative Theorem: If f ∈ R then its antiderivative, if it exists, differs from the indefinite integral
by at most a constant.
48
9.6
Pierson Guthrey
pguthrey@iastate.edu
Higher Dimensional Calculus
9.5.7
Integration Techniques
9.5.7.1 Integration by Substitution
If f : [a, b] → R s.t. f ∈ R, and g : [c, d] → [a, b] is a C 1 diffeomorphism (continuously differentiable bijection
with g 0 (x) > 0 ∀ x) then
Z b
Z d
f (y)dy =
f (g(x))g 0 (x)dx
a
c
9.5.7.2 Integration by Parts
If f, g : [a, b] → R are differetiable and f 0 , g 0 ∈ R, then
b
Z
Z
0
f (x)g (x) = f (b)g(b) − f (a)g(a)
a
9.5.8
b
f 0 (x)g(x)dx
a
Improper Integrals
If a < c < b and [a, c] ⊂ [a, b) and f : [a, c] → R s.t. f ∈ R, where either lim supx→b |f (x)| = ∞ or b = ∞,
then we have the Improper Riemann Integral
b
Z
c
Z
f (x)dx = lim
f (x)dx
c→b
a
a
Or equivalently for either or both sides of the integral.
9.6
Higher Dimensional Calculus
• Let Ω ∈ RN be a bounded open set with a ’nice’ boundary
• Define ~n = ~n(x) as the unit outward normal derivative for x ∈ ∂Ω
~ = ∂ , ..., ∂
is known as the dell operator
• ∇
∂x1
∂xN
~ 2 = ∆ is known as the Laplacian
• ∇
9.6.1
Divergence Theorem
Z
~ · ~gdx =
∇
Ω
Z
~g · ~nds
∂Ω
The first integral is an integral over Ω, the second integral is a line integral around the boundary of Ω
9.6.2
Green’s Identity
Z Z
Z
(f ∆g − g∆f )dx =
ΩN
∂Ω
∂h
∂f
f
−h
∂n
∂n
ds
The first integral is an integral over Ω, the second integral is a line integral around the boundary of Ω
49
9.6
9.6.3
Pierson Guthrey
pguthrey@iastate.edu
Higher Dimensional Calculus
Derivations
Starting with a basic formula:
Z
∂gi
f
dx = −
∂x
i
Ω
Z
∂f
gi
+
∂x
i
Ω
Z
f gi ni ds
∂Ω
Summing over i gives
Z X
Z
N
N
N
X
X
∂f
∂gi
dx = −
+
f
f
gi
gi ni ds
∂xi
∂xi
Ω i=1
∂Ω
Ω
i=1
i=1
Z
Which by various calculus definitions is
Z
Z
Z
~ · ~gdx = − ~g · ∇f
~ +
f∇
Ω
Ω
f~g · ~nds
∂Ω
~ for some scalar h means
Choosing f = 1 yields divergence theorem. Alternatively, choosing ~g = ∇h
2
~
~
∇ · ~g = ∇ h = ∆h, so the above becomes
Z
Z
Z
∂h
~ · ∇f
~ +
ds
∇h
f
f ∇hdx = −
∂Ω ∂n
Ω
Ω
Reversing f, h and then subtracting that from the original gives Green’s Identity
50
Pierson Guthrey
pguthrey@iastate.edu
Part IV
Differential Equations
10
10.1
Solutions
General Solutions
General Solutions are the set of all solutions to a DE. Generally, a nth order D’s general solution has n
arbitrary constants.
Normalized Solutions: The solution set (for example y(x) = c1 y1 (x) + c2 y2 (x) to a DE such that when
y(x = 0) = 0 and y 0 (x = 0) = 1.
10.1.1
Well-Posed Problems
A problem is well posed if
• There is one solution (existence)
• The solution is unique (uniqueness)
• The solution depends continuously on the data (stability condition)
Small changes in the intial or boundary conditions lead to small changes in the solution
Wronksian: The determinant of the Fundamental Matrix of a set of solutions to a differential equation.
A set of solutions to a DE are linearly independent if the Wronskian identically vanishes for all x ∈ I. Note
that W ≡ 0 does not imply linear dependence.
For f, g, W (f, g) = f g 0 − gf 0 . For n real or complex valued functions f1 , f2 , ..., fn which are n − 1 times
differentiable on an interval I, the Wronksian W (f1 , ..., fn ) as a function on I is defined by
f1 (x)
...
fn (x) f10 (x)
...
fn0 (x) W (f1 , ..., fn )(x) = ..
..
x ∈ I
..
.
.
.
(n−1)
(n−1)
f
(x) (x) . . . fn
1
11
Ordinary Differential Equations
An nth order ODE for y = y(t) has the form
F (t, y 0 , y 00 , .., y (n) ) = 0
(Implicit Form)
Usually it can be written
y (n) = f (t, y 0 , y 00 , .., y (n−1) )
(Explicit Form)
n
A solution y is defined on y : I → R with y ∈ C (I) for some I ⊆ R such that
y (n) (t) = f (t, y 0 (t), y 00 (t), .., y (n−1) (t))
∀t∈I
• If the solution is defined on whole of R then we call it a global solution
• If the solution is defined on a subinterval of R then we call it a local solution
51
11.1
11.1
Pierson Guthrey
pguthrey@iastate.edu
Senses of Solutions
Senses of Solutions
11.1.0.1 Classical Solution
u0 = f in a classical sense if u ∈ C 1 and u0 (x) = f (x) ∀ x
11.1.0.2 Weak Solution
u0 = f in a weak sense if u ∈ L1loc and u0 = f in D0 sense.
Classical solutons are always also weak solutions
11.1.0.3 Distributional Solution
u0 = f in a distributional sense if u ∈ D0 and u0 = f in D0 sense.
Classical solutions and weak solutions are always also distributional solutions
11.1.0.4 Regularity of Solutions For u0 = 0 all solutions are classical, weak, and distributional solutions.
For xu0 = 0 the solution u = δ is neither classical nor weak.
Thus, the regularity of the solution depends on the DE.
11.1.1
Initial Value Problems (IVP)
A problem is an IVP if it is given in the form
y (n) = f (t, y 0 , y 00 , .., y (n−1) )
y(a) = γ0
y 0 (a) = γ1
..
.
y (n−1) = γn−1
where a is the lower boundary of the domain.
• Linear IVPs have a unique solution.
