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
$|H_n|=\int_{D} \sqrt{g} du_1 \ldots du_n ,$
and
$ |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 $g=\det[g_{ij}]$.
Thanks.
P.S. I search for formulas which contain only coefficients $g_{ij}$ and derivatives $\frac{\partial u_i}{\partial t_j}$.