01006

Week 9, Long Day: Stokes’ Theorem and Potentials

Today we will finalize our syllabus of the course with a theorem that in many ways can be compared to – and is just as fantastic as – Gauss’ Theorem. It is a theorem about the tangential line integral of a vector field along a closed curve, also called the circulation of the vector field along the closed curve. Stokes’ *Curl Theorem is a relation between this circulation and the flux of the curl of the vector field through an arbitrary surface that has the closed curve as its boundary curve.

An ordinary continuous, real function of one variable has an antiderivative, i.e a function whose derivative is equal to the function itself. When we introduced the tangential line integral we saw that some vector fields can be expressed as the gradient of a function – we called those scalar potentials. Today we will introduce the equivalent concept of vector potentials of given vector fields about which it applies that if we take the curl of the vector potential, we get the vector field itself.

Note that today we will have the last lecture and exercise session in the course. On Short Day this week you have Theme exercise 7, and after Easter the project period will be running for the rest of the semester.

Today’s Key Concepts

Circulation $\mathrm{Circ}(\mV,\mathcal K)$. Flux of the curl of a vector field through an open surface. Stokes’ Curl Theorem. The right-hand rule. Torsion of a surface. Scalar potential. Vector potential.

Preparation and Syllabus

Today’s subjects are from eNote 29 Stokes’ Theorem. And with that, all eNotes have been covered.

Maple Ressources

Today’s Maple demo (the last on of the course) is 33_Stokes.

Activity Program

10.00 – 12.00: $\,$ Lecture (aud. 42, b. 303A) (link to streaming)
12.00 – 17.00: $\,$ Group exercises in the study areas (b. 302, bottom floor)
13.00 – 16.00: $\,$ Your teachers are present

Group Exercises