multivariable calculus – Can we obtain the one parameter function that represents the arguments progression on gradient progression of a multiple parameters function?

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?