• Existence of Solution
– Local Existence Theorem or Peano Existence Theorem: If f is continuous on Rn , then every
(n−1)
(t0 , u0 , ..., u0
) there exists an open interval (t0 − , t0 + ) = I ⊂ R with > 0 that contains t0
and there exists a continuously dfferentiable function u : I → R that satisfies the IVP.
– Local Existence Theorem If f is continuous in a neighborhood of (a, γ0 , ..., γn−1 ) there exists
an open interval (t0 − , t0 + ) = I ⊂ R with > 0 that contains t0 and there exists a continuously
dfferentiable function u : I → R that satisfies the IVP.
• Uniqueness of Solution
– Uniqueness by Continuous Differentiability of f : If ∇f is continuous (if f is continuously
differentiable), then the solution is unique.
– Uniqueness by Lipschitz: If f (u, t) is Lipschitz continuous in u then the solution is unique.
• Gronwall’s Inequality: For u(t) ≥ 0 continous and φ(t) ≥ 0 continous defined on 0 ≤ t ≤ T and
u0 ≥ 0 is a constant, if u(t) satisfies
Z t
u(t) ≤ u0 +
φ(s)u(s)ds
for t ∈ [0, T ]
0
Z
then,
u(t) ≤ u0 exp
0
52
t
φ(s)ds
for t ∈ [0, T ]
11.2
Pierson Guthrey
pguthrey@iastate.edu
Linear Equations
11.1.2
Boundary Value Problems (BVP)
• A BVP with separated conditions affect multiple endpoints such as in the form
ga (y(a)) = 0, gb (y(b)) = 0
• A BVP with unseparated conditons affect the endpoints simultaneously, such as in the periodic
conditions
y(a) − y(b) = 0
11.1.3
Systems of ODEs
One could also consider solutions to systems of ODEs.
Any ODE can be converted into a first order system of ODEs. Example:
 0  

x
y2

y3
x000 = t + cos(x00 )ex
becomes
y 0 =  x00  = 
y2
000
t + cos y3 e
x
11.2
Linear Equations
Linear equations are linear in y, and have the form
n
X
aj (t)D(j) y(t) = g(t)
j=0
Ly = g(t)
Otherwise the ODE is nonlinear for some a0 , a1 , ..., an , g and Dk =
11.2.1
dk
. Note that aj (t) need not be linear.
(dt)k
General solutions
For (a1 , a2 , ..., an continuous; g continuous; an 6= 0), linear ODEs have infinitely many solutions of the form
y(t) = yp (t) +
n
X
cj yj (t)
j=1
Where (y1 , y2 , ..., yn ) are linearly independent solutions to Ly = 0 and yp (t) is a particular solution to Ly = g.
• Linear IVPs have a unique solution.
11.2.1.1
Linear Systems of ODEs Any system of linear ODEs can be viewed as the matrix equation
~y0 = A~y
with solution
~y = eAt~c
where x(t) = y1 (t)
For a vector of arbitrary constants c determined by intial or boundary values.
• If A is diagonalizable , A = V DV −1 with its eigenvectors V , then eAt c = V eDt V −1~c. Since V =
(~v1 . . . ~vn ) and we can define arbitrary constants ~d = V −1~c, this becomes
~y(t) = eAt c = d1 eλ1 t~v1 + d2 eλ2 t~v2 + ... + dn eλn t~vn
Alternatively you can simply evaluate V eDt V −1~c and take the first component.
53
11.3
11.3
Pierson Guthrey
pguthrey@iastate.edu
Nonlinear Equations
Nonlinear Equations
Nonlinear equations such as
y 0 = y 2 with u(t0 ) = u0
May have a unique solution, but usually only local solution.
11.3.1
Exactly Solvable Cases
First Order Linear Equations
y 0 + p(t)y = q(t)
The general solution is
1
y=
M (t)
for M (t) = e
Rt
s0
p(s)ds
t
Z
q(u)M (u)du +
t0
C
M (t)
and any constant C.
Linear Equations with Constant Coefficients
Ly =
n
X
aj Dj y = 0
j=0
Solutions exist in the form
y(t) = eλt
where λ is a root of the characteristic polynomial
p(λ) =
n
X
aj λj
j=0
If roots are repeated, the solutions associated with the same root must take on forms that are orthogonal to
one another, such as
y(t)1 = eλt , y(t)2 = teλt , y(t)3 = t2 eλt
The pair of solutions associated with a pair of complex roots must be real, and so for a pair of roots
(λ ± (α + iβ))
y1 (t) = eαt cos(βt)
y2 (t) = eαt sin(βt)
Euler Type Equations
Ly =
n
X
aj (t − t0 )j Dj y = 0
j=0
Solutions exist in the form
y(t) = (t − t0 )λ
t 6= t0
Where λ can be found by using this solution form in the equation, which forms the indicial equation
n
X
Aj t j = 0
j=0
Where An = an but the other coefficients depend on the nature of the ODE. If the indicial equation has...
54
11.4
Pierson Guthrey
pguthrey@iastate.edu
Nonlinear ODEs
• two roots, then
y(t) = c1 (t − t0 )λ1 + c2 (t − t0 )λ2
• one root, then one must perform reduction of order. However, solutions typically have a solution that
looks something like
y(t) = c1 (t − t0 )λ + c2 (t − t0 )λ ln |t − t0 |
For higher algebraic muliplicities of the root, you will have additional solutions (t − t0 )λ (ln |t − t0 |)2 , ..., (t − t0 )λ (ln |t − t
• a complex pair of roots r = λ ± iω, one must solve for the real solutions. Typically you end up with a
solution that looks something like
y(t) = c1 (t − t0 )λ cos(ω ln |t − t0 |) + c2 (t − t0 )λ sin(ω ln |t − t0 |)
For higher algebraic multiplicities you can solve for real valued solutions of the form
(t − t0 )λ cos(ω ln |t − t0 |) ln |t − t0 | ,(t − t0 )λ sin(ω ln |t − t0 |) ln |t − t0 | , ...
