I'm stuck determining the integrals of the partial derivatives of a scalar function $F:\mathbb{R}^3\times\mathbb{R}^3\rightarrow\mathbb{R}$ with respect to the components of a vector. In the case of the problem I am trying to solve, $F=F(\vec x, \vec y)$ and the following relations hold

$$ y_i=\frac{\partial F}{\partial x_i} \tag{1}$$ $$ x_i=\frac{\partial F}{\partial y_i}\tag{2} $$

My attempt at a solution

Performing the "partial integration" over $(1)$ and $(2)$,

$$F(\vec x, \vec y)=\int\frac{\partial F}{\partial x_i}dx_i=\int y_idx_i+g_i(x_j, \vec y)=x_iy_i +g_i(x_j, \vec y) \tag{3}$$ $$ F(\vec x, \vec y)=\int\frac{\partial F}{\partial y_i}dy_i=\int x_idy_i+h_i(\vec x,y_j)=x_iy_i +h_i(\vec x,y_j) \tag{4}$$

Where $j\neq i$ in the functions $g_i$ and $h_i$. I see that adding for $i=1,2,3$ the RHS $(3)$ and $(4)$ we can write

$$3F(\vec x, \vec y)=\vec x · \vec y + \sum_{i=1\\j\neq i}^3 g_i(x_j, \vec y) \tag{5}$$

$$3F(\vec x, \vec y)=\vec x · \vec y + \sum_{i=1\\j\neq i}^3 h_i(\vec x,y_j) \tag{6}$$

Obtaining this relation for the functions $g_i$ and $h_i$

$$\sum_{i=1\\j\neq i}^3 g_i(x_j, \vec y)=\sum_{i=1\\j\neq i}^3 h_i(\vec x,y_j) \tag{7}$$

Would this approach be correct? Could we tell something else about the integrating "constants" $g_i$ and $h_i$ in addition to $(7)$?



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