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Given $\frac{\partial{P}}{\partial{x}}$ and $\frac{\partial{P}}{\partial{y}}$ how do we integrate to find $P(x, y)$? I'm having difficulty finding out the rules for this because I'm not familiar with the vocabulary for multivariable calculus.

In actuality the problem I'm pondering involves knowing $\frac{\partial^2{P}}{\partial{x}^2}$ and $\frac{\partial{P}}{\partial{y}}$.

I really don't know where to start. I'm familiar with how to find the partial derivatives, but not how to integrate them. Thanks!

Edit: I should add that the problem I have is where $\frac{\partial^2{P}}{\partial{x}^2}$ and $\frac{\partial{P}}{\partial{y}}$ are both constants. Because of this, I was able to reason the solution, $P(x, y) = \frac{1}{2}\frac{\partial^2{P}}{\partial{x}^2}x^2 + C_1x + C_2 + \frac{\partial{P}}{\partial{y}}y + C_3$, but I can't quite figure out how to generalize the solution. Thanks!

Thor Correia
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Do a line integral. It is the content of fundamental theorem of calculus for line integrals that line integral of the gradient of $P$ recovers the function $P$.

Explicitly, you start at a point $(x_0,y_0)$ and choose a value for $P(x_1,y_1)$ (you have to choose, because two functions with the same gradient may differ by a constant). Then for any other point $(x_1,y_1)$, pick an arbitrary smooth path $\gamma:[0,1]\to\Bbb R^2$ with $\gamma(0)=(x_0,y_0)$ and $\gamma(1)=(x_1,y_1)$. Write $\gamma(t)=(\gamma_1(t),\gamma_2(t))$ Then $P(x_1,y_1)$ is equal to $\int_0^1\gamma_1'(t)\frac{\partial P}{\partial x}(\gamma(t))+\gamma_2'(t)\frac{\partial P}{\partial y}(\gamma(t))dt$.

In your problem, $\frac{\partial P}{\partial x}$ is not explicitly given, but you know the following: the gradient of the function $\frac{\partial P}{\partial x}$ is known, i.e. $\frac{\partial^2 P}{\partial x^2}$ and $\frac{\partial^2 P}{\partial y\partial x}$ are known. $\frac{\partial^2 P}{\partial y\partial x}$ is equal to $\frac{\partial^2 P}{\partial x\partial y}$, which is zero (hopefully $P$ is $C^2$, otherwise there is nothing we can do to recover $P$). So you can first do a line integral to recover $\frac{\partial P}{\partial x}$ first and then do another line integral to recover $P$.

edm
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