..., (t − t0 )λ cos(ω ln |t − t0 |)(ln |t − t0 |)m−1 ,(t − t0 )λ sin(ω ln |t − t0 |)(ln |t − t0 |)m−1
11.3.1.1
Example
ax2 y 00 + bxy 0 + cy = 0
Yields the indicial equation
aλ2 + (b − a)λ + c = 0
Say a = 1, b = −6, c = 10. Then λ1,2 = 2, 5 and
y(t) = c1 x2 + c2 x5
Say a = 1, b = −9, c = 25. Then λ = 5 and we must additionally solve
y(x) = x5 u(x)
v = u0
which has the solution
y(x) = x5 (c1 ln |x| + c2 )
Say a = 1, b = −3, c = 20. Then λ = 2 ± 4i
y(x) = c1 x2 cos(4 ln |x|) + c2 x2 sin(4 ln |x|)
11.3.2
Relation between Euler Equations and Constant Coefficient Equations
Let y : (t0 , ∞) → R for and Y : (−∞, ∞) → R be functions of t and x respectively. Assume they are related
by a substitution x = et . That is, y(t) = Y (x). Then the Euler equation for y can be related to the constant
coefficient equation for Y .
11.4
Nonlinear ODEs
y (n) = f (t, y 0 , y 00 , .., y (n−1) )
y(a) = γ0
..
.
y (n−1) (a) = γn−1
55
11.4
Pierson Guthrey
pguthrey@iastate.edu
Nonlinear ODEs
11.4.1
11.4.1.1
Given
Integraton Methods
Integrating Factor
y 0 (x) + p(x)y(x) = q(x)
Multiplying by
R
y 0 (x)e
pdx
+ p(x)e
R
pdx
y(x) = q(x)e
R
pdx
Integrating both sides is used with reverse product rule
Z
R
R
y(x)e pdx = q(x)e pdx dx + c1
11.4.1.2
Given
Variation of Parameters
y 00 + q(t)y 0 + r(t)y = g(t)
We find the solutions to the associated homogenous equation (y 00 + q(t)y 0 + r(t)y = 0)
yc = (t) = c1 y1 (t) + c2 y2 (t)
And we want to find a particular solution to y 00 + q(t)y 0 + r(t)y = g(t) in the form
yp = (t) = u1 (t)y1 (t) + u2 (t)y2 (t)
We let
u01 (t)y1 (t) + u02 (t)y2 (t) = 0
Condition 1
and so
yp0 = (t) = u01 (t)y1 (t) + u1 (t)y10 (t) + u02 (t)y2 (t) + u2 (t)y20 (t)
yp0 = u1 (t)y10 (t) + u2 (t)y20 (t)
Differentiating
yp00 = (t) = u01 (t)y10 (t) + u1 (t)y100 (t) + u02 (t)y20 (t) + u2 (t)y200 (t)
Plugging this into the original equation and cancelling gives
u01 y10 + u02 y20 = g(t)
Condition 2
Solving the system given by the two conditions gives
y2 g(t)
y1 y20 − y2 y10
u02 =
y2 g(t)
dt
y1 y20 − y2 y10
u2 =
u01 = −
so
Z
u1 = −
y1 g(t)
y1 y20 − y2 y10
Z
y1 g(t)
dt
y1 y20 − y2 y10
And so our particular solution is
Z
yp (t) = −y1 (t)
y2 g(x)
dx + y2 (t)
y1 y20 − y2 y10
So our general solution is
y = yc (t) + yp (t)
56
Z
y1 g(x)
dx
y1 y20 − y2 y10
Pierson Guthrey
pguthrey@iastate.edu
12
Integral Equations
For a domain Ω ⊆ Rn , we have a linear integral operator
Z
L {u} (x) = K(x, y)u(y)dy
Ω
with a kernel k defined for each (x, y) ∈ Ω, i.e. on Ω × Ω
L has associated IEs typically of the form
L {u} (x) = λu(x) + g(x)
(λ − L)u(x) = g(x)
12.0.2
Solutions
As you solve the IE, any time you encounter
(λ − L)u(x) = g(x)
In the case of λ = L,
• g(x) = 0, then there are an infinite number of solutions
• g(x) 6= 0, then there are no solutions
In the case of λ 6= L,
u(x) =
g(x)
(λ − L)
And you can continue to solve the IE
12.1
Fredholm IE of the First Kind; λ = 0
Z
L {u} (x) =
K(x, y)u(y)dy = g(x)
Ω
Fredholm IE of the Second Kind; λ ≤ 0
12.2
Z
L {u} (x) =
K(x, y)u(y)dy = λu(x) + g(x)
Ω
Note that you can often convert between IEs and other DEs.
12.2.0.1
Solutions via Fixed Point Method Fredholm IEs can be recast as fixed point problem
Z b
T u(x) = u(x)
with T u(x) = g(x) +
k(x, y)u(y)dy
a
If c = sup
a≤x≤b
R
b
a
|k(x, y)| dy < 1, we know
kT u1 − T u2 k∞
Z
!
b
= sup k(x, y)(u1 (y) − u2 (y))dy ≤ cku1 − u2 k∞
a≤x≤b a
57
12.3
Pierson Guthrey
pguthrey@iastate.edu
Special Kernels
so we know u is a unique solution and it can be obtained via the limit
u = lim T n u0
n→∞
We express the inverse of the operator as the Neumann series
−1
u = (I − L)
=
∞
X
Ln
n=0
Using only partial sums is called the Born approximation
2
λI + L + L + ... u(x) = u(x) +
Z
b
Z
b
Z
b
k(x, y)k(y, z)u(z)dydz + ...
k(x, y)u(y)dy +
a
a
a
The inverse of a differential operator A is an integral operator whose kernel is the Green’s Function of
A.
12.3
Special Kernels
Symmetric Kernels have the property k(y, x) = k(y, x)
Convolution Kernels have the form k(x, y) = k(x − y)
Singular Kernels have the property k(x, y) → ∞ for certain x, y (often x = y)
12.4
Volterra Equations
Equations s.t. k(x, y) = 0 for y > x have the form
Zx
K(x, y)u(y)dy = λu(x) + g(x)
a
12.4.1
Volterra IE of the First Kind (λ = 0)
Zx
K(x, y)u(y)dy = g(x)
a
by FTC,
u(x) = g 0 (x)
Solvability condition: g(a) = 0 and g(x) is differentiable on Ω.
12.4.2
Volterra IE of the Second Kind (λ 6= 0)
Zx
K(x, y)u(y)dy = λu(x) + g(x)
a
by FTC,
u(x) = λu0 (x) + g 0 (x)
which can be solved using ODE methods.
