Showing that a function is the gradient of another function

Question

How do I show that this function;

$ f = \frac{\vec{r}-\vec{X}t}{|\vec{r}-\vec{X}t|^3}$

$\vec{X} = (x_1,x_2,x_3)$ and $\vec{r} = (x,y,z)$

is the gradient of another function?

like so: $ f = \nabla F $

Answer 1

Your $f$ is: $$ f(\vec r)=\dfrac{x-x_1t}{\left(\sqrt{(x-x_1t)^2+(y-y_1t)^2+(z-z_1t)^2}\right)^3}\vec i+$$ $$+\dfrac{y-y_1t}{\left(\sqrt{(x-x_1t)^2+(y-y_1t)^2+(z-z_1t)^2}\right)^3}\vec j+$$ $$+\dfrac{z-z_1t}{\left(\sqrt{(x-x_1t)^2+(y-y_1t)^2+(z-z_1t)^2}\right)^3}\vec k$$

Use: $$ F(\vec r)=\dfrac{-1}{\sqrt{(x-x_1t)^2+(y-y_1t)^2+(z-z_1t)^2}}=\dfrac{-1}{|\vec r-\vec Xt|} $$ and find $\dfrac {\partial F}{\partial x}$,$\dfrac {\partial F}{\partial y}$ and $\dfrac {\partial F}{\partial z}$