Since I have always been at war with polar coordinates, I wanted to find a way to compute spherical Hausdorff integrals in a different way. I know that this will get incredibly tedious if I try to solve the problem for all dimensions, so I am currently working on some dimension $d+1$, where I want to relate an integral over the $d$-sphere via change of variables to an integral over the $d-1$-sphere and the interval $(-1,1)$. This will always be possible due to the theorem about change of multiple variables. In fact, copied from Wikipedia,
Theorem. Let $X$ be a locally compact Hausdorff space equipped with a finite Radon measure $\mu$, and let $Y$ be a $\sigma$-compact Hausdorff space with a $\sigma$-finite Radon measure $\rho$. Let $\phi\colon X\to Y$ be an absolutely continuous function (where the latter means that $\rho(\phi(E)) = 0$ whenever $\mu(E) = 0$). Then there exists a real-valued Borel measurable function $w$ on $X$ such that for every Lebesgue integrable function $f\colon Y\to \mathbb{R}$, the function $(f\circ\phi)\cdot w$ is Lebesgue integrable on $X$, and $$\int _{Y}f(y)\,d\rho (y)=\int _{X}(f\circ \varphi )(x)\,w(x)\,d\mu (x).$$ Furthermore, it is possible to write $$w(x)=(g\circ \varphi )(x)$$ for some Borel measurable function $g$ on $Y$.
Now let $Y=\mathbb{S}^d$ and $X=\mathbb{S}^{d-1}\times (-1,1)$ with the measures $\rho=\sigma_{d}$ and $\mu=\sigma_{d-1}\times\lambda_1$, where $\sigma$ is the uniform Hausdorff measure on the sphere and $\lambda$ the standard Lebesgue measure. Let $$\phi(y,s)=(\sqrt{1-s^2}\,y_1,\dots,\sqrt{1-s^2}\,y_{d-1},s),$$ then $\phi$ is a bijection between $X$ and $Y$ except for some nullsets, which don't matter to the integral. My question is: How do I compute $w$ in general, if I don't want to use spherical coordinates? I know that $\phi$ is somewhat just using spherical coordinates but not accepting that $s$ is simply $\cos(\theta)$ for some angle, but I would actually like to know whether I can avoid getting trig functions into my equation. Even though $w$ will most likely be something like a trig function. Any ideas?