MTH4101 CALCULUS II
REVISION NOTES
1.
COMPLEX NUMBERS (Thomas Appendix 7 + lecture notes)
1.1
Introduction
Types of numbers (natural, integers, rationals, reals)
The need to solve quadratic equations:
ax2 + bx + c = 0
Solution
x=
−b ±
√
b2 − 4ac
2a
What happens if b2 < 4ac? The need for complex numbers.
1.2
Complex Numbers
i=
√
−1
z = x + iy
x is the real part, y is the imaginary part.
x = Re(z)
i2 = −1,
i−1 = −i,
Argand diagram
y = Im(z)
i3 = −i,
i−2 = −1,
i4 = 1,
i−3 = i,
etc.
etc.
Connection with polar coordinates (x = r cos θ, y = r sin θ).
z = x + iy = r(cos θ + i sin θ)
|z| = r =
p
x2 + y 2 is the modulus
arg z = θ = tan−1 (y/x) is the argument
Complex conjugate: if z = x + iy then z̄ = x − iy
1.3
Operations with Complex Numbers
z1 = x1 + iy1
z2 = x2 + iy2
Addition:
z1 + z2 = (x1 + x2 ) + i(y1 + y2 )
Geometric interpretation (addition of vectors in parallelogram in Argand diagram)
Subtraction:
z2 − z1 = (x2 − x1 ) + i(y2 − y1 )
Geometric interpretation (completion of parallelogram in Argand diagram)
Multiplication:
z1 z2 = (x1 x2 − y1 y2 ) + i(x1 y2 + x2 y1 )
z1 z2 = r1 r2 {cos(θ1 + θ2 ) + i sin(θ1 + θ2 )}
(i.e. multiply moduli and add arguments)
Division:
x1 x2 + y1 y2
x1 y2 − x2 y1
z2
+i
=
2
2
z1
x1 + y1
x21 + y12
z2
r2
= {cos(θ2 − θ1 ) + i sin(θ2 − θ1 )}
z1
r1
(i.e. divide moduli and subtract arguments)
1.4
Loci and Regions
Recognise circles (including displaced ones), lines, regions defined by inequalities; use of
modulus and argument to define loci, regions.
1.5
Trigonometric Functions and Hyperbolic Functions
Recognise power series for ez , sin z, cos z.
eiθ = cos θ + i sin θ
(Euler’s relation)
z = r ei(θ+2kπ)
eiπ/2 = i,
cos θ =
eiθ + e−iθ
2
sin θ =
eiθ − e−iθ
2i
eiπ = −1,
e3iπ/2 = −i,
e2iπ = 1
cosh x = ex + e−x
2
sinh x = ex − e−x
2
Other hyperbolic functions and identities.
1.6
de Moivre’s Theorem
z n = (cos θ + i sin θ)n = cos nθ + i sin nθ
Applications:
Expansion of cosn θ, sinn θ, cos nθ, sin nθ
Solution of equations (e.g. n complex roots of z n − 1 = 0)
Evaluation of integrals
2.
2.1
PARTIAL DERIVATIVES
Functions of Several Variables (Thomas 14.1)
Domain and range
Sketch surfaces defined by z = f (x, y)
Draw and label curves in the domain in which f has a constant value
Level curves
Level contours
2.2
Limits and Continuity in Higher Dimensions (Thomas 14.2)
Definition of limit for f (x, y)
Calculate limts of polynomials and rational function by evaluating the function at the limit
point
Definition of continuous function
Two-Path Test for non-existence of a limit (if a function has different limits along two
different paths then the limit does not exist); use of polar coordinates if necessary
2.3
Partial Derivatives (Thomas 14.3)
Definition of partial derivative
Notation (difference between, e.g. d/dx and ∂/∂x; meaning of fx , fy etc.)
Higher derivatives
Mixed Derivatives Theorem:
fxy (a, b) = fyx (a, b)
Definition of differentiability
etc.
