Changing variables in integration over spheres Suppose we would like to change variables in the integral
$$I:=\int_{\mathbb{S}^{n-1}}f(\omega_1,\omega_2,...,\omega_{n})d\sigma_{n-1},$$
where
$\mathbb{S}^{n-1}$ is the standard unit sphere in $\mathbb{R}^{n}$, $n\geq 2$,   $d\sigma_{n-1}$ is the surface measure induced by the Lebesgue measure on  $\mathbb{R}^{n}$, and
$\left(\omega_{1}(\theta_{1},...,\theta_{n-1}),
\omega_{2}(\theta_{1},...,\theta_{n-1}),...,\omega_{n}(\theta_{1},...,\theta_{n-1})\right)$ is a unit vector that gives the parametric spherical representation of every point $(x_1,...,x_n)$ that lies on the sphere.
So, for example, every $(x,y)\in\mathbb{S}^{1}$ has the representation
$(x,y)=(\omega_1,\omega_2)=(\cos{\theta_{1}},\sin{\theta_1})$, $\theta_{1}\in [0,2\pi]$, and every $(x,y,z)\in\mathbb{S}^{2}$ has the representation
$(x,y,z)=(\omega_1,\omega_2,\omega_3)=(\sin{\theta_{1}}\cos{\theta_2},\sin{\theta_1}\sin{\theta_{2}},\cos{\theta_{1}})$, $\theta_{1}\in[0,\pi], \theta_{2}\in[0,2\pi]$.
Question: How to change variables in the integral $I$? My question is about the Jacobian. Precisely, if we change variables $\omega_{i}=\phi_{i}(\omega_{1},\omega_{2},...,\omega_{n})$ where $\phi_{i}$ are continuously differentiable and invertible, is it correct that
$$I=\int_{\cup_{\theta_1,\theta_2,...,\theta_{n-1}}{(\phi_{1},...,\phi_{n})}} f(\phi_{1},...,\phi_{n})\det\left(\frac{\partial(\omega_1,...,\omega_n)}{\partial(\phi_1,...,\phi_n)}\right)\,d\sigma_{n-1} ?$$
 A: Let's forget about $\theta$ notation here, which confuses. Situation is as follows: There is a diffeomorphism $R^n \to R^n$ which we think of as taking $ (\phi_1,...,\phi_n) \to w=(w_1,...,w_n)  $. We are trying to "pull back" an integration in $w$ variables to $\phi$ variables. The suggested formula would gives give change of variables for integration over open subsets of $R^n$. That is very important. Notice how your Jacobian is the full Jacobian on the whole space.
However, you are integrating over a submanifold. For example, it should not matter how the map $\phi \to w$ distorts along radii perpendicular to the sphere -- an example diffeomorphism (around an annulus containing the sphere) is $x \to |x|^2 x$ which when restricted to sphere is the identity map.
A correct change of variable will involve the Jacobian of the restriction of the map $\phi \to w$ to the sphere. I think to compute the Jacobian you can look at the derivative map (matrix) and then restrict it to the tangent plane to the sphere and then it is linear map $R^{n-1} \to R^{n-1}$. The determinant of the latter should be the right Jacobian.
Here is a concrete example in $S^1 \subset R^2$:

