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M05.10 · Vector fields, divergence and curl

A free, illustrated module of 4 lessons and one interactive in M05 Multivariable and Matrix Calculus: Vector fields and their flow lines; Line integrals and conservative fields; Divergence: sources and sinks; Curl: rotation and Green’s theorem.

M05.10 L2 · Intermediate 4 lessons · about 100 min Live · in review

Part of M05 Multivariable and Matrix Calculus in the Mathematics track.

A vector field puts an arrow at every point: a gradient, the velocity of a fluid, the joint update of two players in a game. These four lessons follow the arrows along flow lines, exactly and by Euler steps, add up a field’s push along a path and find that gradients do work that depends only on the ends, then measure what a field does at a single point: how much it spreads out, the divergence, and how much it turns, the curl. They end with Green’s theorem, which ties the turning inside a region to the circulation round its edge, and with the antisymmetric part of a Jacobian that makes two-player training circle.

What you need first. Gradients and the chain rule along a path (M05.2, M05.4), the Jacobian (M05.4.1), Schwarz’s theorem on mixed partial derivatives (M05.5.1), double integrals (M05.6) and Euler’s method (M04.9.4). The code uses NumPy and SciPy (quad and solve_ivp), explaining each call.

Lessons

1. Vector fields and their flow lines M05.10.1 25 min

   A vector field puts an arrow at every point of the plane, and a flow line is the path that always moves along the arrow under its feet. Draw fields by hand, solve their flow lines exactly and by Euler steps, and see gradient descent as one such path.

2. Line integrals and conservative fields M05.10.2 25 min

   The work of a field along a path adds up its push in the direction of travel. Compute it for the rotation round a circle and the bowl’s gradient along three paths, prove that a gradient’s work is the change in its potential, and use a loop to show a field is no gradient at all.

3. Divergence: sources and sinks M05.10.3 25 min

   Divergence measures how much a field flows out of a tiny square, per unit area. Derive it from the four sides of a square, read it as the trace of the Jacobian, check the divergence theorem on the unit square, and see why continuous normalising flows need it.

4. Curl: rotation and Green’s theorem M05.10.4 25 min

   Curl measures how much a field circulates round a tiny square, per unit area: the spin of a paddle wheel. Derive it, prove Green’s theorem for a rectangle, see why gradients have no curl, and find the rotation that makes two-player training circle.

The interactive

M05.10-X1 · Field explorer. Choose the source, the rotation, the bowl’s gradient, the descent-ascent field of x y, a mix of source and rotation, or the cubic field. Click to release particles that follow the arrows by Euler steps, and drag a small square to compare the flux out of it and the circulation round it, per unit area, with the divergence and the curl at its centre. It lives in Divergence: sources and sinks.