How Can You Determine the Potential Function of a Conservative Force?

[slonopotam]

to know that a F is a conservative i need to prove that
rot $$\vec{F}=0$$
or that grad $$U=\vec{F}$$
$$\vec{F}=\frac{\vec{r}}{r}$$

how to know U (potential of F)
??

 

[gabbagabbahey]

to know that a F is a conservative i need to prove that
rot $$\vec{F}=0$$
or that grad $$U=\vec{F}$$
$$\vec{F}=\frac{\vec{r}}{r}$$

how to know U (potential of F)
??

Well, if $\textbf{F}=\mathbf{\nabla}U=\partial_x U \hat{x}+\partial_y U \hat{y}+\partial_z U \hat{z}=F_x\hat{x}+F_y\hat{y}+F_z\hat{z}$ (are you familiar with this notation?), then $U=\int F_x dx$, $U=\int F_y dy$ and $U=\int F_z dz$ must all be true. An important note is that in multi-variable calculus, the 'constants' of integration are only constant with respect to the integration variable, so, for example $\int3x^2 dx=x^3+f(y,z)$