Assume that $(a_1,a_2,...,a_n) \in \Bbb R^n$ and the
$F: \Bbb R^n \rightarrow \Bbb R$
$(x_1,x_2,...,x_n) \longmapsto F(x_1,x_2,...,x_n)$
is differentiable function at all parameters. Is there any math tool (operator, method...) that gives
$f: \Bbb R \rightarrow \Bbb R$
$p \longmapsto f(p) = F(x_1(p),x_2(p),...,x_n(p))$
such that $p=0$ acts like $a_1,a_2,...,a_n$
$x_i(0) = a_i, i=1,2,...,n \Longrightarrow$
$f(0) = F(x_1(0),x_2(0),...,x_n(0)) = F(a_1,a_2,...,a_n)$
and other $p$ values describes the arguments progress on gradient like parametric function
$(x_1'(p),x_2'(p),...,x_n'(p)) = \nabla F(x_1(p),x_2(p),...,x_n(p)), p \in \Bbb R$?
Is it line integral? Is it gradient flow? How to do it?
What you're asking for is the solution to the system of $n$ coupled first-order ordinary differential equations (ODEs) given by: \begin{align*} \frac{dx_1}{dp} &= F_1(x_1, x_2, \dots, x_n) \\ \frac{dx_2}{dp} &= F_2(x_1, x_2, \dots, x_n) \\ &\vdots \\ \frac{dx_n}{dp} &= F_n(x_1, x_2, \dots, x_n) \end{align*} where $F_i \equiv \partial F/\partial x_i$, subject to the initial conditions $x_i(0) = a_i$.
There is no general method to find closed-form solutions of systems of coupled ODEs. In certain cases, it is possible; e.g., if $F$ is a quadratic function of the $x_i$ variables, then the equations are linear and a general analytic solution exists. However, it is definitely possible to solve these equations numerically. Any mathematical software worth its salt (Mathematica, MATLAB, SciPy, GSL, R, Maple, etc.) contains methods and algorithms to find a numerical approximation to $x_i(p)$ over some finite range of $p$.