0
0
Solvability conditions: u(a) = −g(a)
λ , u exists, g exists
58
12.5
Pierson Guthrey
pguthrey@iastate.edu
Famous IEs
12.5
Famous IEs
Fourier Transform has the form
Z
eixy u(y)dy
Lu(x) =
Rn
Laplace Transform has the form
Z∞
Lu(x) =
e−xy u(y)dy
0
Hilbert Transform has the form
Lu(x) =
1
π
Z∞
u(y)
dy
x−y
−∞
Abel Integral Operator has the form
Zx
Lu(x) =
u(y)
√
dy
x−y
0
13
Partial Differential Equations
13.1
Multiindex notation
A multiindex α = (α1 , α2 , ...., αn ) ∈ Zn+ denotes
Dα =
∂ |α|
(∂x1 )α1 (∂x2 )α2 ...(∂xn )αn
xα =
n
Y
i
xα
i
i=1
|x| =
q
v
u n
uX
x2i
x21 + ... + x2n = t
i=1
xα f = {y : y = xα f (x)}
For multiindices α and β,
• |α| = α1 + α2 + ... + αn
• α!
n
Q
αi !
i=1
• α + β = (α1 + β1 , ..., αn + βn )
• α ≥ β if and only if αi ≥ βi for i = 1, ..., n
13.2
Senses of Solution
Let Lu =
P
aα (x)Dα u were aα (x) ∈ C ∞ (Ω) so LT ∈ D0 (Ω) for T ∈ D0 (Ω)
|α|≤M
Example: If Lu = utt − uxx then the general solution is u(x, t) = F (x + t) + G(x − t)
59
13.3
Pierson Guthrey
pguthrey@iastate.edu
Distributional Solutions
13.2.0.1 Classical Solution
Lu = f in a classical sense if u ∈ C M (Ω) and u0 (x) = f (x) ∀ x ∈ Ω Example: We require F, G ∈ C 2 (Ω)
13.2.0.2 Weak Solution
Lu = f in a weak sense if u ∈ L1loc and Lu = f in D0 sense.
Classical solutons are always also weak solutions
Example: We require F, G ∈ L1 (Ω)
13.2.0.3 Distributional Solution
Lu = f in a distributional sense if u ∈ D0 and Lu = f in D0 sense.
Classical solutions and weak solutions are always also distributional solutions
Example: We require F, G = δ ∈ D0 (Ω)
13.3
Distributional Solutions
13.3.1
Fundamental Solutions
We say E ∈ D0 (RN ) is a fundamental solution of
X
L=
aα (x)Dα
aα (x) ∈ C ∞
|α|≤M
if LE = δ.
• Usually fundamental solutions can be found by taking the FT of LE = δ which is
∧

X

∧
Dα E  = (δ) →
|α|≤M
X
(ik)α Ê(k) =
|α|≤M
1
N
(2π) 2
• Fundamental solutons are never unique since you can add solutions to the homogenous equation H
and the equation L(E + H) = f will still be valid
13.3.1.1 Translational Invariance If E is a fundamental solution to and aα (x) = aα (constant coefficients) and the domain of E is all of RN , then u = E ∗ f is a solution formula for Lu = f , and E is translationally invariant which means it commutes with translation. That is, given L : C0∞ (RN ) → C ∞ (RN ),
τh Lφ = Lτh φ
Reason:
∀ φ ∈ C0∞ , ∀ h ∈ RN
Lτh φ = T (τx (τhˇφ)) = T (τx−h φ̌) = (T ∗ φ)(x − h) = Lφ(x − h) = τh Lφ
13.3.1.1.1 Convergence If L : C0∞ (RN ) → C ∞ (RN ) such that L is translation invariant and continuous then there exists a unique T ∈ D0 (RN ) such that Lφ = T ∗ φ
60
13.3
Pierson Guthrey
pguthrey@iastate.edu
Distributional Solutions
13.3.2
Green’s Function
Given a PDE, a fundamental solution that satisfies the boundary conditions is known as a Green’s Function for that BVP. Given
X
aα (x)Dα
L=
aα (x) ∈ C ∞
|α|≤M
Then E = E(x, y) is a fundamental solution of Lx E(x, y) = δ(x − y) if
Z
Z
Lu(x) =
Lx E(x, y)f (y)dy =
δ(x − y)f (y)dy = f (x)
RN
RN
E(x, y) can be of the form E(x − y) but need not be in general.
13.3.2.1
Example
(
y(x − 1) 0 ≤ y ≤ x ≤ 1
K(x, y) =
x(y − 1) 0 ≤ x ≤ y ≤ 1
|x|
2 ,
IF Lu = u00 with u(0) = u(1) = 0 then Lx K = δ(x − y). If E(x) =
LE = δ(x)
then
and LE(x − y) = δ(x − y)
so they should differ by a solution to the homogenous equation. If H(x, y) = K(x, y) − E(x − y) then
(
y(x − 1) − 21 (x − y) 0 ≤ y ≤ x ≤ 1 n
H(x, y) =
= y − 21 x − 12 y 0 ≤ x, y ≤ 1
1
x(y − 1) − 2 (y − x) 0 ≤ x ≤ y ≤ 1
so Lx H = 0 ∀ y. If u(x) =
R1
0
K(x, y)f (y)dy then u00 = f but also
Z
1
Z
K(0, y)f (y)dy = 0 and u(1) =
u(0) =
0
1
K(1, y)f (y)dy = 0
0
so we have solved the BVP using the Green’s function K(x, y)
13.3.2.2
Example: For N = 3, ∆
Further, if f ∈
C0∞ (RN )
−1
4π|x|
= δ.
then let u = E ∗ f and so
Lu = L(E ∗ f ) = LE ∗ f = δ ∗ f = f
Example: Let E(x) =
−1
4π|x|
so ∆E(x) = δ and so if ∆u = f then
u = (E ∗ f )(x) =
−1
4π
Z
R3
f (y)
dy
|x − y|
which can be check to also be a classical solution since E ∗ f ∈ C ∞ .
61
13.3
Pierson Guthrey
pguthrey@iastate.edu
Distributional Solutions
13.3.2.3
Example: In Ω = R2 ,
Lu = utt − uxx
u(x, 0) = h(x)
ut (x, 0) = g(x)
Let E(x, t) = 21 H (() t − |x|), where H is the Heaviside function. So
(
0 t < |x|
(Indicator function for the forward light cone)
E(x, t) = 1
t > |x|
2
If f ∈ C0∞ , u = E ∗ f is a soluion of Lu = f .
