Let $\phi: \partial B_1(0) \to \mathbb{R}$ be a differentiable function where $\partial B_1(0) = \{x \in \mathbb{R}^n : |x| = 1\}$ .
Defining:
$$S = \{\phi(x)x : x\in \partial B_1(0)\}$$
I am interested in calculating the 'surface' integral:
$$\int_S f(z)\,ds(z)$$
I wonder if there is a way to use the change of variable $z = \phi(x)x$ to transform this integral into a surface integral over $\partial B_1(0)$. I want something that resembles the following:
$$\int_S f(z)\,ds(z) = \int_{\partial B_1(0)} f(z(x)) \left|\frac{dz}{dx}\right| \,ds(x)$$
The problem is that I am not sure how to interpret the 'Jacobian' $\left|\frac{dz}{dx}\right|$ in this context.
Until now, I had tried to do it in the case of $n = 2$, and here is what I achieved.
Parameterizing with $\theta \in [0, 2\pi]$, we have:
\begin{align} x & = (\cos\theta, \sin\theta) \\ z & = (\phi\cos\theta, \phi\sin\theta) \end{align}
This implies that:
\begin{align} \frac{dx}{d\theta} & = (-\sin\theta, \cos\theta) \\ \frac{dz}{d\theta} & = \left(\frac{d\phi}{d\theta}\cos\theta - \phi\sin\theta, \frac{d\phi}{d\theta}\sin\theta + \phi\cos\theta\right) \end{align}
And after doing the calculations, it is not difficult to see that:
\begin{align} \left\|\frac{dx}{d\theta}\right\| & = 1\\ \left\|\frac{dz}{d\theta}\right\| & = \sqrt{\left(\frac{d\phi}{d\theta}\right)^2 + \phi^2} \end{align}
With this parameterization, we have that:
\begin{align} \int_S f(z)\,ds(z) & = \int_0^{2\pi} f(\phi\cos\theta, \phi\sin\theta)\sqrt{\left(\frac{d\phi}{d\theta}\right)^2 + \phi^2} \, d\theta \\ & = \int_0^{2\pi} f(\phi\cos\theta, \phi\sin\theta)\sqrt{\left(\frac{\partial\phi}{\partial x_1}\frac{dx_1}{d\theta} + \frac{\partial\phi}{\partial x_2}\frac{dx_2}{d\theta}\right)^2 + \phi^2} \, d\theta \\ & = \int_0^{2\pi} f(\phi(\cos\theta, \sin\theta))\sqrt{\left(-\frac{\partial\phi}{\partial x_1}\sin\theta + \frac{\partial\phi}{\partial x_2}\cos\theta\right)^2 + \phi^2} \, d\theta \\ & = \int_{\partial B_1(0)} f(\phi x) \sqrt{\left(-\frac{\partial\phi}{\partial x_1}x_2 + \frac{\partial\phi}{\partial x_2}x_1\right)^2 + \phi^2} \, ds(x) \end{align}
I'm not 100% sure that my calculations have been correct. In any case, I would like to be able to interpret the term $\sqrt{\left(-\frac{\partial\phi}{\partial x_1}x_2 + \frac{\partial\phi}{\partial x_2}x_1\right)^2 + \phi^2}$ so that I can generalize it to higher dimensions.