I have only the first partial derivatives of a multivariable function, and I am trying to recover the original function at a particular maximum. I have very little knowledge of PDEs, so sorry if the question is pretty basic.
In the one variable case, for some $f:\mathbb{R}\to \mathbb{R}$ that reaches a unique maximum at $f(x\text{*})$, we can write $$f(x\text{*}) = f(0) + \int_0^{x\text{*}} \frac{\partial f}{\partial x} dx$$
In the two variable case, for some $f: \mathbb{R}^2 \to \mathbb{R}$ that reaches a unique maximum at $f(x\text{*},y\text{*})$, is there a way to write $f(x\text{*},y\text{*})$ in the form as above? What about in $\mathbb{R}^n$?
For the sake of argument, let $f$ be increasing in $(x,y)$ and component-wise as well until $(x\text{*},y\text{*})$.
Thanks so much!
This is just the fundamental theorem of calculus, and has nothing to do with the fact that $f$ has a unique maximum.
For higher dimensions, see this. Though I've only glossed over it, this article also seems to have a well-written exposition to Green's theorem.