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 http://mathoverflow.net/questions/57025/down-to-earth-uses-of-de-rham-cohomology-to-convince-a-wide-audience-of-its-usefu
For a non-standard application of the failure of the Stokes theorem, there's the odd case of the Purcell Swimmer: http://iopscience.iop.org/1367-2630/10/6/063016/fulltext/ Rendering this accessible to a multivariable calculus class may take some work, depending on your students.