Unit 5 of 6

Vector Calculus

  • 10 lessons
  • Parts 65 to 84

Vector fields assign a vector to every point, like the velocity of a fluid or a force field. This unit integrates scalar functions and vector fields along curves, studies conservative fields and potential functions, and measures how a field spreads out and rotates with divergence and curl.

Green's theorem connects a line integral around a closed curve with a double integral over the region inside. After parametric surfaces, surface area and flux, the theorems of Stokes and of divergence extend the same idea to surfaces and solids in space.

Every lesson will be a pair of videos, a theory part that explains the idea and a practice part that works through problems. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Vector Fields and Line Integrals

  1. Vector Fields

    What a vector field is, how velocity and force fields model flowing fluids and pulling forces, and how gradient fields, potentials and field lines depend on where the field is defined.

    Theory part 65Practice part 66

  2. Scalar Line Integrals

    How to integrate a function along a curve with respect to arc length, why the speed factor appears, and why the answer ignores both the parametrization and the direction of travel.

    Theory part 67Practice part 68

  3. Work, Circulation, and Vector Line Integrals

    How to integrate a vector field along an oriented curve, how that integral measures work and circulation, and why reversing the direction flips its sign.

    Theory part 69Practice part 70

  4. Conservative Fields and the Fundamental Theorem for Line Integrals

    When the work done by a field depends only on the endpoints, how to compute it from a potential, how to find that potential, and exactly which conditions on the domain make the cross-partial test reliable.

    Theory part 71Practice part 72

Divergence, Curl, and Green’s Theorem

  1. Divergence and Curl

    How divergence measures net outflow per unit volume and curl measures circulation per unit area at a point, how to compute both with del, and why the identities need smooth fields on open sets.

    Theory part 73Practice part 74

  2. Green’s Theorem

    How the circulation and the outward flux of a plane field around a closed boundary equal double integrals of curl and divergence, which hypotheses this needs, and how holes and areas fit in.

    Theory part 75Practice part 76

Surfaces and Flux

  1. Parametric Surfaces and Surface Area

    How a map from a flat parameter region describes a surface in space, how its partial derivatives give tangent planes and normals, and how their cross product measures surface area.

    Theory part 77Practice part 78

  2. Scalar Surface Integrals and Flux

    How to add up a quantity spread over a curved surface using the true area factor, and how to measure flux through a surface once a side has been chosen by a unit normal.

    Theory part 79Practice part 80

The Theorems of Stokes and Divergence

  1. Stokes’ Theorem

    How the circulation of a field around the edge of a surface equals the flux of its curl through the surface, with the orientation, smoothness and domain conditions that make this true.

    Theory part 81Practice part 82

  2. The Divergence Theorem

    How the net outward flux through a closed surface equals the total divergence inside it, and how to handle cavities, point singularities and Gauss's law with the right signs.

    Theory part 83Practice part 84