Answer to Question #224786 in Calculus for Unknown346307

Question

Question:

If 𝜑 = ln$|\vec r|$

, then show that ∇𝜑 = $\vec r$ /𝑟2

Answer 1

$\overline{r}$(x,y,z)=(x,y,z);

| $\overline{r}$(x,y,z)|= $\sqrt{x^{2}+y^{2}+z^{2}}$ ;

$\phi$(x,y,z)= ln| $\overline{r}$(x,y,z)|= 1/2$\cdotp$ ln(x²+y²+z²);

$\nabla$$\phi$(x,y,z)=( d$\phi$(x,y,z)/dx, d$\phi$(x,y,z)/dy, d$\phi$(x,y,z)/dz);

d(x,y,z)/dx=1/2$\cdotp$ dln(x²+y²+z²)/dx=1/2$\cdotp$ 2$\cdotp$ x/(x²+y²+z²)=x/$\overline{r}$²;

d(x,y,z)/dy=1/2$\cdotp$ dln(x²+y²+z²)/dy=1/2$\cdotp$ 2$\cdotp$ y/(x²+y²+z²)=y/$\overline{r}$²;

$\nabla$ $\phi$(x,y,z)=(x/r²,y/r²,z/r²)=(x,y,z)/r²= $\overline{r}$/$\overline{r}$²;