# A question regarding orthogonal parametrizations

This passage is part of the solution of an exercise in Differential Geometry. Let $$S$$ be a surface, and let $$X: U \longrightarrow S$$ be an orthogonal parametrization. If $$N^X = \frac{X_u \wedge X_v}{|X_u \wedge X_v|}$$, then $$\langle X_{uu}, N^X \rangle \langle X_{vv}, N^X \rangle = \langle X_{uu}, X_{vv} \rangle - \langle X_{uu}^T, X_{vv}^T\rangle, \qquad \qquad \qquad (*)$$ where $$T$$ denotes the "part tangent to the surface", according to the book.

The exercise asks the reader to compute the Gaussian curvature for a surface parametrized by an orthogonal parametrization.

My questions are the following:

What does "part tangent to the surface" mean? What is the image of the second partial derivative, say $$X_{uu}$$? How to prove $$(*)$$?

Thanks in advance and kind regards

$$X_{uu}$$ lives in $$\Bbb{R}^3$$, so we can write it as $$X_{uu} = \langle X_{uu}, X_u \rangle X_u + \langle X_{uu}, X_v \rangle X_v + \langle X_{uu}, N^X \rangle N^X,$$ and analogously for $$X_{vv}$$: $$X_{vv} = \langle X_{vv}, X_u \rangle X_u + \langle X_{vv}, X_v \rangle X_v + \langle X_{vv}, N^X \rangle N^X.$$ Recall that $$\Bbb{R}^3 = T_XS \oplus \langle N^X \rangle$$.
From these expressions, $$(*)$$ easily follows, sufficing to write the inner product $$\langle X_{uu}, X_{vv} \rangle$$.