I have an electromagnetic field with a Poynting vector that has the following form in spherical coordinates:(adsbygoogle = window.adsbygoogle || []).push({});

$$\bar{P}(R,\phi,\theta)=\frac{f(\phi,\theta)}{R^2}\bar{e}_{r}$$

The exact nature of [itex]f(\phi,\theta)[/itex] 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 [itex]\alpha[/itex] the (known) angle between detector surface and Poynting vector and [itex]XY[/itex] the detector plane in the detector reference frame [itex]XYZ[/itex] for which we know the relation to the [itex]\bar{P}[/itex] 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 [itex]\bar{P}[/itex]

$$\begin{bmatrix}x'\\y'\\z'\\1 \end{bmatrix} =L\cdot\begin{bmatrix}x\\y\\z\\1\end{bmatrix}$$

where [itex]L[/itex] 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 [itex]XY[/itex]. 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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Flux in different coordinate systems

**Physics Forums | Science Articles, Homework Help, Discussion**