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Appendix H

Vector Analysis

Rectangular Coordinates

f = f(x, y, z), ~g = g i + g j + g k
x
z
y
∂f ∂f ∂f
∇f = i + j + k
∂x ∂y ∂z

∂g x ∂g y ∂g z
∇ · ~g = + +
∂x ∂y ∂z

i j k

∂ ∂ ∂
∇ × ~g =
∂x ∂y ∂z
g x g y g z

2
2
2
∂ f ∂ f ∂ f
2
∆f = ∇ f = + +
∂x 2 ∂y 2 ∂z 2
Spherical Coordinates

x = r cos θ sin φ, y = r sin θ sin φ, z = r cos φ

f = f(r, θ, φ), ~g = g r + g θ + g φ
r
θ
φ
Divergence Theorem.

ZZ I
∇ · u dx dy = u · n ds

Stoke’s Theorem.
ZZ I
(∇ × u) · ds = u · dr

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