# 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.