2.4
The Chain Rule (Thomas 14.4)
Chain Rule:
If w = f (x, y) then
dw
∂f dx ∂f dy
=
+
dt
∂x dt
∂y dt
If w = f (x, y, z) then
dw
∂f dx ∂f dy ∂f dz
=
+
+
dt
∂x dt
∂y dt
∂z dt
If w = f (x, y), x = g(r, s), y = h(r, s) then
∂w
∂w ∂x ∂w ∂y
=
+
∂r
∂x ∂r
∂y ∂r
∂w
∂w ∂x ∂w ∂y
=
+
∂s
∂x ∂s
∂y ∂s
If w = f (x), x = g(r, s) then
∂w
dw ∂x
=
∂r
dx ∂r
and
∂w
dw ∂x
=
∂s
dx ∂s
Tree diagrams
Formula for implicit differentiation of F (x, y):
dy
Fx
=−
dx
Fy
2.5
Directional Derivatives and Gradient Vectors (Thomas 14.5)
Definition of directional derivative, (Du f )P0 .
Definition of gradient vector,
∇f =
∂f ∂f ∂f
,
,
∂x ∂y ∂z
Relationship with directional derivative:
Du f = ∇f · u
.
The gradient of f is normal to the level curve.
2.6
Tangent Planes and Differentials (Thomas 14.6)
The tangent plane at P0 (x0 , y0 , z0 ) on the level surface f (x, y, z) = c is the plane through
P0 normal to ∇f |P0 . Equation is
fx (P0 )(x − x0 ) + fy (P0 )(y − y0 ) + fz (P0 )(z − z0 ) = 0
The normal line of the surface at P0 is line through P0 parallel to ∇f |P0 . Equation is
x = x0 + fx (P0 )t,
y = y0 + fy (P0 )t,
z = z0 + fz (P0 )t .
Linearisation
L(x, y, z) = f (x0 , y0 , z0 )+
+fx (x0 , y0 , z0 )(x − x0 ) + fy (x0 , y0 , z0 )(y − y0 ) + fz (x0 , y0 , z0 )(z − z0 )
Total differential resulting from change (dx, dy, dz) is:
df = fx (x0 , y0 , z0 )dx + fy (x0 , y0 , z0 )dy + fz (x0 , y0 , z0 )dz
2.7
Extreme Values and Saddle Points (Thomas 14.7)
Definitions of local minimum, local maximum, critical point, saddle point.
Second derivatives test for local maximum, local minimum, saddle point; when is test
inconclusive.
2
Use of second derivatives and discriminant, fxx fyy − fxy
.
2.8
Lagrange Multipliers (Thomas 14.8)
Method of Lagrange Multipliers: To find the local maximum and minimum values of
f (x, y, z) subject to the constraint g(x, y, z) = 0 we find the values of x, y and z that
simultaneously satisfy
∇f = λ∇g
3.
3.1
and g(x, y, z) = 0 .
MULTIPLE INTEGRALS
Double Integrals (Thomas 15.1, 15.2 + lecture notes)
Definition of double integral of function f (x, y) over region R:
Z Z
f (x, y) dx dy .
R
Connection with calculation of volume beneath surface using z = f (x, y).
Iterated or repeated integral. Importance of order of integration; Fubini’s theorem (two
forms).
Method for double integrals: (i) sketch, (ii) find y-limits of integration, (iii) find x-limits
of integration.
Bounded rectangular regions; bounded non-rectangular regions; unbounded regions.
Reversing order of integration; importance of finding new limits.
3.2
Area (Thomas 15.3 + lecture notes)
Area enclosed by a region R is the double integral,
Z Z
dx dy .
R
Average value of a function f (x, y) over a region R is
1
area of R
Z Z
f (x, y) dx dy .
R
3.3
Change of Variables in Double Integrals
(Thomas 15.8, 15.4 + lecture notes)
Change of variables from (x, y) to, say (u, v).
Definition of Jacobian matrix:
∂x/∂u
∂y/∂u
∂x/∂v
∂y/∂v
Definition of Jacobian (or Jacobian determinant):
∂(x, y)
∂x/∂u
=
∂y/∂u
∂(u, v)
∂x/∂v
= (∂x/∂u)(∂y/∂v) − (∂y/∂u)(∂x/∂v) .
∂y/∂v
Using Jacobian of transformation from Cartesian to polar coordinates to get
dx dy = r dr dθ
Use of
∂(x, y)
=
∂(u, v)
∂(u, v)
∂(x, y)
−1
.
Evaluation of
Z
∞
e−x
2
/2
dx
−∞
by making it a double integral and then transforming to polar coordinates; connection
with normal distribution and error function.
3.4
Triple Integrals (Thomas 15.5 + lecture notes)
Definition of triple integral of function f (x, y, z) over volume V :
Z Z Z
f (x, y, z) dx dy dz .
