Flux in different coordinate systems

[Wox]

I have an electromagnetic field with a Poynting vector that has the following form in spherical coordinates:
$$\bar{P}(R,\phi,\theta)=\frac{f(\phi,\theta)}{R^2}\bar{e}_{r}$$
The exact nature of f(\phi,\theta) is not known. Suppose I measure the flux of this vector field by a flat area detector. The pixel values of an image acquired by such a detector are given by
$$I_{\text{pixel}}(x,y)=\iint_{\text{pixel}} \bar{P} \cdot \bar{\text{d}S}=\iint_{\text{pixel}} \left\|\bar{P}(x,y)\right\|\cos(\alpha(x,y))\ \text{d}x\text{d}y$$
where \alpha the (known) angle between detector surface and Poynting vector and XY the detector plane in the detector reference frame XYZ for which we know the relation to the \bar{P} coordinate system. From this I want to know the pixel values as would have been measured when the detector had another orientation and position with respect to the origin of \bar{P}
$$\begin{bmatrix}x'\\y'\\z'\\1 \end{bmatrix} =L\cdot\begin{bmatrix}x\\y\\z\\1\end{bmatrix}$$
where L a know composition of rotations and translations or even for a spherical detector. How can I do this? The Jacobian can be used when the transformation only involved the detector plane XY. However this is not the case and I can also not use the divergence theorem, since we don't have a closed surface. Any ideas on how to approach this?

 

[chiro]

Hey Wox.

Have you encountered unfixed co-ordinate system mathematics and calculus through tensors?

 

[Wox]

I'm not sure what you mean by "unfixed co-ordinate system mathematics". You mean vector spaces without choosing a basis?

As for tensors, this is what I know: a tensor product of two vector spaces V and W both over field K is a pair (T,\otimes) where T a vector space over K and \otimes\colon V\times W\rightarrow T a bilinear map with the property that for any bilinear map B_{L}\colon V\times W\rightarrow X with X a vector space over K, there exists a unique linear map F_{\otimes}\colon T\rightarrow X so that B_{L}=F_{\otimes}\circ\otimes. Furthermore if (T,\otimes) and (T',\otimes') are two tensor products of V and W then there exists a unique isomorphism F\colon T\rightarrow T' such that \otimes'=F\circ\otimes. Although I understand what all this says (I know vector spaces, bilinear maps, linear maps, bijective linear maps = vector space isomorphisms), I don't really grasp the idea or the practical implications. But any suggestions you have are welcome, if I don't understand I'll try to learn it.

 

[chiro]

The application physically is widespread:

http://en.wikipedia.org/wiki/Application_of_tensor_theory_in_physics

It has to do with what you are saying, but basically you have identities that deal with things like del, grad, differentiation and integration in arbitrary co-ordinate systems by relating them back to Euclidean since we have a developed theory in R^n.

The isomorphism is the best attribute that captures the description but instead of just thinking about vector spaces, add the results of calculus to that and you have a good idea of how its used.