Differential forms, abstrakt algebra

[rayman123]

Homework Statement

Here is my problem
http://i51.tinypic.com/34dihx5.jpg

However my teacher had some suggestions of solving this problem in a nice mathematical way. Here is the plan of solving which I would like to follow



1) find the Hodge dual to f
2) compute df
3) is f exact?
4) is f closed?
5) compute $$d^+f$$ (I do not know what he means with this one)
6) integrate f over a sphere
7) conclusions? f is a harmonical non-trivial 2-form.

I would appreciate if someone could help me with these steps.
I start with 1)
Finding a Hodge dual is to operate with the operator * on f, right?
$$*f=\frac{E}{4\pi}(\frac{x}{(x^2+y^2+z^2)^{3/2}}dy\wedge dz+\frac{y}{(x^2+y^2+z^2)^{3/2}}dz\wedge dx+\frac{x}{(x^2+y^2+z^2)^{3/2}}dx\wedge dy)= \frac{E}{4\pi}}(\frac{x}{(x^2+y^2+z^2)^{3/2}}dx+ \frac{y}{(x^2+y^2+z^2)^{3/2}}dy$$
$$+\frac{z}{(x^2+y^2+z^2)^{3/2}}dz$$ so we have showed that our 2-form is represented now by the dual 1-form.
this can be somehow writter in a shorter way $$*f=\frac{E}{\pi}\frac{\hat{r}}{r^2}$$


2) find df , should we simply differentiate f with respect to x,y,z? by the way it must be $$df=0$$

Please help

 

[lippyka]

me with the rest steps.Homework Equations The Attempt at a Solution 1) *f=\frac{E}{4\pi}}(\frac{x}{(x^2+y^2+z^2)^{3/2}}dx+ \frac{y}{(x^2+y^2+z^2)^{3/2}}dy+\frac{z}{(x^2+y^2+z^2)^{3/2}}dz so we have showed that our 2-form is represented now by the dual 1-form.this can be somehow writter in a shorter way *f=\frac{E}{\pi}\frac{\hat{r}}{r^2} 2) df= \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy + \frac{\partial f}{\partial z} dz = \frac{E}{4\pi}(-\frac{3x^2}{(x^2+y^2+z^2)^{5/2}}dx\wedge dy\wedge dz-\frac{3y^2}{(x^2+y^2+z^2)^{5/2}}dy\wedge dz\wedge dx-\frac{3z^2}{(x^2+y^2+z^2)^{5/2}}dz\wedge dx\wedge dy)3) Since df does not equal 0, f is not exact.4) Since df does not equal 0, f is not closed.5) I'm not sure what this means. I'm assuming it means to calculate the exterior derivative of f, which is given by d^+f=df.6) To integrate f over a sphere, we need to use the formula ∫Sf=∫S∗f. We can use this formula to calculate the integral. We know that ∗f=\frac{E}{\pi}\frac{\hat{r}}{r^2}, so the integral becomes: