Riemannian volume form on surface of a smooth function

Question

It should be easy calculation exercise in my text, but I am afraid I am a little bit stuck on the concept of the question.

Let $f:\Bbb R^n\to \Bbb R$ be a smooth function. Consider graph $X$ of $f$ in $\Bbb R^{n+1}$. It is $n$-dimensional smooth manifold with chart $(X,\Bbb R^n,\varphi)$ where $\varphi: \Bbb R^n\to X$ by $\varphi(x)=(x,f(x))$. If $\sigma$ is Riemannian volume form on $X$ prove that

$$\varphi^*\sigma=\sqrt{1+\sum_{k=1}^n{(\frac{\partial f}{\partial x_k})^2}}dx_1\wedge dx_2 \wedge ... \wedge dx_n.$$

Answer 1

If $g$ is the Riemannian metric on $X$ induced by the Euclidean metric in $\mathbb R^{n+1}$ then $dV_g(x)=f(x)dx_1\wedge\dots\wedge dx_n$ where $f(x)$ is the Euclidean measure of the parallelotope generated by $\phi_\ast\partial_{x_1},\dots,\phi_\ast\partial_{x_n}.$
Now you have just to take the Euclidean norm of the vector in $\mathbb R^{n+1}$ whose components are the minors of the $(n+1)\times n$-matrix $(\phi_\ast\partial_{x_1},\dots,\phi_\ast\partial_{x_n})$.

Answer 2

Use the formula for the Riemannian volume form $\sigma = vol_g$ in the coordinates $(x^1,\dots,x^n)$ in $\Bbb R^n$ $$ vol_g = \sqrt{|\det{g}|} dx^1 \wedge \dots \wedge dx^n $$ where $g$ is the first fundamental form of the surface $X$ which is given by $$ g_{i j} \colon = \langle\varphi_i, \varphi_j\rangle $$ where $\langle\cdot,\cdot\rangle$ denotes the standard inner product ("dot-product") in $R^{n+1}$, and vectors $\varphi_i$ are defined as $$ \varphi_i \colon = \varphi_* (\partial_i) $$

Notice, that your $\varphi$ is usually referred to as a parametrization, not as a chart. Using a parametrization we identify the surface with the parameter space (locally).

More explicitly your parametization looks as $$ \varphi \colon \Bbb R^n \to \Bbb R^{n+1} \colon \, x \mapsto (x,f(x)) \colon \, (x_1, \dots, x_n) \mapsto (x_1, \dots, x_n, f(x_1, \dots, x_n)) $$

In the given parametrization we have $$ \varphi_i = \begin{pmatrix} 0 \\ \dots \\ 1 \\ \dots \\ 0 \\ f_i \end{pmatrix} $$ where $f_i \colon = \frac{\partial f}{\partial x^i}$

Calculate the first fundamental form and its determinant, and you are done!