$M$ is a Riemannian manifold ,and $g_{ij}$ is Remannian metric. Let $x=(x^1...x^d)$ $(i.e. x:U_x\rightarrow R^d)$ be a local coordinates ,and $v,w\in T_pM$ with coordinate representations $(v^1...v^d)$ and $(w^1...w^d)$$(i.e.v=v^i\frac{\partial}{\partial x^i},w=w^j\frac{\partial}{\partial x^j})$.
Let $y=f(x)$ $(i.e. y:U_y\rightarrow R^d)$ define different local coordinates.In these coordinates,$v$ and $w$ have representations $(\tilde{v}^1...\tilde{v}^d)$ and $(\tilde{w}^1...\tilde{w}^d)$. Let the metric in the new coordinates be given by $h_{kl}(y)$,then,we have: $$ h_{kl}(f(x))\frac{\partial f^k}{\partial x^i}\frac{\partial f^l}{\partial x^j}=g_{ij}(x) $$
Let $g=det(g_{ij}),h=det(h_{kl})$,and $\Phi$ is a function on $M$, $U=U_x\bigcap U_y$
Show that the integral of a function $\Phi$ on $M$ is invariant ,namely $$ \int _U \Phi(f(x))\sqrt{g(x)}dx^1...dx^d=\int _U\Phi(y)\sqrt{h(y)}dy^1...dy^d $$