First fundamental form of a surface patch $\sigma$

Question

I have its answer, but I cannot understand again. Please explain its solution clearly. Thanks a lot.

Answer 1

It depends on the theorems you've been given and the definitions you're working from, but assuming you've met it already, the easiest way to show this is from the matrix multiplcation as follows.

We have $$\langle \mathbf v,\mathbf w\rangle = \left\langle \binom{du(\mathbf v)}{dv(\mathbf{ v})}, \binom{du(\mathbf w)}{dv(\mathbf{ w})}\right\rangle = (du(\mathbf v),dv(\mathbf{ v}))\begin{pmatrix}E&F\\ F&G\end{pmatrix}\binom{du(\mathbf w)}{dv(\mathbf{ w})}\\ = (Edu(\mathbf v)+Fdv(\mathbf{ v}),Fdu(\mathbf v)+Gdv(\mathbf{ v}))\binom{du(\mathbf w)}{dv(\mathbf{ w})}\\ = Edu(\mathbf v)du(\mathbf{ w})+F(dv(\mathbf{ v})du(\mathbf{ w})+du(\mathbf v)dv(\mathbf{ w}))+Gdv(\mathbf{ v})dv(\mathbf{ w}). $$

As you can see, the text you're using got the very last term involving $G$ wrong (but given the complexity of the TeX to write this out, I don't blame them for making a mistake).