Z
Z
Z
1 ∞ ∞
H(t − s − |x − y|)f (y, s)dyds
u(x, t) = E ∗ f =
E(x − y, t − s)dyds =
2 −∞ −∞
R2
So
1
u(x, t) =
2
Z
t
x+s−t
Z
f (y, s)dyds
−∞
(Indicator function for backward light cone)
x+t−s
If we assume f (x, t) = 0 for t < 0, this becomes
Z Z
1 t x+s−t
u(x, t) =
f (y, s)dyds
2 0 x+t−s
(Integral of cone with vertex at x, y )
Note u(x, 0) = ut (x, 0) = 0
Also (E ∗ g) =
R∞
(x)
and
∂
∂t (E ∗ h)(x, t)
(x)
−∞
E(x − y, t)g(y)dy =
1
2
R x+t
x−t
g(y)dy (Part of D’Alembert’s Solution Formula)
= 12 (h(x + t) + h(x − t))
so solution of PDE is
u(x, t) = (E ∗ f )(x, y) + (E ∗ g) +
(x)
13.3.3
∂
(E ∗ h)(x, t)
∂t (x)
Symbol of Operator L
Define the Symbol of L to be
N
P (k) = (2π) 2
X
(ik)α
|α|≤M
with P (k)Ê(k) = 1.
• The coefficients of the operator L yield a unique symbol and vice versa
P
N
(ik)α is the principal symbol of L and excludes lower order terms
• Pm (k) = (2π) 2
|α|=M
L is elliptic if Pm (k) = 0 if and only if k = 0 for k ∈ RN : that is Pm (k) has no real roots except 0
• A fundamental solution should satisfy (assuming P (k) 6= 0)
Ê(k) =
But this is not clear if
1
p
1
1
=⇒ E(x) =
N
P (k)
(2π) 2
Z
RN
eikx
dk
P (k)
∈
/S
• Malgrange Ehrenpreis Theorem If L 6= 0 everywere then a fundamental solution to Lu = f exists.
• Theorem If L is elliptic, f ∈ C ∞ (RN ), u ∈ D0 (RN ) and Lu = f , then u ∈ C ∞ (RN )
62
13.4
Pierson Guthrey
pguthrey@iastate.edu
Characteristic Curves
13.3.3.1
N
P
N
Example: L = ∆ then P (k) = −(2π) 2
j=1
kj2 so P (k) = 0 when k1 = k2 = 0 so is therefore
elliptic
13.3.3.2 Example: Lu = utt − uxx then P (k) = 2π(kx2 − kt2 ) so P (k) = 0 when kt = ±kx and is therefore
not elliptic.
13.3.4
Duhamel’s Principle
Duhamel’s principle is a general method for obtaining solutions to inhomogeneous linear evolution equations
(Ex: see heat equation) by convolution of a fundamental solution with the inhomogenous term.
13.3.4.1 Regularity of Solutions For ∆u = 0 all solutions are classical, weak, and distributional solutions.
Thus, the regularity of the solution depends on the PDE.
13.4
Characteristic Curves
of a PDE are the lines on which the solution is constant.
13.5
Types of PDEs
13.5.1
Linear PDEs
A PDE of order m is linear if it can be expressed as
X
Lu(x) =
aα (x)Dα u(x) = f (x)
|α|≤m
An order k linear PDE over Rn will have
13.5.2
n+k−1
k−1
distinct terms
Semilinear PDEs
A PDE of order m is semilinear if it can be expressed as
X
aα (x)Dα u(x) + a0 Dk−1 u, ..., Du, u, x = f (x)
|α|=m
13.5.3
Quasilinear PDEs
A PDE of order m is quasilinear if it can be expressed as
X
aα (Dα u(x), ..., Du, u, x) + a0 Dk−1 u, ..., Du, u, x = f (x)
|α|=m
13.5.4
Nonlinear PDEs
A PDE of order m is nonlinear if it depends nonlinearly upon the highest order derivatives.
PDEs are classified by order and type (elliptic, hyperbolic, parabolic)
63
13.6
Pierson Guthrey
pguthrey@iastate.edu
First Order Equations (m = 1)
13.6
First Order Equations (m = 1)
13.6.1
Linear Equations
13.6.1.1
Constant Coefficients
~ · ~θ = ∂u = aux + buy = 0
∇u
∂~θ
A directional derivative vanishes for all points (x, y).
A characteristic curve though point (x0 , 0) obeys ay =
b(x −x0 ), so x0 =
If u(x, 0) = f (x), then u(x, y) = u(x0 , 0) = f (x0 ) = f
13.6.1.2
bx−ay
b
bx−ay
b
Nonconstant Coefficients
a(x, y)ux + b(x, y)uy = 0
A directional derivative at each point vanishes.
A characteristic curve parameterized by (x(t), y(t)) satisfying the ODE system
dx
= a(x, y)
dt
dy
= b(x, y)
dt
d
dx
dy
u(x(t), y(t)) = ux
+ uy
= aux + buy = 0
dt
dt
dt
u is a solution of the PDE =⇒ the curves are characteristics =⇒ u is constant along these curves
Let f (x) = u(x, 0), if (x, y) and (x0 , 0) lie on the same characteristic then u(x, y) = u(x0 , 0) = f (x0 )
13.6.1.3
Cauchy Problem A solution u of the PDE is specified by f (s) on a curve γ
• γ can be defined parametrically by x = φ(s) and y = ψ(s) then u(φ(s), ψ(s)) = f (s)
• γ can be nowhere characteristic nor can it be tangent to the characteristic direction (x0 , y 0 ) = (a(x, y), b(x, y))
because if f = 0 anywhere, then f touches a characteritic, which means it is a characteristic
• Solution: Existence isn’t guaranteed. If γ is nowhere tangent to (a, b) and (a, b) 6= (0, 0) then a unique
solution exists locally (near γ)
Solve the ODE for (x(t, s), y(t, s)), a fixed s which is a characteristic through (φ(s), ψ(s))
dx
= a(x, y)
dt
dy
= b(x, y)
dt
x(0, s) = φ(s)
y(0, s) = ψ(s)
Perform a transformation (x(t, s), y(t, s)) → (t(x, y), s(x, y))
Which means u(0, s) = f (s) becomes u(x, y) = f (s(x, y))
Solution is valid at a point (x, y) if it can be conncted to γ by a characteristic curve
64
13.7
13.6.2
Pierson Guthrey
pguthrey@iastate.edu
Second Order Equations (m = 2)
Semilinear Equations
a(x, y)ux + b(x, y)uy = c(x, y, u)
So we have an ODE system
dy
= b(x, y)
dt
dx
= a(x, y)
dt
d
(u(x(t), y(t))) = a(x, y)ux + b(x, y)uy = c(x, y, u) = c(x(t), y(t), u(x(t), y(t)))
dt
Parameterize γ (see Cauchy Problem), solve system to get t(x, y) and s(x, y) and solve
u0 = c(x, y, u)
u(s, 0) = f (s)
using previously obtained x(t, s) and s(t, s) to set u(t, s) and substitute to get u = u(x, y)
13.7
Second Order Equations (m = 2)
Typically made easier using a linear transformation of coordinates
Let η = φ(x, y), ξ = ψ(x, y) be an invertible transformation (x, y) → (η, ξ). We need to find ux and uxx in
terms of the new system. Using chain rule,
ux = uη φx + uξ ψx
uxx = uηη φ2x + uη + uξη ψx φx + uξ + uηξ φx ψx + uη + uξξ ψx2 + uξ
uxx = uηη φ2x + 2uξη ψx φx + uξξ ψx2 + 2uη + 2uξ
Now we must find uy and uyy in the new system.
