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

## Footnote 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