I was wondering what types of integrals are studied? Here are my naive view:
- Multiple integral: If I understand correctly, it is just a plain generalization of Riemann integral on $\mathbb{R}$ to on $\mathbb{R^n}$.
- Integral of differential forms: It is something I am not able to truly understand. Is it a special kind of Lebesgue integral? Does its definition rely on measure?
When trying to compare them together, I have some further questions:
Are multiple integrals and integrals of differential forms two different types of integrals? Do they belong to some common type of integral, similarly to that Riemann integral and Lebesgue-Stieltjes integrals both belong to Lebesgue integral? How are they related?
Is it correct that multiple integrals have no orientation involved, but an integral of differential forms does, in the sense of changing the order of dummy variables in $dx_1 dx_2$ will or will not change the integral?
What type of integral is used in vector calculus, for topics such as gradient, divergence, curl, Laplacian, the gradient theorem, Green's theorem, Stokes' theorem, divergence theorem? Are the line integral, surface integral and volume integral defined as belonging to Lebesgue integrals or Riemann integrals, integrals of differential forms, or something else? Do their definitions rely on measure?
Thanks and regards!