uy = uη φy + uξ ψy
uyy = uηη φ2y + uη + uξη ψy φy + uξ + uηξ φy ψy + uη + uξξ ψy2 + uξ
uyy = uηη φ2y + 2uξη ψy φy + uξξ ψy2 + 2uη + 2uξ
Now we must compute the mixed derivative uxy based on ux
uxy = uηη φy φx + uη + uξη φy ψx + uη + uηξ ψy φx + uξ + uξξ ψy ψx + uξ
uxy = uηη φy φx + uξη (φy ψx + ψy φx ) + uξξ ψy ψx + 2uη + 2uξ
Adding everything together, a second order equation in the original coordinates
auxx + 2buxy + cuyy + dux + f uy + hu = g(x, y)
becomes in the new coordinate system:
Auηη + Buηξ + Cuξξ + Duη + F uξ + Hu = g(η, ξ)
where
A(η, ξ) =aφ2x + bφx φy + cφ2y
B(η, ξ) =2aφx ψx + b(φy ψx + φx ψy ) + 2cψy φy
C(η, ξ) =aψx2 + bψx ψy + cψy2
D(η, ξ) =dφx + f φy + 2a + 2b + 2c
F (η, ξ) =dψx + f ψy + 2a + 2b + 2c
H(η, ξ) =h
65
13.7
Pierson Guthrey
pguthrey@iastate.edu
Second Order Equations (m = 2)
If the transformation η = φ(x, y), ξ = ψ(x, y) is linear, it can be expressed in the form
η
α β
x
=
ξ
γ δ
y
with αδ − βγ 6= 0 to ensure invertibility. This means φx = α, φy = β, ψx = γ, ψy = δ, so
A(η, ξ) =aα2 + bαβ + cβ 2
B(η, ξ) =2aαγ + b(βγ + αδ) + 2cβδ
C(η, ξ) =aγ 2 + bγδ + cδ 2
D(η, ξ) =dα + f β + 2a + 2b + 2c
F (η, ξ) =dγ + f δ + 2a + 2b + 2c
H(η, ξ) =h
13.7.1
Types of Equations
13.7.1.1 Parabolic Equations b2 −ac = 0 With equations of this type, one can choose b(η, ξ) = c(η, ξ) =
0 to obtain
uηη = 0
or equivalently b(η, ξ) = a(η, ξ) = 0 to get
uξξ = 0
13.7.1.2 Hyperbolic Equations b2 − ac > 0 With equations of this type, one can choose a(η, ξ) =
c(η, ξ) = 0 to obtain
uξη + D|1| u = 0
or equivalently b(η, ξ) = 0 to get
uξξ − uηη + D|1| u = 0
Where D|1| u is the lower order terms
If D|1| u = 0, the solutions to these equations are of the form
u(η, ξ) = Φ(η) + Ψ(ξ)
for some functions Φ(η), Ψ(ξ)
13.7.1.3 Elliptic Equations b2 − ac < 0
to obtain
With equations of this type, one can choose a(η, ξ) = c(η, ξ) = 0
uξη + D|1| u = 0
or equivalently b(η, ξ) = 0 to get
uξξ − uηη + D|1| u = 0
Where D|1| u is the lower order terms
13.7.2
Separation of Variables
Seeking separable solutions u(x1 , ..., xn ) = X1 (x1 ) × ... × Xn (xn ) works for many kinds of PDEs in multiple
dimensions. (0 ≤ θ < 2π, r ≥ 0)
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13.7
Pierson Guthrey
pguthrey@iastate.edu
Second Order Equations (m = 2)
13.7.3
Boundary Conditions
For a well-posed mth order PDE, you will have up to m side conditions, usually in the form of
F (u) = 0 on ∂Ω
Boundary conditions are said to be homogenous if they are closed under linear combinations, such as
u = 0 on ∂Ω
The most common boundary conditions are:
13.7.3.1
Dirichlet Conditions (First Type)
u = g on ∂Ω
13.7.3.2
Neumann (Second Type)
∂u
= ∇u · ~n = g on ∂Ω
∂n
where ~n is the unit outward normal
13.7.3.3
Robin (Third Type)
∂u
+ σu = ∇u · ~n + σu = g on ∂Ω
∂n
13.7.4
Common Well-Posed Problems
The following equations are well-posed and have classic solutions
13.7.4.1
Poisson’s Equation
∆u = f
(Poisson’s Equation)
∆u = 0
(Laplace’s Equation)
Solutions are harmonic functions, typically with Dirichlet, Neumann, or Robin conditions
(rectangular coordinates)
uxx + uyy + uzz = f
1
1
urr + ur + 2 uθθ = f
(polar coordinates)
r
r
Requires product solution families to be 2π periodic in θ:
(
c1 rn cos (nθ) + c2 rn sin (nθ) + c3 r−n cos (nθ) + c4 r−n sin (nθ) n = 1, 2, 3...
un (θ, r) =
c1 + c2 log(r)
n=0
Coefficients are found using boundary conditions, initial conditions, and Fourier theory
1 ∂
∂u
1 ∂2u ∂2u
r
+ 2 2 + 2 =0
(cylindrical coordinates)
r ∂r
∂r
r ∂φ
∂z
1 ∂
r2 ∂r
r2
∂u
∂r
+
1
∂2
2
r sin(θ) ∂θ2
sin(θ)
∂u
∂θ
+
1
∂2u
=0
2
r2 sin (θ) ∂φ2
67
(spherical coordinates)
13.7
Pierson Guthrey
pguthrey@iastate.edu
Second Order Equations (m = 2)
13.7.4.1.1 Using Distributions
−1
u(x) = 4π|x|
, note u ∈ L1loc (R3 ) since
Z
R
Z
2π
Z
Z
π
|u(x)| dx =
Claim: ∆u = δ. Reason:
−π
0
0
R
1 2
C
r sin(θ)dφdrdθ = R2
4πr
2
Z
Z
u(∆φ) =
u∆φ = lim
→0
u∆φdx
<|x|<R
Since φ has compact support and so for some R φ(|x| > R) = 0. By Green’s Identity this becomes
!
