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
-
u
xv
is perpendicular tou
andv
-
|u
xv| = |u||v|sin(theta)
-
u
xv = -(v
xu)
Triple product
u.(v
xw)
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
: