Solving Tensor Integration on a Unit Sphere: Help Needed

[tim85ruhruniv]

Could someone help me out ??

I tried this integration over the surface of a sphere of unit radii,

$$\[ P_{mn}e_{m}\otimes e_{n}=\frac{1}{D_{pq}e_{p}\otimes e_{q}}\int e_{m}\otimes e_{n}dS_{r=1}\]$$

and I always get $$\[ 4\pi e_{m}\otimes e_{n}\]$$ and the 'D' tensor as it is..

I am expecting additionally a '3' in the denominator, am I wrong ? If i do the integration over unit volume then I get the 3 in the denominator. Sorry for sounding stupid but is there a necessity to consider the unit tensor, i just assume it as a constant under integration.

 

[clamtrox]

Homework assignments and stuff like that should be posted in the appropriate section.

I don't get your indices either; it seems they don't add up on left and right hand sides. $$e_m \otimes e_n$$ definitely needs not be constant. Consider for example usual spherical coordinates.

 

[George Jones]

$$\[ P_{mn}e_{m}\otimes e_{n}=\frac{1}{D_{pq}e_{p}\otimes e_{q}}\int e_{m}\otimes e_{n}dS_{r=1}\]$$

Okay, I'll bite; along with clamtrox's note about indices, I have questions. What does

$$\frac{1}{D_{pq}e_{p}\otimes e_{q}}$$

mean? How does one divide by a tensor (not the component of a tensor), which is an element of a vector space?

Homework assignments and stuff like that should be posted in the appropriate section.

Yes and no. From the Physics Forums rules:

Since graduate level assignments are meant to be more thought provoking (and hence more worthy of discussion), graduate level questions will be allowed in the relevant part of the main section of PF, provided that the graduate student attempts the problem and shows his work.

 

[tim85ruhruniv]

hey !

Thanks guys for looking at my work.


I can't see how the indices don't add up... maybe i am missing something... but

Each component of
$$\[ \mathbf{P}\]$$will be a function of the $$\mathbf{\mathrm{D}^{-1}}$$ tensor.


about division by the tensor..

$$x=\mathbf{D}y$$ for some 'x' and some 'y'

so I hope I can rewrite this as $$y=\mathbf{\mathrm{D}^{-1}}x$$
and probably find the Inverse at a later stage. Which for the time being I believe doesent depend on the co-ordinates of integration.

Like clamtrox said, I use spherical co-ordinates to integrate, should I worry about $$\[ e_{m}\otimes e_{n}\]$$ should I transform the tensor basis ?