Let $F:S^{2}\rightarrow\mathbb{R}^{4}$ be the immersion defined as $(x^{2}-y^{2},xy,xz,yz)$ and consider the metric on $S^{2}$ induced by $F$. Find $g_{ij}(0,0)$ for the upper hemisphere parameterization $\phi(x,y)=(x,y,\sqrt{1-x^{2}-y^{2}})$
For the metric on $S^{2}$ induced by $F$, we can explicity determine $g_{ij}$ (this is just finding $(DF)^{T}(DF)$): $ g_{ij}=\left(\begin{array}{cccc} 4x^{2}+4y^{2} & 0 & 2xz & 0\\ 0 & y^{2}+z^{2} & yz & xz\\ 2xz & yz & x^{2}+z^{2} & xy\\ -2yz & xz & xy & z^{2}+y^{2}\end{array}\right)$
I am wondering for the parameterization $\phi$, would I just substitute $z$ by $\sqrt{1-x^{2}-y^{2}}$ and calculuate $g_{ij}(0,0)$?