• Spherical Coordinates:
• Curl and Divergence:
curl ⃗F = ∇ × ⃗F , div ⃗F = ∇ · ⃗F ,
∇ = < ∂x, ∂y, ∂z>
• Surface Area Element (if the surface is given by ⃗r(u, v)):
dS =||⃗ru× ⃗rv||dA
• For ⃗r(t) =< x(t), y(t), z(t) >,
d⃗r =< dx, dy, dz >=< x′, y′, z′> dt
• Stokes Theorem:
Let S be an oriented piecewise-smooth surface that is bounded by a simple, closed, piecewise-smooth boundary curve C with positive orientation. Let ⃗F be a vector field whose components have continuous partial derivatives on an open region inR3 that contains S. Then
∫
C
F⃗· d⃗r =
∫ ∫
S
(∇ × ⃗F ) · d⃗S
• Divergence Theorem:
Let E be a simple solid region and let S be the boundary surface of E, given with positive (outward) orientation.
Let ⃗F be a vector field whose component functions have continuous partial derivatives on an open region that contains E. Then
∫ ∫
S
F⃗ · d⃗S =
∫ ∫ ∫
E
∇ · ⃗F (x, y, z) dV
