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Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
2
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How should one present curl and divergence in an undergraduate multivariable calculus class?
Well, $\nabla\times\nabla f=0$ if $f \in C^2$ can be proved by contradiction. Suppose the curl wasn't 0: then, if you chose a contour integral around the area with non-zero curl, it will be different …