Indefinite Multiple Integration

Question

In multivariable calculus, definite integrals with multiple variables seem routine. However, I have not seen any example of an indefinite multiple integral. In fact, it seems as if limits of integration are a necessary part of multiple integrals. Is there actually any way to evaluate indefinite integrals of functions with more than one variable?

Answer 1

Yes indeed: you could define an indefinite integral of a function $f$ on (an open subset of) ${\mathbb R}^n$ as a function $F$ such that $$ \dfrac{\partial}{\partial x_1} \ldots \dfrac{\partial}{\partial x_n} F(x_1,\ldots,x_n) = f(x_1,\ldots,x_n)$$ And you can construct such an $F$ by integrating one variable at a time.

Answer 2

You could actually define something similar, as e.g. done in probability

$$ F(x,y) = \int_{-\infty}^x \int_{-\infty}^y f(u,v) \,du\,dv $$

But they are less useful as they can only help you evaluate definite integrals over rectangular domains.