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Unformatted text preview: rientation
(i.e. has counter-clockwise orientation), then
P (x, y ) dx + Q(x, y ) dy =
∂D D ∂ Q ∂P
∂y dA . Stokes’s Theorem:
(∇ × F ) • n dA =
S F • dr .
∂S (If n is pointing right at you then orient ∂S in a positive fashion (i.e. counterclockwise fashion, typically) to make the identity hold.) Spherical Coordinates:
x = ρ sin φ cos θ y = ρ sin φ sin θ
1 z = ρ cos φ dV = ρ2 sin φ dρ dφ dθ .
Second Derivative Test: Suppose the second partial derivatives of f are
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