Unit 6 of 7

Fields, circulation and flux

  • 3 chapters
  • Chapters 25 to 27

These chapters integrate vector fields. A gradient field needs only a path's endpoints, Green's theorem turns circulation into a double integral, and the divergence theorem does the same for flux.

The work integral of y dx plus x dy is the change in the potential xy, so every path from the origin to the point (1, 1) gives 1 and a closed path gives 0. Around the unit circle, the circulation of the field (minus y, x) equals the integral of its scalar curl over the disk, which Green's theorem explains by adding small cells whose shared edges cancel. Through the unit sphere, the outward flux of the field (x, y, z) equals the integral of its divergence over the ball, found in the same way from small boxes. Reversing the orientation changes the sign each time.

Every chapter will be part of one long video. The video is coming soon, and each chapter page already has the definitions and an example for every key term.

Line integrals, Green's theorem and the divergence theorem

  1. When a path integral only needs its endpoints

    Why the work of the field (y, x) from the origin to (1, 1) is 1 on every path, explained with line integrals, exact differentials, potential functions, conservative fields and the fundamental theorem for line integrals.

    Chapter 25

  2. Add the turning inside, find the circulation outside

    How Green's theorem equates the circulation around a counterclockwise closed curve with the double integral of the scalar curl inside it, and why edges shared by neighboring cells cancel.

    Chapter 26

  3. Count outward flow from the sources inside

    How the divergence theorem equates the outward flux through a closed surface with the triple integral of the divergence inside, using surface integrals, outward unit normals and the cancellation of shared faces.

    Chapter 27