1
TABLE OF CONTENTS
Chapter 1 Euclidean space
A.
B.
C.
D.
E.
F.
G.
The basic vector space
Distance
Right angle
Angles
A little trigonometry
Balls and spheres
Isoperimetric inequalities
Chapter 2 Differentiation
A.
B.
C.
D.
E.
F.
G.
H.
I.
J.
K.
L.
M.
Functions of one real variable
Lengths of curves
Directional derivatives
Pathology
Differentiability of real-valued functions
Sufficient condition for differentiability
A first look at critical points
Geometric significance of the gradient
A little matrix algebra
Derivatives for functions Rn → Rm
The chain rule
Confession
Homogeneous functions and Euler’s formula
Chapter 3 Higher order derivatives
A.
B.
C.
D.
E.
F.
G.
H.
I.
Partial derivatives
Taylor’s theorem
The second derivative test for R2
The nature of critical points
The Hessian matrix
Determinants
Invertible matrices and Cramer’s rule
Recapitulation
A little matrix calculus
2
Chapter 4 Symmetric matrices and the second
derivative test
A.
B.
C.
D.
E.
F.
G.
Eigenvalues and eigenvectors
Eigenvalues of symmetric matrices
Two-dimensional pause
The principal axis theorem
Positive definite matrices
The second derivative test
A little matrix calculus
Chapter 5 Manifolds
A.
B.
C.
D.
E.
F.
G.
Hypermanifolds
Intrinsic gradient-warm up
Intrinsic critical points
Explicit description of manifolds
Implicit function theorem
The tangent space
Manifolds that are not hyper
Chapter 6 Implicit function theorem
A.
B.
C.
D.
E.
F.
G.
H.
I.
Implicit presentation
Rank
Implicit function theorem
Results
Parametric presentation
The explicit bridge
The derivative reconsidered
Conformal mapping
Examples of conformal mappings
Chapter 7 Cross product
A.
B.
C.
D.
E.
F.
G.
Definition of the cross product
The norm of the cross product
Triple products
Orientation
Right hand rule
What is SO(3)?
Orientation reversal
3
Chapter 8 Volumes of parallelograms
A.
B.
C.
D.
Volumes in dimensions 1, 2, 3
The Gram determinant
Volumes in all dimensions
Generalization of the cross product
Chapter 9 Integration on Rn
A.
B.
C.
D.
E.
F.
G.
H.
I.
The idea of Riemann sums
Step functions
The Riemann integral
Sufficient conditions for integrability
The fundamental theorem
The Fubini theorem
Iterated integrals
Volume
Integration and volume
Chapter 10 Further investigation of integration
A.
B.
C.
D.
E.
F.
G.
H.
I.
Topological background
Topological characterization of contentedness
Integration over more general sets
Cavalieri’s principle
Elementary matrices
Linear changes of variables
The general formula for change of variables
The gamma function
Notation for the Jacobian determinant
Chapter 11 Integration on manifolds
A.
B.
C.
D.
E.
The one-dimensional case
The general case
Hypermanifolds
The Cauchy-Binet determinant theorem
Miscellaneous applications
4
Chapter 12 Green’s theorem
A.
B.
C.
D.
E.
F.
G.
The basic theorem of Green
Line integrals
A general Green’s theorem
Areas by means of Green
Conservative vector fields
Sufficiency
A slick proof of sufficiency
Chapter 13 Stokes’ theorem
A.
B.
C.
D.
E.
F.
G.
H.
I.
Orientable surfaces
The boundary of a surface
Inherited orientation
The basic calculation
The basic theorem, curl
Stokes’ theorem
What is curl?
Curlometer
Conservative fields revisited
Chapter 14 Gauss’ theorem
A.
B.
C.
D.
E.
F.
Green’s theorem reinterpreted
Gauss’ theorem for Rn
The proof
Gravity
Other differentiation formulas
The vector potential
Chapter 15 ∇ in other coordinates
A.
B.
C.
D.
E.
F.
G.
H.
I.
Biorthogonal systems
The gradient
The divergence
The curl
Curvilinear coordinates
The gradient
Spherical coordinates
The divergence
The curl
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J. Orthogonal curvilinear coordinates
K. The Laplacian
L. Parabolic coordinates
Index