I am familiar with the integration of a function $f:D\subset\mathbb{R}^n\to \mathbb{R}$ with respect of some variable $x_i \,\,1\leq i \leq n $.
For example: $$\int x\sin (y)\, \mathrm{d}x=\frac{1}{2}x^2 \sin(y) + c(x,y)$$
with $$\frac{\partial }{\partial x}c \equiv 0$$
And this concept is the motivation for my question, so
could we define a map that as argument would have a $(n \times m)$ matrix with entities real valued functions $f_{ij}:D\subset\mathbb{R}^n\to \mathbb{R}, \, 1 \leq i \leq n, \, 1 \leq j \leq m $ and its value it would be a function $F$ such that $\mathrm{D}F=[ f_{ij}]$ ?