Potential for integration

Question

I have the following function inside an integral

$$\frac{2xdx + 2ydy + 2zdz}{x^2 + y^2 + z^2}$$

I need to find the potential for solving the integral, but I don't know how to transform it into a vector.

Answer 1

Hint: If $$F = \left\langle \frac{2x}{x^2 + y^2 + z^2},\frac{2y}{x^2 + y^2 + z^2},\frac{2z}{x^2 + y^2 + z^2}\right\rangle$$

then $$ F \dot \, d\bf {x} = \frac{1}{x^2 + y^2 + z^2}\left(2x dx + 2y dy + 2z dz \right)$$

Answer 2

You provided $$ dF = \frac{2 r}{\lVert r \rVert^2} \cdot dr = f \cdot dr $$ for the position vector $r = (x,y,z)$ and $$ f = \frac{2r}{\lVert r \rVert^2} = \frac{2}{\lVert r \rVert} e_r $$

If $f$ is the gradient of some scalar field (potential function): $$ f = \mbox{grad } \Phi $$ then we had $dF = \mbox{grad }\Phi \cdot dr = d\Phi$.

We guess sharp and see that $$ \Phi = 2 \ln \lVert r \rVert + C \Rightarrow \\ \partial_i \Phi = \frac{2}{\lVert r \rVert}\frac{1}{2}\frac{1}{\lVert r \rVert}2x_i = \frac{2x_i}{\lVert r \rVert^2} $$

Answer 3

Consider the potential function

$$f(x,y,z)=\log{(x^2+y^2+z^2)}$$