Interesting applications of the classical Stokes theorem?

Question

When students learn multivariable calculus they're typically barraged with a collection of examples of the type "given surface X with boundary curve Y, evaluate the line integral of a vector field Y by evaluating the surface integral of the curl of the vector field over the surface X" or vice versa. The trouble is that the vector fields, curves and surfaces are pretty much arbitrary except for being chosen so that one or both of the integrals are computationally tractable.

One more interesting application of the classical Stokes theorem is that it allows one to interpret the curl of a vector field as a measure of swirling about an axis. But aside from that I'm unable to recall any other applications which are especially surprising, deep or interesting.

I would like use Stokes theorem show my multivariable calculus students something that they enjoyable. Any suggestions?

Answer 1

If you don't mind specializing Stokes theorem to Green's theorem, then one of the most practical applications is computation of the area of a region by integrating around its contour. I am old enough to have used a planimeter, a delightful physical embodiment of Green's theorem:
                
One can also derive an (otherwise non-obvious) formula for the area of a planar polygon via Green's theorem: $A = \frac{1}{2} \sum_{i=0}^{n-1} x_i y_{i+1} - x_{i+1} y_i$.

Sorry—no swirling vector fields in these examples!

Answer 2

One could argue that complex function theory (the fact that analytic functions integrate to zero around contours) is an application, and a nice one.

Answer 3

The proof of Brouwer's fixed point theorem using Stokes' theorem is a nice application I think.

Answer 4

In the theory of electromagnetism, the classical Stokes Theorem moves between the differential and integral forms of two of Maxwell's four equations; see the relevant Wikipedia entry for discussion. Note that the integral forms may be directly interpreted using classical physical intuition, while the differential forms give us differential equations that we might solve, so it is important that we can switch between them.

ETA: I think that Wikipedia's discussion is a little vague, although possibly appropriate in that context. So here is more detail, looking at Faraday's Law. In terms of physically observable quantities, the law states that the rate of change of the magnetic flux through a stationary surface is proportional to the electromotive force around the boundary of the surface. The magnetic flux is the surface integral of the magnetic field $ \vec H $, and the EMF is the line integral of the electric field $ \vec E $, so we have $$ \oint _ { \partial S } \vec E \cdot \mathrm d \vec r = - \frac { \mathrm d } { \mathrm d t } \iint _ S \vec H \cdot \mathrm d ^ 2 \vec A $$ using standard units and sign conventions. Applying the classical Stokes Theorem on the left and using that $ S $ is stationary on the right, this becomes $$ \iint _ S ( \nabla \times \vec E ) \cdot \mathrm d ^ 2 \vec A = - \iint _ S \frac { \partial \vec H } { \partial t } \cdot \mathrm d ^ 2 \vec A \text ; $$ since this holds for arbitrarily small surfaces, we conclude that $$ \nabla \times \vec E = - \frac { \partial \vec H } { \partial t } \text , $$ a differential equation. (The argument in reverse is even easier, since you don't have to worry about arbitrarily small surfaces.)

Answer 5

You can tell your students that a clever use of Stokes theorem can give you a Fields medal. Indeed, the proof that the formality map given by M. Kontsevich is a $L_\infty$-morphism, is nothing else than Stokes theorem. A detailed account of this can be found in Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66 (2003), no. 3, 157–216 or with more details in Déformation, quantification, théorie de Lie, 123–164, Panor. Synthèses, 20, Soc. Math. France, Paris, 2005 which is in English, contrary to its title.

Answer 6

A student may also learn about the content from Stokes theorem from instances where it failed to hold as expected. For example, one has to exercise care when trying to use the theorem on domains with holes. Turn this around: the failure of Stokes to hold as expected tells you about the cohomology of the domain. I think it is possible via concrete examples to illustrate this point in a multivariate calculus class without using the more technical phraseology.

A similar discussion occured at Down-To-Earth Uses of de Rham Cohomology to Convince a Wide Audience of its Usefulness

For a non-standard application of the failure of the Stokes theorem, there's the odd case of the Purcell Swimmer:¹ https://iopscience.iop.org/article/10.1088/1367-2630/10/6/063016 Rendering this accessible to a multivariable calculus class may take some work, depending on your students.

¹J E Avron and O Raz: A geometric theory of swimming: Purcell's swimmer and its symmetrized cousin, 2008 New J. Phys. 10 063016 DOI 10.1088/1367-2630/10/6/063016

Answer 7

A nice application in fluid mechanics is Kelvin's circulation theorem. You could also discuss how it fails to hold, if there are obstacles in the fluid flow. In the same spirit stokes theorem is applied in the canonical formalism of classical mechanics to find the poincare-cartan integral invariants.

Answer 8

I find interesting the fact that divergence theorem (that is a corollary of the general Stokes theorem on differentiable varieties) in a vectorial form (one integral for each cartesian coordinate) gives a proof of Archimedes' principle.

Given a body immersed on a (incompressible) fluid, let $S$ be its surface and for each point $p\in S$ let $\vec{n}:=(n_x(p), n_y(p), n_z(p))$ be the normal versor to $S$ in $p$, and (symbolically) put $$ d\vec{S}:=\vec{n}\cdot dS. $$ Then the total force exerted by the fluid on the body is the following (vector) integral $$ \int_S\mu\circ \big(l-z(p)\big)\cdot(-\vec{n})\cdot dS $$ which in componentwise form is: $$ -\mu \begin{pmatrix} \displaystyle\int_S (l-z(p))\cdot n_x(p) \circ dS,\\ \displaystyle\int_S (l-z(p))\circ n_y(p) \circ dS,\\ \displaystyle\int_S(l-z(p))\circ n_z(p) \circ dS\\ \end{pmatrix} =-\mu \begin{pmatrix} \displaystyle\int_S (l-z(p))\cdot \vec{i}\circ d\vec{S}\\ \displaystyle\int_S (l-z(p))\circ \vec{j} \circ d\vec{S}\\ \displaystyle\int_S(l-z(p))\circ \vec{k} \circ d\vec{S} \end{pmatrix}$$ where

• $l$ is the fluid level,
• $\mu$ its density and
• $\vec{i},\ \vec{j},\ \vec{k}$ are the usual cartesian versors.

Then from the divergence theorem (on each of the three components) and from the identities $$ \begin{align} \nabla\circ\big((l-z)\overrightarrow{i}\big)&=\frac{\partial}{\partial x} (l-z)=0 \\ \nabla\circ\big((l-z)\overrightarrow{j}\big)&=\frac{\partial}{\partial y} (l-z)=0\\ \nabla\circ\big((l-z)\overrightarrow{k}\big)&=\frac{\partial}{\partial z} (l-z)=-1 \end{align} $$ it follow that total force on the body is
$$ \mu\cdot \int_VdV\circ \overrightarrow{k} =\mu\cdot V\circ \overrightarrow{k} $$ where $V$ in the volume measure of the body bounded by the surface $S$.

Answer 9

Here's a Math Olympiad question that has an idea similar to Green's theorem in its solution. Unfortunately, I didn't find the source other than my recollection.

The depicted planar graph has 24 triangles formed by its edges and 19 nodes. Each node is arbitrarily assigned a distinct integer; the assignment gives each edge orientation from the smaller integer to the larger one. Prove that at least 7 of the triangles are oriented clockwise.