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Appendix H
Vector Analysis
Rectangular Coordinates
f = f(x, y, z), ~g = g x i + g y j + g z k
∂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 r + g θ θ + g φ φ
Divergence Theorem.
ZZ I
∇ · u dx dy = u · n ds
Stoke’s Theorem.
ZZ I
(∇ × u) · ds = u · dr
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