Math basics › Multivariable calculus

Vector fields and line integrals

A line integral

∫ F · dr adds up how much a field pushes along a path. Write the path as r(t), and it becomes ∫ F(r(t)) · r′(t) dt.

Conservative fields

If F = ∇f for some f, the answer depends only on the endpoints: f(end) − f(start). For F = ⟨P, Q⟩, check whether ∂Q/∂x = ∂P/∂y.

Divergence and curl

Divergence ∇ · F measures how much the field spreads out from a point. Curl ∇ × F measures how much it spins around it.

The big theorems

Green's, Stokes' and the divergence theorem all say one thing: adding up a derivative over a region equals adding up the field around that region's boundary.