21-259  Calculus in 3-D  Spring 2003

Formula Sheet for Test 3

Cylindrical Coordinates:

x = r cosθ, y = r sin θ, z = z, r = \sqrt{x^{2} + y^{2}}, dV = r dzdrdθ

Spherical Coordinates: x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ, ρ = \sqrt{x^{2} + y^{2} + z^{2}}, dV = ρ^{2} sin φ dρdφdθ Green’s Theorem:

∫_{C} P (x, y)dx + Q(x, y)dy = ∫_{D} ∫ (\frac{∂Q}{∂x} − \frac{∂P}{∂y} ) dA

Here C is a simple closed curve in the x − y plane with counterclockwise orientation and D is the region enclosed by C. Surfaces Described by r⃗(u, v):

dS = |\frac{∂r⃗}{∂u} × \frac{∂r⃗}{∂v} | dudv, n⃗dS = ±\frac{∂r⃗}{∂u} × \frac{∂r⃗}{∂v} dudv

Surfaces Described by z = f(x, y):

dS = \sqrt{1 + f_{x}^{2} + f_{y}^{2}} dydx, n⃗dS = ± (f_{x}i⃗ + f_{y}j⃗ − k⃗) dydx

Sphere of Radius a > 0 Centered at (0, 0, 0):

r⃗(φ, θ) = (a sin φ cos θ)i⃗ + (a sin φ sin θ)j⃗ + (a cos φ)k⃗ dS = a^{2} sin φdφdθ