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I have taught multivariable calculus exactly once, to engineering students at Concordia University in Montreal. I found the course to be replete with expository challenges like the one you mention: namely, to explain what is going on with the various concepts of vector analysis in something like geometric terms, but of course without introducing anything like differential forms. [Note that conversely Conversely, it is possible to know Stokes' Theorem in the form $\int_{\partial M} \omega = \int_M d \omega$ and still not have any real insight into flux, divergence and other such geometric and physical notions. I myself spent about 10 years in this position.]

I thought hard and often found explanations that were much more satisfactory than the textbook, which was amazingly laconic. Or rather, I found explanations which were much more satisfactory to me. The students had a lot of trouble conceptualizing the material, to the extent that my lectures almost certainly would have been more successful if I hadn't tried to give geometric explanations and intuition but simply concentrated on the problems. Thus Gerald Edgar's comment rings true to me. But let me proceed on the happier premise that you want to give more motivation to the bright student who approaches you outside of class.

One thing which was useful for me was to read the "physical explanations" that the book sometimes gave and try to make some kind of mathematical sense of them. For context, I should say that I have never taken any physics classes at the university level and that I have rarely if ever met a mathematician who has less physics background than I. Moreover, when I took introductory multivariable calculus myself (around the age of 17), I found the physical explanations to be so vague and so far away from the mathematics as to be laughable. For instance, the geometric intuition for a curl involved some story about a paddlewheel.

So when I taught the class, I tried to make some mathematical sense out of the names "incompressible" (zero divergence) and "irrotational" (zero curl), and to my surprise and delight I found that this was actually rather straightforward once I stopped to think about it.

Let me also tell you about my one "innovation" in the course (I am sure it will be commonplace to many of the mathematicians here). It seems strange that there are two versions of Stokes' theorem in three-dimensional space (one of them is called Stokes' theorem and one of them is called Gauss' Theorem or -- better! -- the Divergence Theorem) whereas in the plane there is only Green's Theorem. Stokes' Theorem is about curl, whereas Gauss' theorem is about Divergencedivergence. What about Green's Theorem?

The answer is that Green's theorem has a version for curl divergence -- i.e., a flux version involving normal line integrals -- and a version for divergence curl -- a divergence circulation version involving tangent line integrals -- but these two versions are formally equivalent. Indeed, one gets from one to the other by applying the "turning" operators L and R: L applied to a planar vector field rotates each vector 90 degrees counterclockwise, and L R is the inverse operator. Then (with the convention that the curl of a planar vector field should always be pointing in the vertical direction, so we can make a scalar function out of it)

curl(L(F)) = div(F)

and by making this formal substitution one gets from one version of Green's Theorem to the other.

See

http://math.uga.edu/~pete/handoutfive.pdf

http://math.uga.edu/~pete/handouteight.pdf

http://math.uga.edu/~pete/reviewnotes.pdf

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I have taught multivariable calculus exactly once, to engineering students at Concordia University in Montreal. I found the course to be replete with expository challenges like the one you mention: namely, to explain what is going on with the various concepts of vector analysis in something like geometric terms, but of course without introducing anything like differential forms. [Note that conversely it is possible to know Stokes' Theorem in the form $\int_{\partial M} \omega = \int_M d \omega$ and still not have any real insight into flux, divergence and other such geometric and physical notions. I myself spent about 10 years in this position.]

I thought hard and often found explanations that were much more satisfactory than the textbook, which was amazingly laconic. Or rather, I found explanations which were much more satisfactory to me. The students had a lot of trouble conceptualizing the material, to the extent that my lectures almost certainly would have been more successful if I hadn't tried to give geometric explanations and intuition but simply concentrated on the problems. Thus Gerald Edgar's comment rings true to me. But let me proceed on the happier premise that you want to give more motivation to the bright student who approaches you outside of class.

One thing which was useful for me was to read the "physical explanations" that the book sometimes gave and try to make some kind of mathematical sense of them. For context, I should say that I have never taken any physics classes at the university level and that I have rarely if ever met a mathematician who has less physics background than I. Moreover, when I took introductory multivariable calculus myself (around the age of 17), I found the physical explanations to be so vague and so far away from the mathematics as to be laughable. For instance, the geometric intuition for a curl involved some story about a paddlewheel.

So when I taught the class, I tried to make some mathematical sense out of the names "incompressible" (zero divergence) and "irrotational" (zero curl), and to my surprise and delight I found that this was actually rather straightforward once I stopped to think about it.

Let me also tell you about my one "innovation" in the course (I am sure it will be commonplace to many of the mathematicians here). It seems strange that there are two versions of Stokes' theorem in three-dimensional space (one of them is called Stokes' theorem and one of them is called Gauss' Theorem or -- better! -- the Divergence Theorem) whereas in the plane there is only Green's Theorem. Stokes' Theorem is about curl, whereas Gauss' theorem is about Divergence. What about Green's Theorem?

The answer is that Green's theorem has a version for curl -- i.e., a flux version involving normal line integrals -- and a version for divergence -- a divergence version involving tangent line integrals -- but these two versions are formally equivalent. Indeed, one gets from one to the other by applying the "turning" operators L and R: L applied to a planar vector field rotates each vector 90 degrees counterclockwise, and L is the inverse operator. Then (with the convention that the curl of a planar vector field should always be pointing in the vertical direction, so we can make a scalar function out of it)

curl(L(F)) = div(F)

and by making this formal substitution one gets from one version of Green's Theorem to the other.

See

http://math.uga.edu/~pete/handoutfive.pdf

http://math.uga.edu/~pete/handouteight.pdf

http://math.uga.edu/~pete/reviewnotes.pdf