Suppose we have a function $g: \mathbb{R}^m \to \mathbb{R}^m$ which is defined as follows: $$ [g(x)]_i= \exp \left( \int_0^{x_i} (e_i)^{t} f(t) dt_i \right) \qquad (i=1,\dots,m) $$ where $t_j=x_j$ for $j \neq i$ and $f: \mathbb{R}^m \to \mathbb{R}^m$ is another function. Here, $[ \cdot ]_i$ denotes the $i$-th element of the vector and $\{e_i :\, i=1,\dots,m \}$ is the standard basis of $\mathbb{R}^m$.
Question: Can this be written in vector notation such that the basis vectors are avoided?
For example, can this be written in terms of a line integral?
Note that inside the exponential you're integrating the $i$th component of $f(x_1,x_2,\dots,x_{i-1},t_i,x_{i+1},\dots,x_m)$ with respect to $t_i$. If you had a sum of such integrals, rather than a vector with those components, it would be tempting to say you were doing a single line integral from $0$ to $x$. But, in fact, for this you would need to do a sequence of line segments all getting you from the origin to $x$ (e.g., $(t_1,0,0) \rightsquigarrow (x_1,t_2,0) \rightsquigarrow (x_1,x_2,t_3)$). But, as things stand, everything is totally dependent on the standard coordinate system. I've never seen anything like this before.