This is a page from Evan's PDEs where the Radon transform is defined.
I have three brief questions: (1) It says the integrals $\int_{\Pi(s,\omega)}\nabla u\cdot b_{i}\,dS$ vanish because the vectors $b_{i}$ are parallel to the plane $\Pi(s,\omega)$ and $u$ has compact support.
I am a little confused because I understand that the vectors $b_{i}$ are parallel to the plane $\Pi(s,\omega)$; they generate the plane. It is also evident that $\nabla u\cdot b_{i}$ is the length of the projection of the vector $\nabla u$ in the direction $b_{i}$, and $|\nabla u|$ compact support (because $u$ is assumed to have compact support). Now, the scalar function $\nabla u\cdot b_{i}$ can live entirely or partly in the support of $\nabla u$. So why must the integral vanish.
(2) why does not $\int_{\Pi(s,\omega)}\nabla u\cdot A\, dS$ vanish for any vector $A$ ? (after all the gradient $\nabla u$ is compactly supported)
(3) Finally, it is stated that $\tilde{u}_{s}=\int_{\Pi(s,\omega)}\nabla u\cdot \omega\, dS$. How come ? I am guessing by a change of variables (precisely a scaling $y\rightarrow y/s$) but not sure how to formulate it.
Apologies for the long question, and thanks.
For (1), work out the case when $b_i=e_1$, so you're integrating $\partial u/\partial x_1$. Integrate over a giant rectangle containing the support of $u$ in $\Pi(s,\omega)$. By Fubini's Theorem, integrate first with respect to $x_1$, and you end up with $0$.
For (2), imagine $A=e_n$ create a compactly supported function $u$ with $\int_{x_n=c} \partial u/\partial x_n\,dS > 0$.
For (3), parametrize $\Pi(s,\omega)$ by taking any parametrization of $\Pi(0,\omega)$ and adding $s\omega$. Now write down the integral over $\Pi(0,\omega)$ and differentiate with respect to $s$ under the integral sign.