# 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?