How do we find the function of radial distance $f(r)$ from an equation of the form $\int_a^b \nabla f(r)\cdot d\vec{r}=c$ for some constant $c$? $f(r)$ is radially symmetric.
Thank you.
How do we find the function of radial distance $f(r)$ from an equation of the form $\int_a^b \nabla f(r)\cdot d\vec{r}=c$ for some constant $c$? $f(r)$ is radially symmetric.
Thank you.
Assuming that your function's sole independent variable is r we have:
$\int_a^b \nabla f(r)\cdot d\vec{r}=\int_a^b \frac{df(r)}{dr} \hat{r} \cdot d\vec{r}=\int_a^bdf(r)=f(b)- f(a)=c $.
So your equation becomes: $f(b)-f(a)=c \; \ (1)$.
So your equation has infinite solutions because if you find a function $f$ that satisfies the condition (1) every function of the form $h=f+ct$ will satisfy it too.