Vector Calculus and Complex Variable

Question 1

Triangle inequalities

Answer 1

|x + y| <= |x| + |y|
|x - y| >= | |x| - |y| |

Question 2

Even vs. Odd functions

Answer 2

Even function: f(-x) = f(x)
Odd function: f(-x) = -f(x)
Otherwise, neither

Question 3

Dot product properties

Answer 3

• Associativity
• Commutitivity
• Distributivity
• |u| = sqrt(u.u)
• u.v = |u||v|cos(theta)

Question 4

Cross product properties

Answer 4

• uxv is perpendicular to u and v
• |uxv| = |u||v|sin(theta)
• uxv = -(vxu)

Question 5

Triple product

Answer 5

u.(vxw)

Question 6

Gradient

Answer 6

Scalar field -> Vector field

Question 7

Divergence

Answer 7

Vector field -> Vector field

Question 8

Curl

Answer 8

Vector field -> Vector field

(Always has zero divergence)

Question 9

Laplacian

Answer 9

Scalar field -> Scalar

Question 10

Normal to a surface

Answer 10

Question 11

Length integral

Answer 11

Question 12

Line integral of a scalar field

Answer 12

Question 13

Line integral of a vector field

Answer 13

Question 14

Surface integral

Answer 14

Question 15

Surface integral of a scalar field

Answer 15

Question 16

Flux of a non-constant vector field

Answer 16

Question 17

Path-independant vector field

Answer 17

Given two curves joining A and B,
a vector field is path-independant if the line integrals over each curve are euqal

Question 18

Connected domain

Answer 18

A domain U is connected if any A,B in U can be joined by a curve in U

Question 19

Simple connected domain

Answer 19

A domain U is simply connected if it is connected,
and any loop can be shrunk to a point

Question 20

Divergence (Gauss) Theorem

Answer 20

Requires a closed, orientable, and piecewise smooth surface.
For a bounded, solid region in R^3 with boundary surface S:

Question 21

Principle value

Answer 21

-pi < Arg(z) <= pi

Question 22

cosh(x) and sinh(x) in terms of exp(x)

Answer 22

Question 23

cos(x) and sin(x) in terms of exp(x)

Answer 23

Question 24

Double angle formula

Answer 24

cos(2x) = 2.cos(x)^2 - 1

Question 25

Holomorphic

Answer 25

A function is holomorphic on S if it is differentiable for all x in S

Question 26

Entire

Answer 26

A function is entire if it is holomorphic on C

(The complex set)

Question 27

Cauchy-Reimann equations

Answer 27

Question 28

Result from the Cauchy-Reimann equations

Answer 28

If a function satisfies the C-R equations,
and each of the C-R equations are continuous,
then f is differentiable

Question 29

Estimation Lemma

Answer 29

For all z on a contour gamma with length L,

Question 30

Cauchy’s theorem

Answer 30

If f is holomorphic on and inside a closed contour gamma,

Question 31

Deformation theorem

Answer 31

If f is holomorphic in D, and two closed contours in D can be continuously deformed into each other in D,
then the integrals of f(z) over each contours are equal

Question 32

Cauchy’s integral formula

Answer 32

If f is holomorphic on and inside a simple, closed contour gamma,

Question 33

Cauchy’s integral formula for the nth derivative

Answer 33

If f is holomorphic on and inside a simple, closed contour gamma, and if f is n-times differentiable at a inside gamma,

Question 34

Cauchy’s inequalities

Answer 34

If f is holomorphic on and inside a circle gamma, centre a, radius R, then:

Question 35

Liouville’s theorem

Answer 35

If f is holomorphic and bounded on C,
then f is a constant

f holomorphic on C == f entire

Question 36

Analytic

Answer 36

A complex function that is analytic at z0 can be represented as a power series near z0.

Equivalent to Holomorphic.

Question 37

Laurent series coefficient formula

Answer 37

Cauchy’s integral formula for the nth degree, without the n!

Question 38

Types of singularities

Answer 38

• Removable
• Pole
• Essential

Question 39

Removable singularity

Answer 39

When the Laurent series around a has no negative powers of (z - a)

Question 40

Pole

Answer 40

When the Laurent series around a has a finite number of negative powers of (z - a).
The order of the pole is the most negative power

Question 41

Essential singularity

Answer 41

When the Laurent series around a has infinitely negative powers of (z - a)

Question 42

Calculating Laurent series

Answer 42

If f is analytic (holomorphic/analytic at a), use Taylor’s series around a,
otherwise use the formula for coefficients

Question 43

Residue of a removable singularity

Answer 43

residue = 0

Question 44

Residue of an essential singularity

Answer 44

c/-1 coefficient

Question 45

Real integrals of rotational, periodic functions

Answer 45

Substitute in z = e^(i.theta)

Question 46

Conservative vector fields

Answer 46

If a differentiable vector field F is conservative in a connected region U, then:
* there exists a C^2 function phi in U s.t. f = grad.phi
* F is path independant
* the line integral of F on any closed loop wholly contained in U is zero
* the curl of F is zero

If U is simply connected and the curl of F is zero, then F is conservative

Question 47

Stoke’s theorem

Answer 47

For an orientable, smooth, bounded surface S in R^3, with boundary delta S. Let F be a vector field whose domain includes S, then:

Question 48

Irrotational

Answer 48

A vector field with 0 curl

Question 49

Laplace equation

Answer 49

Question 50

Cauchy’s Residue Theorem

Answer 50

Question 51

When Stoke’s theorem vs Divergence theorem can be used

Answer 51

Stoke’s: smooth, orientable, and bounded
Divergence: smooth, orientable, and closed