Suppose we have a smooth injection $\Phi(x,y)$ from an open set $\Omega \subset \mathbb{R}^2$ to $S = \Phi(\Omega) \subset \mathbb{R}^3$. It is well-known that the integration by substitution formula holds:
$$ \int_{S} f(x,y,z) \, \text{d}\sigma(x,y,z) = \int_{\Omega} f(\Phi(x,y)) \, J(x,y) \, \text{d}x \, \text{d}y $$
where $J(x,y)$ denotes the Jacobian.
Recently, I have been working on projects related to integration on a sphere. For example, consider a smooth bijection $\Phi(\omega)$ from $S^2$ to $S^2$, the sphere in 3-dimensional space. I am curious if there exists a formula like:
$$ \int_{S^2} f(\omega) \, \text{d}\sigma(\omega) = \int_{S^2} f(\Phi(\omega)) \, T(\omega) \, \text{d}\omega $$
where $T(\omega)$ is only determined by the transition function $\Phi$.
I tried to use the Jacobian, but it is evidently incorrect because $T$ is not even differentiable on the sphere in 3-dimensional space (it is only defined on the sphere!).
I read some papers and found something like:
$$ \int_{S^2} f(\omega) \, \text{d}\sigma(\omega) = \int_{S^2} f(\Phi(\omega)) \, ||DT_\omega(\vec s)\times DT_\omega(\vec t)|| \, \text{d}\omega $$
where(copied from paper):
$\vec s$ and $\vec t$ are orthonormal tangent vectors of the unit sphere at $\omega$ and the $DT_{\omega}$ is the 'differential' of $T$ with respect to the vector $\omega$, The norm of the cross product of transformed tangent vectors accounts for the distortion in the integration domain, similar to the Jacobian determinant for a change of variables in ambient space.
But it doesn't provide any precise definition. I don't know how to define the 'differential' of $T$ w.r.t. $\omega$.
I would be more than grateful if you could provide any clues, thank you!
Update:
I found something like:
$$\int_\mathcal{S^2}{f(\omega)}\text d\sigma(\omega)=\int_\mathcal{S^2}{f(\Phi(\omega))\frac{|\omega^TJ^*(\omega)F(\omega)|}{|F(\omega)|^3}}\text d\sigma(\omega)$$
where $\Phi((x,y,z)^T)=\frac{F((x,y,z)^T)}{|F((x,y,z)^T)|}$, $\Phi$ is bijection from sphere to sphere, $F$ is smooth and non-zero on the sphere.$J$ denotes the Jacobian of $F$ and $J^*$ is the adjugate matrix.