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?

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# Flux in different coordinate systems

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