Mathematics Educators Asked by mirams on November 19, 2020

I find it useful to give physical examples of theorems, especially in vector calculus – for example $nabla f$ being the direction of maximum ascent on a surface $f$.

What is a good example for Stokes’ Theorem?

I always think of $int_D nabla times v = int_{partial D} v$ in terms of water flow. You have a bunch of water flowing around: It's velocity at a given position $(x,y)$ is given by the vector field $v(x,y)$. Stick a tiny paddle wheel into the water at position $(x,y)$ and let it start spinning: Its speed is $nabla times v$. (Probably times the length of the paddle wheel arms or something, since this answer doesn't have the right units.)

Now take a great big inner tube and plop it down on the surface of the water. Let $D$ be the area enclosed by the inner tube. It will rotate at $int_{partial D} v$. (Again, I am probably missing some physical factor that makes the units work out.) So the result is that the rotation of the inner tube is the total of the rotations from all the paddle wheels within it.

I remember a good discusson of this near the beginning of Volume 2 of the *Feynmann Lectures*, and in Courant *Differential and Integral Calculus*.

Alternate answer (source):

Correct answer by David E Speyer on November 19, 2020

The **divergence theorem or Gauß’s theorem** can be nicely linked to reality:

In terms of **hydrodynamic flows** you could start with the following statement:

The total signed water flow through the border of an area or surface of a volume (without sinks and sources) is zero – what goes in, must come out.

Then you can add sinks and sources, i.e., points with a divergence that is not zero to the scope.

A little bit more abstract are **force fields**, such as the gravitational or electrical field:

For those, sinks and sources correspond to masses or charges inside your surface and the vector field itself corresponds to the local field strength, which in turn is proportional to the force experienced by an object of a given mass or charge, respectively.

In particular, if your mass or charge distribution is radially symmetric, the force experienced by an object only depends on the total amount of masses or charges which are closer to the center of symmetry but not on their distribution. So just going by the gravitational field we cannot decide whether the earth corresponds to a solid sphere with a homogenuous mass distribution or a hollow one with all the mass agglomerating close to the surface. Also, if you are inside a hollow ball, you do not experience any gravitational force from it, even if you are close to the surface.

Answered by Wrzlprmft on November 19, 2020

I don't know if this is what you are looking for, but for Green's theorem, when you discuss how you can change the integral $int 1 dA$ into the line integral $int x dy - y dx$, you can discuss several real life applications.

You can approximate the area of a lake by walking around it and recording your GPS coordinates $(x_i,y_i)$, then adding up $Sigma (x_i Delta y_i-y_i Delta x_i)$.

Similarly, you can record the mouse cursor position of a computer and use the same approximation as in 1) to write a program that finds the area (in pixels) of any loop that you draw on a computer (my roommate did this once within 10 minutes after I told him the idea).

Then you can *ask* your students if you can use this same technique to find the area of general 3-d surfaces using Stoke's theorem; it could be an interesting discussion.

Answered by Brian Rushton on November 19, 2020

6 Asked on September 6, 2021

6 Asked on September 5, 2021 by occhima

2 Asked on September 5, 2021 by rusty-core

logic mathematical pedagogy middle school proofs secondary education

0 Asked on September 5, 2021 by w-g

4 Asked on September 5, 2021 by d-r

1 Asked on September 5, 2021

2 Asked on September 5, 2021

2 Asked on September 5, 2021 by lex_i

2 Asked on September 5, 2021 by dirac-academy-of-self-study

middle school resource request secondary education self learning textbooks

1 Asked on September 5, 2021 by ritesh-singh

online online instruction problem solving secondary education undergraduate education

2 Asked on September 5, 2021

graduate education secondary education undergraduate education

8 Asked on September 5, 2021 by user14243

4 Asked on September 5, 2021

7 Asked on September 5, 2021 by user13544

3 Asked on January 19, 2021

1 Asked on January 14, 2021 by cab

4 Asked on January 13, 2021 by john-clever

Get help from others!

Recent Questions

Recent Answers

- Peter Machado on Why fry rice before boiling?
- Joshua Engel on Why fry rice before boiling?
- Jon Church on Why fry rice before boiling?
- haakon.io on Why fry rice before boiling?
- Lex on Does Google Analytics track 404 page responses as valid page views?

© 2022 AnswerBun.com. All rights reserved. Sites we Love: PCI Database, MenuIva, UKBizDB, Menu Kuliner, Sharing RPP, SolveDir