Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

For a function $f\in C^{\infty}(\mathbb{R^n})$ with support contained in some ball of radius $R$, how to express $f$ as an integral involving $\partial f/\partial x_1$? Some kind of fundamental theorem of calculus in higher dimensions?

share|cite|improve this question
up vote 0 down vote accepted

You don't need calculus in higher dimensions, since you only want to use the derivative with respect to a single variable; just apply the fundamental theorem of calculus directly:

$$ f(x_1,\dotsc,x_n)=f(x_1,\dotsc,x_n)-f(-\infty,\dotsc,x_n)=\int_{-\infty}^{x_1}\frac{\partial f}{\partial x_1}(x_1',\dotsc,x_n)\mathrm dx_1'\;, $$

where you can replace $-\infty$ by some value outside the support of $f$ to make things more rigorous.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.