Index calculation

Question

Let $\mathbf{r} = (x,y,z)$ in cartesian coordinates. Calculate $div(r^4 \mathbf{r})$. My solution: $$div(r^4 \mathbf{r}) = \frac{\partial}{\partial_i}(r^4 \mathbf{r}) = x_i\frac{\partial r^4}{\partial_i}+r^4\frac{\partial x_i}{\partial_i} = x_i\frac{\partial r^4}{\partial_i}+3r^4 $$
So as a final step I wish to simplify the term $$x_i\frac{\partial r^4}{\partial_i}$$ How can I do this?

Answer 1

Hint: What is the difference of $r$ and bold $r$? Magnitude and vector?Btw you should not write $\partial_i$ but rather $\partial_{x_i}$.

Now if $r=\sqrt{x^2+y^2+z^2}=\sqrt{x_{j}x_j}$ (Note, that I use Einstein summation convention, which implies summation over indices that appear twice) , what is $\frac{\partial r^4}{\partial x_i}$ in this case? It is just a simple partial derivative.

$$\frac{\partial (x_{j}x_j)^2}{\partial x_i}=2(x_j x_j)\left(\dfrac{\partial{x_j x_j}}{\partial x_i} \right)=2(x_j x_j)\left(\delta_{ji}x_j+x_j\delta_{ji}\right)=2(x_jx_j)(x_i+x_i).$$

In the last step I used $\delta_{ji}x_j=x_i$