Calculus 4: Vector Calculus

Picture vector fields, find gradient fields, and measure curl and divergence.

1. Vector Fields
2. Gradient Fields and Potentials
3. Divergence and Curl
4. What Curl and Divergence Measure

Integrate along curves, compute work, and recognize conservative fields.

1. Parametrizing Curves for Integration
2. Line Integrals of Functions
3. Line Integrals of Vector Fields
4. Line Integrals in the Form ∫P dx + Q dy
5. Circulation and Flux across Plane Curves
6. The Fundamental Theorem for Line Integrals
7. Testing for Conservative Fields
8. Finding a Potential Function
9. Path Independence and Closed Loops

Turn line integrals around closed curves into double integrals.

1. Green's Theorem: Circulation Form
2. Green's Theorem: Flux Form
3. Area with Green's Theorem
4. Green's Theorem for Regions with Holes

Parametrize surfaces and integrate over them, including flux.

1. Parametric Surfaces
2. Surface Area of a Parametric Surface
3. Surface Integrals of Functions
4. Oriented Surfaces
5. Flux Integrals

Relate integrals over a region to integrals over its boundary.

1. Stokes' Theorem
2. Using Stokes' Theorem to Switch Surfaces
3. The Divergence Theorem
4. Using the Divergence Theorem
5. One Idea, Four Theorems

Use the theorems in physics and engineering.

1. Fluid Flow, Sources and Circulation
2. Gauss's Law and Electric Flux
3. Maxwell's Equations in Two Forms
4. Mass and Centers of Mass of Thin Shells