Vector Calculus and Complex Variable Flashcards
Triangle inequalities
|x + y| <= |x| + |y||x - y| >= | |x| - |y| |
Even vs. Odd functions
Even function: f(-x) = f(x)
Odd function: f(-x) = -f(x)
Otherwise, neither
Dot product properties
- Associativity
- Commutitivity
- Distributivity
|u| = sqrt(u.u)u.v = |u||v|cos(theta)
Cross product properties
-
uxvis perpendicular touandv -
|uxv| = |u||v|sin(theta) -
uxv = -(vxu)
Triple product
u.(vxw)
Gradient
Scalar field -> Vector field
Divergence
Vector field -> Vector field
Curl
Vector field -> Vector field
(Always has zero divergence)
Laplacian
Scalar field -> Scalar
Normal to a surface
Length integral
Line integral of a scalar field
Line integral of a vector field
Surface integral
Surface integral of a scalar field
Flux of a non-constant vector field
Path-independant vector field
Given two curves joining A and B,
a vector field is path-independant if the line integrals over each curve are euqal
Connected domain
A domain U is connected if any A,B in U can be joined by a curve in U
Simple connected domain
A domain U is simply connected if it is connected,
and any loop can be shrunk to a point
Divergence (Gauss) Theorem
Requires a closed, orientable, and piecewise smooth surface.
For a bounded, solid region in R^3 with boundary surface S:
Principle value
-pi < Arg(z) <= pi
cosh(x) and sinh(x) in terms of exp(x)
cos(x) and sin(x) in terms of exp(x)
Double angle formula
cos(2x) = 2.cos(x)^2 - 1
Holomorphic
A function is holomorphic on S if it is differentiable for all x in S
Entire
A function is entire if it is holomorphic on C
(The complex set)
Cauchy-Reimann equations
Result from the Cauchy-Reimann equations
If a function satisfies the C-R equations,
and each of the C-R equations are continuous,
then f is differentiable
Estimation Lemma
For all z on a contour gamma with length L,
Cauchy’s theorem
If f is holomorphic on and inside a closed contour gamma,
Deformation theorem
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
Cauchy’s integral formula
If f is holomorphic on and inside a simple, closed contour gamma,
Cauchy’s integral formula for the nth derivative
If f is holomorphic on and inside a simple, closed contour gamma, and if f is n-times differentiable at a inside gamma,
Cauchy’s inequalities
If f is holomorphic on and inside a circle gamma, centre a, radius R, then:
Liouville’s theorem
If f is holomorphic and bounded on C,
then f is a constant
f holomorphic on C == f entire
Analytic
A complex function that is analytic at z0 can be represented as a power series near z0.
Equivalent to Holomorphic.
Laurent series coefficient formula
Cauchy’s integral formula for the nth degree, without the n!
Types of singularities
- Removable
- Pole
- Essential
Removable singularity
When the Laurent series around a has no negative powers of (z - a)
Pole
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
Essential singularity
When the Laurent series around a has infinitely negative powers of (z - a)
Calculating Laurent series
If f is analytic (holomorphic/analytic at a), use Taylor’s series around a,
otherwise use the formula for coefficients
Residue of a removable singularity
residue = 0
Residue of an essential singularity
c/-1 coefficient
Real integrals of rotational, periodic functions
Substitute in z = e^(i.theta)
Conservative vector fields
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
Stoke’s theorem
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:
Irrotational
A vector field with 0 curl
Laplace equation
Cauchy’s Residue Theorem
When Stoke’s theorem vs Divergence theorem can be used
Stoke’s: smooth, orientable, and bounded
Divergence: smooth, orientable, and closed
