I am trying to understand the Fourier transform of the surface measure. Here is the definition: If $d\sigma$ is the surface measure on the sphere $\mathbb{S}^{n-1}$, then $$\widehat{d\sigma}(\xi):=\int_{\mathbb{S}^{n-1}} e^{-\imath \theta\cdot\xi}\,d\theta.$$
So, in dimension $n=2$, $$\widehat{d\sigma}(\xi):=\int_{0}^{2\pi} e^{-\imath \left(\xi_{1}\cos{\theta}+\xi_{2}\sin{\theta}\right)} \,d\theta,$$
and in dimension $n=3$,
$$\widehat{d\sigma}(\xi):=\int_{0}^{2\pi}\int_{0}^{\pi} e^{-\imath \left(\xi_{1}\sin{\theta}\cos{\varphi}+\xi_{2}\sin{\theta}\sin{\varphi}+ \xi_{3}\cos{\varphi}\right)} \,\sin{\theta} d\theta\,d\varphi$$
and so on. Correct?