V
Volume enclosed by a volume V is the triple integral,
Z Z Z
dx dy dz .
V
Average value of a function f (x, y, z) over a volume V is
1
volume of V
3.5
Z Z Z
f (x, y, z) dx dy dz .
V
Change of Variables in Triple Integrals (Thomas 15.8 + lecture notes)
Change of variables from (x, y, z) to, say (u, v, w).
Definition of Jacobian matrix:

∂x/∂u
 ∂y/∂u
∂z/∂u
∂x/∂v
∂y/∂v
∂z/∂v

∂x/∂w
∂y/∂w  .
∂z/∂w
Definition of Jacobian (or Jacobian determinant):
∂x/∂u ∂x/∂v
∂(x, y, z)
= ∂y/∂u ∂y/∂v
∂(u, v, w)
∂z/∂u ∂z/∂v
4.
4.1
INFINITE SEQUENCES AND SERIES
Sequences (Thomas 10.1)
Lists of numbers {an }
Convergence and divergence of sequences
Limit of a sequence
Sandwich Theorem for sequences
Continuous Function Theorem for sequences
Non-decreasing sequences
Sequences bounded from above
Upper bound
Least upper bound
∂x/∂w
∂y/∂w .
∂z/∂w
4.2
Infinite Series (Thomas 10.2)
Sequence of partial sums
Convergent series and their sum; divergent series
Geometric series:
∞
X
n
2
a r = a + ar + ar + · · · + ar
n−1
n=0
+ ··· =
∞
X
a rn−1
n=1
(ratio r)
∞
X
a rn−1 =
n=1
a
1−r
n-th Term Test for Divergence:
P∞
n=1 an diverges if limn→∞ an fails to exist or is different from zero.
4.3
The Integral Test (Thomas 10.3)
P∞
Divergence of the harmonic series, n=1 1/n
P∞
Convergence of the series, n=1 1/n2
R∞
P∞
Integral test: If an = f (n) then n=N an and N f (x) dx both converge or both diverge
(proof using graphs for the case n = 1).
4.4
Ratio Tests (Thomas 10.5)
Ratio Test:
If
an+1
=ρ
n→∞ an
lim
then (i) the series converges if ρ < 1, (ii) the series diverges if ρ > 1 or ρ is infinite and
(iii) the test is inconclusive if ρ = 1.
4.5
Power Series (Thomas 10.7)
Power series about x = 0:
∞
X
cn xn = c0 + c1 x + c2 x2 + · · · + cn xn + · · ·
n=0
Power series about x = a:
∞
X
cn (x − a)n = c0 + c1 (x − a) + c2 (x − a)2 + · · · + cn (x − a)n + · · ·
n=0
Radius of convergence, interval of convergence.
Alternating Series Test:
∞
X
(−1)n+1 un = u1 − u2 + u3 − u4 + · · ·
n=1
converges if the following hold:
The un ’s are all positive
un ≥ un+1 for all n ≥ N , for some integer N
un → 0
Absolute convergence
Conditional convergence
4.6
Taylor and Maclaurin Series (Thomas 10.8)
Taylor series generated by function f at x = a:
∞
X
f (k) (a)
k=0
f (a) + f 0 (a)(x − a) +
k!
(x − a)k =
f (n) (a)
f 00 (a)
(x − a)2 + · · · +
(x − a)n + · · ·
2!
n!
Maclaurin series generated by function f at x = 0:
∞
X
f (k) (0)
k=0
k!
xk =
f (0) + f 0 (0)x +
f 00 (0) 2
f (n) (0) n
x + ··· +
x + ···
2!
n!
Taylor polynomial of order n
4.7
Convergence of Taylor Series;
Error Estimates (Thomas 10.9)
Taylor’s formula
Remainder R of order n (error term)
Remainder Estimation Theorem:
|Rn (x)| ≤ M
4.8
|x − a|n+1
(n + 1)!
Applications of Power Series (Thomas 10.10)
Binomial series:
(1 + x)m = 1 + mx +
m(m − 1) 2 m(m − 1)(m − 2) 3
x +
x + ···
2!
3!
(1 + x)
m
∞ X
m k
=1+
x
k
k=1
Solving differential equations:
Assume a power series solution of the form,
y = a0 + a1 x + a2 x2 + · · · + an xn + · · ·
substitute in the differential equation and solve for the coefficients.
Evaluating non-elementary integrals
Evaluating indeterminate forms