Vector Expressions Homework: Evaluate 1-8

[Pual Black]

Homework Statement

let $\vec{r}$ be a vector from origin to the point (x,y,z) and let ${\vec{r}\,}'$ be a vector from the origin to the point (x',y',z'). Evaluate the following expressions

$1. \triangledown r$

$2.\triangledown\frac{1}{r}$

$3.\triangledown\cdot r$

$4.\triangledown\times r$

$5.\triangledown^{2} r$

$6.\triangledown^{2} \frac{1}{r}$

$7.\triangledown\frac{1}{\mid\vec{r}-{\vec{r}\,}'\mid}$

$8.\triangledown\frac{1}{{\mid\vec{r}-{\vec{r}\,}'\mid}^2}$

The attempt at a solution

i have added my solution for the first few equations. My problem is page 3 and page 4. I can't get the right answer.

 

[BvU]

Well, can you pick the one that's wrong ? 2 out of 3 say 3, so maybe the odd one out is the 'ccs'. What could be the problem here ?

 

[Pual Black]

Yes my problem is $3.\triangledown\cdot r$ in C.c.s. I don't know how to get 3 as result like R.c.s which is the correct answer.
and my solution for the S.c.s I am not sure if its correct. Its seems wrong.
I added the others so you can see how i solve the equations.

 

[BvU]

Well, the expressions in your thread under

Homework Equations

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don't help you much further at the moment . I seem to remember quite different expressions for $\nabla\cdot \vec A$. Could you check, e.g. here ?

 

[Pual Black]

Thank you. You are great. I think i get it.
I have uploaded an image with the solution. Sorry that i don't use Latex but I am on mobile phone right now.
Is this correct?

 

[BvU]

Well done. A few more exercises are waiting ...

 

[Pual Black]

thank you again. Than my solution for spherical is also wrong. ( on page 4)

 

[BvU]

Indeed, the first two lines were some kind of inspiration ? After that, it looks pretty convincing !