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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?

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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.

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