I need a function $F$ of $(x,y)$ such that, for specific function $G$ of $(x,y)$, I have:
$\partial F/\partial x=\partial G/\partial x$ and $\partial F/\partial y=0$.
How can I construct it?
I thinked about $F(x,y)=G(x,y)-G(0,y)+C,\qquad C$ constant... So,
$\partial F/\partial x=\partial G/\partial x\cdot 1-\partial G/\partial x\cdot 0=\partial G/\partial x$
and
$\partial F/\partial y=\partial G/\partial y\cdot 1-\partial G/\partial y\cdot 1=0$.
Am I correct?
Many thanks.
You cannot do it except in special circumstances.
Because $\frac{\partial F}{\partial x} = \frac{\partial G}{\partial x}$, we can hold $y$ constant and integrate with respect to $x$ to get $F(x,y) = G(x,y) + C_y$ where the constant term can change with $y$, which in fact makes it a function of $y$: $$F(x,y) = G(x,y) + c(y)$$ Since both $F$ and $G$ are differentiable with respect to $y$, $c$ must be differentiable as well. Differentiating:
$$\frac{\partial F}{\partial y} = \frac{\partial G}{\partial y} + c'(y)$$ But $\frac{\partial F}{\partial y} = 0$, so $$\frac{\partial G}{\partial y} = -c'(y)$$ Integrating this equation with respect to $y$ gives $$G(x,y) = -c(y) + d(x)$$
And therefore for such an $F$ to exist, it must be possible to express $G(x,y)$ as the sum of functions in $x$ and $y$:
$$G(x,y) = g_1(x) + g_2(y)$$
Most functions of two variables will not satisfy that condition. If $G$ does satisfy that condition, then $F(x,y) = g_1(x) + C$ for some constant $C$.