Vector Calculus Master Flashcards

Flashcards covering key concepts and definitions from vector calculus, including line integrals, fundamental theorems, and theorems related to vector fields.

What is a line integral?

∫C F · dr → measures work along a curve

Scalar line integral formula

∫C f ds = ∫ f(r(t)) |r'(t)| dt

Vector line integral formula

∫C F · dr = ∫ F(r(t)) · r'(t) dt

Work done

W = ∫C F · dr

Conservative field

F = ∇f

Fundamental Theorem

∫C F · dr = f(B) − f(A)

Check conservative (2D)

∂P/∂y = ∂Q/∂x

Check conservative (3D)

∇ × F = 0

Curl definition

∇ × F measures rotation

Divergence definition

∇ · F measures outward flow

Flux definition

Flux = ■ F · dS

Surface normal

ru × rv

Green’s Theorem

■ (P dx + Q dy) = ■ (∂Q/∂x − ∂P/∂y) dA

Stokes’ Theorem

■ F · dr = ■ (∇ × F) · dS

Divergence Theorem

■ F · dS = ■ (∇ · F) dV

Surface area (z=f)

■ sqrt(1 + fx^2 + fy^2) dA

Surface area (parametric)

■ |ru × rv| dudv

When to use Fundamental Theorem

Any path → endpoints only

When to use Green’s Theorem

2D closed curve

When to use Stokes’ Theorem

3D curve

When to use Divergence Theorem

Closed surface

Key idea

Convert hard integrals into easier ones

Curl vs Divergence

curl = rotation, divergence = expansion