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Let $\Phi:S \to T$ be a map between surfaces in $\mathbb{R}^n$. What precisely does this mean:

Let $\text{det}(\mathbf{D}_S \Phi(.))$ denote the Jacobian determinant of the matrix representation of $\mathbf{D}_S \Phi(.)$ with respect to orthogonal bases of the respective tangent space

I assume $\mathbf{D}_S \Phi$ refers to the derivative/differential of $\Phi$. Why the subscript $S$? Does $\mathbf{D}_S \Phi(a)$ mean the derivative at the point $a$ in $S$?

Why does he say "matrix representation?" Since we're in Euclidean space won't the derivative automatically be a matrix? Also I don't know what the orthogonal bases of the tangent space line means.

Please explain if you can. Thanks.

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I believe the subscript $S$ does signify that $a$ is a point in $S$ and not in $T$.

Linear operators need not be represented by matrices, but they still have determinants regardless. I suspect the author is merely reminding you that the derivative is a matrix.

The tangent space to an N-dimensional surface is an N-dimensional flat space whose basis vectors are the same as those at a given point in the surface. The author is saying that the basis vectors used to take derivatives are local and may change with position.

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