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Is there a built-in way to integrate a vector or differential form field on an embedded submanifold of Euclidean space?

Reasons for asking: I'm doing calculations related to some problems in electromagnetism, and want to be able to compute fluxes, circulations, and similar integrals of vector fields over submanifolds of R^3. Ideally, I'd just define the surface or region of interest, then pass the (existing) field and domain to some built-in function to compute the relevant integral. Does such a thing exist at present?