Vector Differentiation Rules

1. ∇ · ∇ × F = 0

2. ∇ × ∇f = 0

Spherical Coordinates x = ρ cos θ sin ϕ x = ρ sin θ sin ϕ x = ρ cos ϕ Change of Variables Formula

∫ ∫_{D} f(x, y) dA_{(x,y)} = ∫ ∫_{D∗} f(T(u, v)) |\frac{∂(x, y)}{∂(u, v)}| dA_{(u,v)}

Line, Path, Surface, and Volume Integrals

1. ∫_{C} f ds = ∫_{a}^{b} f(c(t))∥c′(t)∥ dt

2. ∫_{C} F · ds = ∫_{a}^{b} F(c(t)) · c′(t) dt

3. ∫ ∫_{S} F · dS = ∫ ∫_{D} F(ϕ(u, v)) · (T_{u} × T_{v}) dA

4. A(D) = ∫ ∫_{D} 1 dA = \frac{1}{2} ∫_{∂D} x dy − y dx

5. A(S) = ∫ ∫_{S} 1 dS = ∫ ∫_{D} ∥T_{u} × T_{v}∥ dA

6. ∫_{C} P dx + Q dy = ∫ ∫_{D} \frac{∂Q}{∂x} − \frac{∂P}{∂x} dA

7. ∫ ∫_{S} ∇ × F · dS = ∫_{∂S} F · ds

8. ∫ ∫ ∫_{Ω} ∇ · F dV = ∫ ∫_{∂Ω} F · dS

Conservative Fields

The following are equivalent: 1. for any oriented simple closed curve C, ∫_{C} F · ds = 0;

2. for any two oriented simple closed curves C_{1} and C_{2} which have the same endpoints, ∫_{C_{1}} F · ds = ∫_{C_{2}} F · ds;

3. F is the gradient of some function, f (that is, F = ∇f);

4. ∇ × F = 0.