Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $H_n$ be a $n$-dimensional hypersurface covered by a parametrization $\Phi=\Phi(u_1,\ldots, u_n)$, for $(u_1, \ldots, u_n) \in D$, and let $H_k$ be a k-dimensional hypersurface contained in $H_n$, where $1 \leq k \leq n$, given by $\Psi=\Phi(u_1(t_1, \ldots, t_k), \ldots, u_n(t_1, \ldots, t_k))$ for $(t_1, \ldots, t_k) \in \Delta$.

I want to know what is a formula for area $|H_k|$ of $H_k$.

I know only that

$$|H_n|=\int_{D} \sqrt{g} du_1 \ldots du_n ,$$


$$ |H_1|= \int_{\Delta} \sqrt{\sum_{i,j=1}^k g_{ij}(u_1(t_1), \ldots, u_n(t_1)) \frac{du_i}{dt_1} \frac{du_j}{dt_1}} \ dt_1 ,$$

where $g_{ij}= \langle \Phi|_i , \Phi|_j \rangle$ for $i,j=1, \ldots, n$, and $g=\det[g_{ij}]$.


P.S. I search for formulas which contain only coefficients $g_{ij}$ and derivatives $\frac{\partial u_i}{\partial t_j}$.

share|cite|improve this question

1 Answer 1

up vote 2 down vote accepted

The metric $g$ on $H_n$ is the pull-back of the metric in the ambient space through $\Phi$, i.e. as you just have said: $g=g_{ij}du^i du^j=\langle\frac{\partial\Phi}{\partial u^i},\frac{\partial\Phi}{\partial u^j}\rangle du^i du^j$.

Now similarly the metric $\overline{g}$ on $H_k$ is the pull-back of the metric $g$ on $H_n$ through the map $t\mapsto u(t)$, i.e. $\overline{g}=\overline{g}_{hk}dt^h dt^k=g_{ij}\frac{\partial u^i}{\partial t^h}\frac{\partial u^j}{\partial t^k}dt^h dt^k$.

So $\overline{g}_{hk}=g_{ij}\frac{\partial u^i}{\partial t^h}\frac{\partial u^j}{\partial t^k}$, and $|H_k|=\int_{\Delta}\sqrt{\overline{g}}dt_1\dots dt_k$.

share|cite|improve this answer
Thanks for answer. – Richard Oct 17 '11 at 11:57

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.