# Motivation usage of Gramian Matrix for Integration on Submanifolds

I am struggling to understand the motivation for the definition of the Gram Matrix and its corresponding role for the definition of the Lebesgue measure on submanifolds.

$$M \subset \mathbb{R}^n\ k$$ dimensional $$C^1$$-Submanifold

$$\phi^{-1}: T \subseteq \mathbb{R}^m \to M$$ local parametrization of M

We define the Gram-Matrix as

$$G_\phi: T \to \mathbb{R}^{n\times n}$$

$$t \mapsto (D(\phi^{-1}(t))^TD(\phi^{-1}(t))$$

and its corresponding determinant at $$t\in T: g_\phi(t)$$.

Using this one defines the m-dimensional-Lebesgue measure $$\lambda^m$$ on the Submanifold M (for simplification assume that there exists a map which already describes all of $$M$$, i.e.:

$$\phi: M \mapsto T \subset \mathbb{R}^m$$, the Atlas of M is only one map.)

Then one defines:

$$\lambda_M: \mathbb{B}^d \cap M \to [0,\infty]$$

$$B \mapsto \int_{\phi(B)} (g_{ \phi^{-1}}(t))^{1/2}d\lambda^m(t)$$

I think the usage of the Gram-Determinant accounts for some deformation of $$\phi$$, similar to the Transformation Rule, but the construction of the Gram-Matrix does not make sense to me yet.

Recall that for linear independent vectors $$v_1, \cdots, v_k \subset \mathbb{R}^n$$ the $$k$$-dimensional volume of the k-parallelepiped $$P$$ is given by $$Vol_{k} = \sqrt{\det{A^TA}}$$ where $$A \subset \mathbb{R}^{n \times k}$$ is the matrix containing the column vectors $$v_1, \cdots, v_k$$ (we assume $$k \leq n$$).
Let $$M \subset \mathbb{R}^n$$ be a $$k$$-dimensional submanifold with one atlas $$\Phi: V \to M$$. Let's assume we devide $$V$$ into finitely many small cubes $$Q_i$$. We might want to write $$Vol_k(M) \approx \sum_i Vol_k(\phi(Q_i))$$ similarly to Riemann-sums. If we make the cubes "small enough" we have $$\phi \approx D\phi$$ since the Jacobi Matrix is the liniarization of $$phi$$.
Now replacing the sum by an integral and $$Vol_k(..)$$ by $$\sqrt{D\phi^T D \phi}$$ yields the representation. I hope this gives some motivation on the usage of the gram matrix.