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Given a function $f:\Delta^n \to Y$ from a simplex into a riemmannian manifold. Furthermore given a point $x \in \Delta^n$ we can send an orthonormal basis at $x$ using $D_x f$ to a set of vectors defining a parallelepiped. I know that the volume of a parallelepiped can be calculated using the absolute value of the determinant of its defining vectors.

So it would be convenient to write $\left|\det(D_x f)\right|$. But the Jacobian $D_x f$ is not necessarily a square matrix.

Is there a notational way to fix this?

EDIT: What I am looking for is a simple notation to write this down.

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  • $\begingroup$ what is $Y$'s dimension? $\endgroup$ – janmarqz Jul 8 '15 at 18:19
  • $\begingroup$ @janmarqz it is only necessary for Y to have a dimension $\geq n$ $\endgroup$ – Loreno Heer Jul 8 '15 at 19:36
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It is possible to calculate the volume by $n$ vectors $\{\vec{v_i}\}$ in a $m$-dimensional coordinate system with $m > n$. Actually they must lie in some $n$ dimensional subspace. Denote the matrix with $\{\vec{v_i}\}$ as rows $V$.

Apply an orthogonal transformation $O$ to get them into the first $n$ coordinates, we get $\{O V\}$, and multiply with a $E_{m,n}$ which is an $m*n$ matrix with $1$ on $(1, 1), (2,2),\cdots$ and $0$ elsewhere. Then the desired volume is $\det(E_{m,n}OV)$.

Now lets get rid of $E_{m,n}$ and $O$.

Remember that $\det(A^T)=\det(A)$, and $\det(A^T A)=\det(A)^2$. So in this case we have $$\mathrm{Volume}=\sqrt{\det((E_{m,n}OV)^T(E_{m,n}OV))} = \\ \sqrt{\det(V^TO^TE_{m,n}^TE_{m,n}OV)}$$

It is nice that $O^TO=E$ and $E_{m,n}^TE_{m,n}=E$, so $$\mathrm{Volume}=\sqrt{\det(V^TV)}$$, which is only dependent of $V$, or $\{\vec{v_i}\}_n$.

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