Vector Identities: Calculate & Surface Integrals

[neworder1]

Homework Statement

1.

Calculate:

[tex] \nabla \times (\frac{\vec{p} \times \vec{r}}{r^{3}})[/tex]

in cartesian and spherical coordinates, where [tex] \vec{p}[/tex] is a constant vector.

2.

Calculate surface integrals:

[tex] \int \vec{r} (\vec{a} \cdot \vec{n}) dS[/tex]
[tex] \int \vec{n} (\vec{a} \cdot \vec{r}) dS[/tex]

where [tex]\vec{a}[/tex] is a constant vector and [tex]\vec{n}[/tex] is a unit vector normal to the surface.

Homework Equations

The Attempt at a Solution

I tried do the first by using some basic vector identities but I didn't get anywhere (the result wasn't by any means neat and short ;)). I was told that Dirac delta is supposed to show up somewhere, but I don't see it.

The second one is probably done using Stokes' Theorem but I don't see any simple fashion in which it can be applied.

 

[Defennder]

? You haven't posted any question at all.

 

[Defennder]

I'm not sure if I understand your questions. Your first one asks to "calculate" that mathematical expression presumably using vector identities involving curl but you didn't specify in what mathematical form do you want the answer to be expressed in? In other words, there's so many equivalent ways you can express that statement, but you didn't specify what vectors, mathematical notation the final answer should contain so we can eliminate the other possibilities. Is this a "Show that the following may be equivalently written as" type of question?

For your second question, you didn't say what vector r is, and you didn't tell us what surface you're integrating over, as such we can't tell if it's bounded by a closed loop or whether it's a closed surface (then we may apply Div theorem) ie. as such we can't tell (or at least I can't tell) how to help you.

Perhaps it's best if you post the exact question the textbook is asking here to clear up the confusion.

 

[neworder1]

1. I mean something like this: we can calculate that
[tex]\nabla \cdot (\frac{\vec{r}}{r^3}) = 4 \pi \delta^{3}(\vec{r})[/tex] (Dirac delta),
and here we have to do something similar, i.e. express the rotation in a more "explicit" way (using Dirac delta etc.). Using vector identities I obtained [tex] \nabla \times (\frac{\vec{p} \times \vec{r}}{r^3}) = 4 \pi \vec{p} \delta^{3}(\vec{r}) + (\vec{p} \cdot \nabla)\frac{\vec{r}}{r^3}[/tex], and I wonder whether the second term (involving [tex]\vec{p} \cdot \nabla[/tex]) can be simplified (expressed using Dirac delta etc.)?


2. We can assume that the surfaces are closed. [tex]\vec{r}[/tex] is the vector connecting the current point and the origin.