Vtyjkyuk

If $v = v(r)$ and $r^2 = sum_i=1^n x_i^2$, then prove $$sum v_x_i x_i = v_r ((n-1)/r) + v_rr$$

I start by:

$$fracpartial rpartial x_i = fracx_ir$$

$$fracpartial vpartial x_i = fracpartial vpartial r fracpartial rpartial x_i = fracpartial vpartial r fracx_ir$$

Now,
$$fracpartial^2 vpartial x^2_i = fracpartialpartial x_i left(fracpartial vpartial r fracx_irright)$$

From here, I did bifurcation into two methods:

Method 1:

$$fracpartial^2 vpartial x^2_i = fracpartialpartial x_i left(fracpartial vpartial r fracx_irright)\
= fracpartial^2 vpartial x_ipartial rfracx_ir + fracpartial vpartial rfrac1r - x_ifracpartial vpartial rfrac1r^2fracpartial rpartial x_i\
= fracpartial^2 vpartial x_ipartial rfracx_ir + fracpartial vpartial rfrac1r - fracpartial vpartial rfracx_i^2r^3$$

Apply sigma

$$sumfracpartial^2 vpartial x^2_i =fracn-1r fracpartial vpartial r + colorredsum fracpartial^2 vpartial x_ipartial rfracx_ir$$

Method 1(A):

Wrote the red term as $$fracpartialpartial rleft(fracpartial vpartial rright) fracpartial rpartial x_i fracx_ir = fracpartial^2 vpartial r^2 fracx_i^2r^2$$
which on summing gives required result!

Method 1(B):

Wrote red part as:

$$fracpartial partial rleft(fracpartial vpartial x_iright)fracpartial r partial x_i = fracpartial ^2 vpartial x_i^2$$

which is obviously not correct, but doesnt appear where error occurs.

Method 2:

Using chain rule:

$$fracpartial^2 vpartial x^2_i= fracpartialpartial x_i left(fracpartial vpartial r fracx_irright) = fracpartialpartial r left(fracpartial vpartial r fracx_irright) fracpartial rpartial x_i\
= fracx_ir left( fracx_ir fracpartial^2 vpartial r^2 + colorred frac1r fracpartial vpartial rfracpartial x_ipartial r-fracpartial vpartial rfracx_i^2r^2right)$$

On applying sigma and noting $sum x_i fracpartial x_ipartial r = r$ we get wrong answer:

$$sumfracpartial^2 vpartial x_i^2 = fracpartial ^2 vpartial r^2 + 0$$

Here what went wrong?

I am really confused here, as to when we use chain rule without encountering error. Here I know $v$ is function of $r$ which is function of all $x_i$. So it is known that $fracpartial vpartial x_i = fracpartial vpartial r fracpartial rpartial x_i$. Where do I go wrong everytime?

Thank you a lot!

Your troubles stem from dubious notation: The same letter $v$ is used for a certain function $rmapsto v(r)$ of one variable $r$, and then also for the nested function $$xmapsto f(x):=vbigl(r(x)bigr)$$ of the vector variable $x$. – We are told to compute the Laplacian $$Delta f(x):=sum_i=1^npartial^2 foverpartial x_i^2 .$$
You started correctly with
$$partial foverpartial x_i=v'(r)partial roverpartial x_i=v'(r)over r,x_i .$$
Now the next step:
$$partial^2 foverpartial x_i^2=dover drv'(r)over r cdot partial rover partial x_icdot x_i+v'(r)over r=r v''(r)-v'(r)over r^2>x_i^2over r+v'(r)over r .$$
Summing over $i$ then gives the result you were asked to prove.