Vector Calculus and Complex Variable Flashcards

1
Q

Triangle inequalities

A

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

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2
Q

Even vs. Odd functions

A

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

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3
Q

Dot product properties

A
  • Associativity
  • Commutitivity
  • Distributivity
  • |u| = sqrt(u.u)
  • u.v = |u||v|cos(theta)
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4
Q

Cross product properties

A
  • uxv is perpendicular to u and v
  • |uxv| = |u||v|sin(theta)
  • uxv = -(vxu)
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5
Q

Triple product

A

u.(vxw)

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6
Q

Gradient

A

Scalar field -> Vector field

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7
Q

Divergence

A

Vector field -> Vector field

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8
Q

Curl

A

Vector field -> Vector field

(Always has zero divergence)

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9
Q

Laplacian

A

Scalar field -> Scalar

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10
Q

Normal to a surface

A
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11
Q

Length integral

A
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12
Q

Line integral of a scalar field

A
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13
Q

Line integral of a vector field

A
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14
Q

Surface integral

A
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15
Q

Surface integral of a scalar field

A
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16
Q

Flux of a non-constant vector field

A
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17
Q

Path-independant vector field

A

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

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18
Q

Connected domain

A

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

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19
Q

Simple connected domain

A

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

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20
Q

Divergence (Gauss) Theorem

A

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

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21
Q

Principle value

A

-pi < Arg(z) <= pi

22
Q

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

A
23
Q

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

A
24
Q

Double angle formula

A

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

25
Q

Holomorphic

A

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

26
Q

Entire

A

A function is entire if it is holomorphic on C

(The complex set)

27
Q

Cauchy-Reimann equations

A
28
Q

Result from the Cauchy-Reimann equations

A

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

29
Q

Estimation Lemma

A

For all z on a contour gamma with length L,

30
Q

Cauchy’s theorem

A

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

31
Q

Deformation theorem

A

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

32
Q

Cauchy’s integral formula

A

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

33
Q

Cauchy’s integral formula for the nth derivative

A

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

34
Q

Cauchy’s inequalities

A

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

35
Q

Liouville’s theorem

A

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

f holomorphic on C == f entire

36
Q

Analytic

A

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

Equivalent to Holomorphic.

37
Q

Laurent series coefficient formula

A

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

38
Q

Types of singularities

A
  • Removable
  • Pole
  • Essential
39
Q

Removable singularity

A

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

40
Q

Pole

A

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

41
Q

Essential singularity

A

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

42
Q

Calculating Laurent series

A

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

43
Q

Residue of a removable singularity

A

residue = 0

44
Q

Residue of an essential singularity

A

c/-1 coefficient

45
Q

Real integrals of rotational, periodic functions

A

Substitute in z = e^(i.theta)

46
Q

Conservative vector fields

A

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

47
Q

Stoke’s theorem

A

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:

48
Q

Irrotational

A

A vector field with 0 curl

49
Q

Laplace equation

A
50
Q

Cauchy’s Residue Theorem

A
51
Q

When Stoke’s theorem vs Divergence theorem can be used

A

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