Let $P$ be an arbitrary point on a regular surface $\Phi$. In the tangent plane $T \Phi_{P}$ we take two vectors $\vec{\lambda}$ and $\vec{\mu}$. Let $\lambda^{1}, \lambda^{2}$ and $\mu^{1}, \mu^{2}$ be coordinates of these vectors $\vec{\lambda}$ and $\vec{\mu}$ in a local basis $\overrightarrow{\boldsymbol{r}}{u}(P)$ and $\overrightarrow{\boldsymbol{r}}{v}(P)$. Find a formula for calculation of a scalar product of $\vec{\lambda}$ and $\vec{\mu}$ in terms of coordinates of these vectors in a local basis. Using our above notation $E=\left\langle\overrightarrow{\boldsymbol{r}}{u}, \overrightarrow{\boldsymbol{r}}{u}\right\rangle, F=\left\langle\vec{r}{u}, \overrightarrow{\boldsymbol{r}}{v}\right\rangle, G=\left\langle\overrightarrow{\boldsymbol{r}}{v}, \overrightarrow{\boldsymbol{r}}{v}\right\rangle$, we obtain
$$\langle\vec{\lambda}, \vec{\mu}\rangle=E \lambda^{1} \mu^{1}+F\left(\lambda^{1} \mu^{2}+\lambda^{2} \mu^{1}\right)+G \lambda^{2} \mu^{2} .$$

In this way a scalar product generates on the surface $\Phi$ (in each tangent plane to Ф) a field of symmetric bilinear forms
$$I(\vec{\lambda}, \vec{\mu})=E \lambda^{1} \mu^{1}+F\left(\lambda^{1} \mu^{2}+\lambda^{2} \mu^{1}\right)+G \lambda^{2} \mu^{2}$$
In particular, the first fundamental form of a surface is defined as
$$I(\vec{\lambda})=I(\vec{\lambda}, \vec{\lambda})=E\left(\lambda^{1}\right)^{2}+2 F \lambda^{1} \lambda^{2}+G\left(\lambda^{2}\right)^{2}$$

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