Suppose one has an integral of the form $\int_{S_1^{d-1}} f(\phi(v)) d \text{vol}_{S_1^{d-1}}(v)$. Here $S_1^{d-1}\subset \mathbb{R}^d$ is the unit sphere. Let $B_1^{d-1}\subset\mathbb{R}^{d-1}$ be the unit ball in $\mathbb{R}^{d-1}$. And $\phi$ maps a vector $v=(v_1, \dots , v_d)\in S_1^{d-1}$ to $z=(v_2, \dots , v_d)\in B_1^{d-1}$.
Can someone please provide me with an explicit formula for $g$ with $ \int_{S_1^{d-1}} f(\phi(v)) d \text{vol}_{S_1^{d-1}}(v) = \int_{B_1^{d-1}} g(z) d \text{vol}_{{d-1}}(z). $
Thank you.