MA 261 Final Exam Review

Gradient (∇f)

The vector of partial derivatives:

Directional derivative

Dᵤf = ∇f · u (u must be a unit vector).

Divergence of F

div F = ∂F₁/∂x + ∂F₂/∂y + ∂F₃/∂z.

Curl of F

curl F = ∇ × F.

Conservative vector field

F = ∇f and curl F = 0 on a simply connected domain.

Scalar line integral

∫_C f ds = ∫ f(r(t))‖r′(t)‖ dt.

Vector line integral

∫_C F · dr = ∫ F(r(t)) · r′(t) dt.

Surface integral (scalar)

∬_S f dS = ∬ f(r(u,v)) ‖rᵤ × rᵥ‖ du dv.

Flux integral

∬_S F · n̂ dS.

Jacobian (cylindrical)

r.

Jacobian (spherical)

ρ² sinφ.

Green’s Theorem

∮C F · dr = ∬R (∂N/∂x − ∂M/∂y) dA.

Divergence Theorem

∬∂V F · n̂ dS = ∭_V div F dV.

Stokes’ Theorem

∮∂S F · dr = ∬S curl F · n̂ dS.

Fundamental Theorem for Line Integrals

If F = ∇f, ∫C F·dr = f(B) − f(A).

Mixed partials theorem

If fₓᵧ and fᵧₓ are continuous, then fₓᵧ = fᵧₓ.

Tangent plane formula

z − z₀ = fₓ(x₀,y₀)(x − x₀) + fᵧ(x₀,y₀)(y − y₀).

Multivariable chain rule

df/dt = fₓ dx/dt + fᵧ dy/dt (+ f_z dz/dt).

Polar double integral

∬ f(x,y) dA = ∫∫ f(r cosθ, r sinθ) r dr dθ.

Cylindrical triple integral

∭ f(r,θ,z) r dr dθ dz.

Spherical triple integral

∭ f(ρ,θ,φ) ρ² sinφ dρ dθ dφ.

Surface area (parametric)

A = ∬ ||rᵤ × rᵥ|| du dv.

When to use polar coordinates

Circular regions or integrands with x² + y².

When to use cylindrical coordinates

Regions with circular symmetry and a vertical z-component.

When to use spherical coordinates

Spherical symmetry, integrands with x²+y²+z².

When to use Green’s Theorem

Converting a line integral to a double integral (closed curve).

When a field is conservative

Check curl F = 0 (in simply connected region).