What are some relevant examples of the application of vector calculus in economics, apart from using gradients and Hessian matrices for optimization? I'm mostly referring to the topics and concepts such as curl, divergence, gradient theorem, divergence theorem and Stokes' theorem.
An example: Vector calculus, in particular the gradient theorem, is used in the characterization of Bayesian incentive compatible social choice functions in mechanism design with multi-dimensional types. See the respectively first result in either of:
Jehiel, Philippe, Benny Moldovanu, and Ennio Stacchetti. "Multidimensional mechanism design for auctions with externalities." Journal of economic theory 85.2 (1999): 258-293.
Jehiel, Philippe, and Benny Moldovanu. "Efficient design with interdependent valuations." Econometrica 69.5 (2001): 1237-1259.
EDIT: In the same field, e.g. Armstrong uses the divergence theorem:
Armstrong, Mark. "Multiproduct nonlinear pricing." Econometrica: Journal of the Econometric Society (1996): 51-75.