Z
Z
Z
∂φ
∂u
lim
u∆φdx = lim
φ(x)∆udx +
− u ds
φ
→0 <|x|<R
→0
∂n
<|x|<R
|x|= ∂n
Note that ∆u ∝ ∆ 1r = 0, and
Z
Z
∂φ ∂φ 1
∂φ 1
u ds ≤
ds =
max 4π2 ≤ C → 0
|x|= ∂n 4π |x|= ∂n
4π
∂n
∂u
since the 4π2 is the surface area of a sphere. We also see that since ∂n
= − ∂u
∂r ,
Z
Z
Z
Z
∂u
1
1
∂u
φ ds = −
φ
ds =
φds
φ ds = −
2
2
∂n
∂r
4π
4π
|x|=
|x|=
|x|=
|x|=
which is theR average of φ over the sphere |x| = which must tend to the value at the center. That is, as
1
→ 0, 4π
φds → φ(0), so
2
|x|=
∆u(φ) = u(∆φ) = φ(0) = δ(φ)
In higher dimensions,
(
u(x) =
−1
2π
AN
log(x) N = 2
1
N ≥3
|x|N −2
Where AN is the surface area of the N -Sphere.
13.7.4.2
Fundamental Solution
E(x) =
 |x|

2


N =1
1
2π log |x|



 CN
|x|N −2
N =2
N ≥3
N
Where CN = N (N???
If ∆ is replaced by the positive operator −∆,
−1)an and an is the volume of B1 (0) in R
the the FS is E− (x) = −E(x)
13.7.4.3
Wave Equation
utt − c2 uxx = 0
Typically with Dirichlet, Neumann, or Robin conditions in space and some
u(~x, 0) =f (~x)
ut (~x, 0) =g(~x)
68
x∈Ω
13.7
Pierson Guthrey
pguthrey@iastate.edu
Second Order Equations (m = 2)
η = x + ct and ξ = x − ct, then
u(x, t) = Φ(x + ct) + Ψ(x − ct)
If initial conditions u(x, 0) = f (x) and ut (x, 0) = g(x) are given,
Rx
then f (x) = Φ(x) + Ψ(x) and g(x) = Φ0 (x) + Ψ0 (x) and so 0 g(y)dy = Φ(x) + Ψ(x) + C
R
x
Combining, F (x) = 21 f (x) + 0 g(y)dy + C giving us
D’Alembert’s Solution Formula
Z x+ct
1
1
g(y)dy
u(x, t) = (f (x + ct) + f (x − ct)) +
2
2c x−ct
13.7.4.4
Fundamental Solution

1


2 H(t − |x|) N = 1


 H(t−|x|)
1
N =2
E(x, t) = 2 √t2 −|x|2




 δ(t−|x|)
N =3
4π|x|
Example: For N = 3
Z
(E ∗ f ) =
R4
δ(s − |y|)
f (x − y, t − s)dsdy =
4π |y|
Assume f (x, t) < 0 for t < 0, so
Z
(E ∗ f ) =
R3 ,|y|<t
1
f (x − y, t − |y|)dy =
4π |y|
Z
R3 ,s=|y|
Z
B(x,t)
1
f (x − y, t − |y|)dy
4π |y|
1
f (y, t − |x − y|)dy
4π |x − y|
So the solution of utt − ∆u = f , u(x, 0) = h(x), ut (x, 0) = g(x) can be shown to be
u(x, t) = (E ∗ g) +
(x)
∂
(E ∗ h) + (E ∗ f )
∂t (x)
13.7.4.4.1 Uniqueness Showing the uniqueness of solutions to this problem is equivalent to showing
that the homogenous problem
utt − c2 uxx = 0
ut (~x, 0) = u(~x, 0) = 0
with homogenous boundary conditions has only a trivial solution.
13.7.4.4.2 Four Point Property for solutions to the wave equation uxx −utt = 0 on Ω ⊂ R2 containing
the tilted rectangle with vertices (x, t), (x + h − k, t + h + k), (x + h, t + h), u(x − k, t + k),
u(x, t) + u(x + h − k, t + h + k) = u(x + h, t + h) + u(x − k, t + k)
13.7.4.4.3 Using Distributions
Let F ∈ L1loc (R), then u(x, t) = F (x + t) is one solution to the wave equation in D0 (R2 ) since
Z Z
T (φ) = (utt − uxx )(φ) = u(φtt − φxx ) =
F (x + t)(φtt (x, t) − φxx (x, t))dxdt
R2
By change of coordinates, ξ = x + t, η = x − t, φtt − φxx = −4φξη , dxdt =
Z ∞
Z ∞
=2
F (η)
φξη dξdη = 0
−∞
−∞
since φ has compact support
69
∂(x,t)
∂(ξ,η)
= − 21 dξdη
13.7
Pierson Guthrey
pguthrey@iastate.edu
Second Order Equations (m = 2)
13.7.4.5
Heat Equation
ut − ∆u = 0
(~x, t) ∈ Ω × [0, ∞)
Typically with Dirichlet, Neumann, or Robin conditions in space and some
u(~x, 0) = f (~x), x ∈ Ω
Look for separable solutions of the form u(t, x) = Φ(t)Ψ(x) where
Φ(t) = c1 ekt + c2 e−kt
Ψ(x) = c1 sin(kx) + c2 cos(kx)
Solutions will usually involve k = kn so the set of product solutions is
n
o∞
2
2
2
2
∞
{Φn (t)Ψn (x)}0 = c1 ekn t sin(kn x) + c2 ekn t cos(kn x) + c3 e−kn t sin(kn x) + c4 e−kn t sin(kn x)
0
Where the c1 , ..., c4 is chosen to satify the boundary and initial conditions using Fourier theory.
13.7.4.5.1 Uniqueness Showing the uniqueness of solutions to this problem is equivalent to showing
that the homogenous problem
ut − ∆u = 0
ut (~x, 0) = u(~x, 0) = 0
with homogenous boundary conditions has only a trivial solution.
