How to evaluate $\nabla \frac 1r$, $\nabla^2 \frac 1r$, $\int_S \nabla \frac 1r . ndS$

Question

Let S be a smooth closed surface in a three-dimensional xyz-space, n, be the unit outward normal vector on S, and r be the distance between the origin and a point (x, y, z). Solve the following problems.

(1) Evaluate $\nabla (\frac 1r)$

(2) Evaluate $\nabla^2 (\frac 1r)$

(3) Evaluate the integral
$\quad \int_S \nabla (\frac 1r).ndS$
when the origin is located outside S.

(4) Evaluate the integral
$\quad \int_S \nabla (\frac 1r).ndS$
when the origin is located intside S.

First of all, I completely have no idea on what topic this problem is about. I have tried learning divergence, differential, projection from a point to a plane, but nothing seems to be applicable to solve this problem.

Edited on 30th Oct 2018
Here is my current solution
$r = \sqrt{(x^2 + y^2 + z^2)}$
$\frac 1r = \frac 1 {\sqrt{x^2 + y^2 + z^2}}$

Answer No. 1
$\nabla (\frac 1r) = {\frac {\partial {\frac 1r}}{\partial x}} + {\frac {\partial {\frac 1r}}{\partial y}} + {\frac {\partial {\frac 1r}}{\partial z}} $
$\nabla (\frac 1r) = - \frac {x \hat i + y \hat i + y \hat k}{{(x^2 + y^2 + z^2)}^\frac 32} $
$\nabla (\frac 1r) = - \frac 1{r^3} {<x,y,z>}$

Answer No. 2
$ \nabla . \nabla \frac 1r = div \nabla \frac 1r = \nabla^2 \frac1r $
$ \nabla^2 \frac1r = 0$

This is as far as I've got Thank you for all the helps and hints! But still I couldn't find solution to next question.

I know that on question 4 we could apply divergence theorem, but how about question 3 when the origin is outside S?

I assume we can't apply the divergence theorem the region is not within the closed surface anymore. Is that right?

Any kind of help would be gladly appreciated!

Answer 1

Here are some hints to help you get started:

• It seems that you think $r$ is a scalar value, but it is not in the context of this problem. $r:\mathbb{R}^3\rightarrow\mathbb{R}$ is a function that maps a point $(x,y,z)\in\mathbb{R}^3$ to a distance $\sqrt{x^2+y^2+z^2}\in\mathbb{R}$. In particular, $r$ is a hypercone in $\mathbb{R}^4$.
• The notation $\nabla f(x)$ refers to the gradient vector of $f$, $\nabla f(x)=\langle f_x,f_y,f_z\rangle$ where $f_x$ is the partial derivative of $f$ with respect to $x$, etc...
• $\int_sf(x,y,z)dS$ is the surface (in this case closed) integral of $f$ over $S$.
• The divergence theorem states that $\iiint\limits_V(\nabla\cdot F)dV=\oint_S(F\cdot \hat{n})dS$

Answer 2

I think people in physics exchange may have a better answer, because those forlumas are not common in undergraduate math.

Bascially what you are having a headache with is coordinate transformation, you may consider your expression in spherical coordicantes. See, http://mathworld.wolfram.com/SphericalCoordinates.html

If you want some examples: https://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates#Del_formula

There are also some useful post about Del and laplas operator in both math and physics exchange. As for the last one, I don't think you are writting it properly, it looks awfully like as if you are trying to find gauss's theorem, in that cas $n$ ought to be a vector point out of the surface.