13.7.4.6
Fundamental Solution
E(x, t) =
13.7.4.7
H(t)
(2πt)
N
2
e
−|x|2
4t
Reason:
13.7.4.7.1
∧
Using FT For Lu = ut − uxx with u(x, 0) = f (x) taking the partial FT in x yields
Z
1
∧
u(x, t)e−ikx dx
and (ut ) = (û)t
û(k, t) =
N
N
2
(2π)
R
2
2
So (−∆u) = |k| û so ût + |k| û = 0, which is an ODE for û for fixed k. Solving,
∧ !
Z
|x−y|2
1
1
−|k|2 t
−1
ˆ
û(k, t) = f e
=⇒ u(x, t) =
F
(f
∗
g)
=
e− 4t f (y)dy
N
N
(2π) 2
(4πt) 2 RN
2
where ĝ(k) = e−|k|
t
so g(x) =
|x|2
4t
N
(2t) 2
e−
. This is valid for f ∈ Lp (RN ) for some 1 ≤ p ≤ ∞, t > 0, and
lim u(x, t) = f (x) a.e.
t→0+
70
13.7
Pierson Guthrey
pguthrey@iastate.edu
Second Order Equations (m = 2)
13.7.4.7.2 Using FT and Duhamel’s Principle By Duhamel’s Principle, we can do the same for
Rt
Lu = h. Let v(x, t, s) satisfy vt − ∆v = 0 x ∈ RN , t > 0, v(x, 0, s) = h(x, s) and let u(x, t) = 0 v(x, t − s, s)ds.
Z
t
t
Z
vt (x, t − s, s)ds = h(x, t) +
ut = v(x, 0, t) +
0
∆v(x, t − s, s)ds = h(x, t) + ∆u
0
So for ut − ∆u = h(x, t) for t > 0 with u(x, 0) = 0, we have
v(x, t, s) =
where E =
Z
1
(2πt)
N
2
−|x|2
4t
N
(2πt) 2
H(t)e
e
−|x−y|2
4t
Z tZ
1
h(y, s)dy =⇒ u(x, t) =
RN
0
RN
(2π(t − s))
N
2
e
−|x−y|2
4(t−s)
h(y, s)dyds = E ∗ h
and h(x, t) = 0 for t < 0. So our solution is
u(x, t) = (E ∗ h)(x, t) + (E ∗ f )(x, t)
(x)
N F √xt if
So E is the fundamental solution to the Heat Equation. Note: for t > 0, E(x, t) = √1t
|x|2
R
−
F (x) = e 4N . Here F ≥ 0, F ∈ L1 , and RN F (x)dx = 1. By previous discussion of approximate identities
let k =
(2π)
1
√
t
2
k N F (kx) → δ as k → ∞ =⇒ E(x, t) → δ ∈ D0 as t → 0+
so (E ∗ f )(x, t) = E(·, t) ∗ f → f as t → 0+ (similar to approximate identities)
(x)
13.7.4.8 Maximum Principle
Dirichlet boundary conditions.
13.7.4.9
The solution u is bounded by the extremes of the initial condition and the
Shrodinger Equation
ut − i∆u = 0
(~x, t) ∈ Ω × [0, ∞)
Typically with Dirichlet, Neumann, or Robin conditions in space and some u(~x, 0) = f (~x), x ∈ Ω
13.7.4.10
Fundamental Solution
E(x, t) =
13.7.4.11
H(t)
(2πt)
N
2
−|x|2
4it
Helmholtz Equation
∆u − k 2 u = 0
13.7.4.12
π
ei(N −2) 4 e
(~x, t) ∈ Ω × [0, ∞)
Fundamental Solution
(
E(x, t) =
1
2π K0 (k |x|)
−e−k|x|
4π|x|
N =2
N =3
Where Ki is the ith order Modified Bessel Function.
Note as k → 0, E(x) tends toward the E(x) corresponding to L = ∆
71
Pierson Guthrey
pguthrey@iastate.edu
13.7.4.13
Euler-Tricomi Equation
uxx − xuyy = 0
Hyperbolic in the half plane x > 0, parabolic at x = 0, elliptic in x < 0
2
Characteristics are along y ± 23 x 3 = C
Particular solutions are
u =c1 xy + c2 x + c3 y + c4
u =c1 3y 2 + x3 + c2 y 3 + x3 y + c3 6xy 2 + x4
13.7.4.14
Biharmonic Oscillator
∆2 u = 0
13.7.4.15
Fundamental Solution ??
13.7.4.16
Klein Gordon Equation
(~x, t) ∈ Ω × [0, ∞)
Lu = utt − uxx + u
13.7.4.17
Fundamental Solution
E(x, t) =
14
(~x, t) ∈ Ω × [0, ∞)
p
1
H(t − |x|)J0 ( t2 − x2 )
2
Distribution Theory in DE
If f 0 (x) = 0 for a ≤ x ≤ b then f (x) = c classically If T 0 = 0 ∈ D0 (a, b) then T = c
Rb
Rb
Reason: Choose φ0 ∈ C0∞ with a φ0 (x)dx = 1. If φ ∈ C0∞ (a, b) let ψ(x) = φ(x) − a φ(y)dyφ0 (x). Note
Rb
Rx
ψ(x)dx = 0. Let ζ = a ψ(s)ds so ζ 0 = ψ(x) since ζ ∈ C ∞ (a, b) and further ζ ∈ C0∞ since
a
R ζ(a) = ζ(b) = 0
b
and ζ 0 = 0 for x < a or x > b. Then 0 = T 0 (ζ) = −T (ζ 0 ) = −T (ψ) = −T (φ) + a φydy T (φ0 ) so
Rb
Rb
T (φ) = a T (φ0 )φ(y)dy = a cφ(y)dy = c in D0 sense.
14.1
ODEs in D0 sense
T0 = f
f ∈ L1loc (R)
Rx
f (s)ds (antiderivative of f ). Claim g 0 = f in D0 (a, b). Reason:
Z b
Z bZ x
Z
Tg0 (φ) = −Tg (φ0 ) = −
g(x)φ0 (x)dx = −
f (s)dsφ0 (x)dx = −
Let g(x) =
a
a
a
a
b
a
So the general solution of T 0 = f in D0 (a, b) is
Z
T =
x
f (s)ds + C
a
72
Z
f (s)
a
Using FTC and compact support of φ,
Z b
Z
Tg0 (φ) = −
f (s)(φ(b) − φ(s))ds =
a
b
f (s)φ(s)ds = Tf (φ)
s
b
φ0 (x)dxds