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## Main Question or Discussion Point

Let $$\phi(x^1,x^2....,x^n) =c$$ be a surface. What is unit Normal to the surface?

I know how to find equation of normal to a surface. It is given by:

$$\hat{e_{n}}=\frac{\nabla\phi}{|\nabla\phi|}$$

However the answer is given using metric tensor which I am not able to derive. Here is the answer

$$({g^{\alpha\beta}}{\frac{\partial\phi}{\partial x^{u}}}{\frac{\partial\phi}{\partial x^{v}}})^{-1/2} {g^{r\alpha}}{\frac{\partial\phi}{\partial x^{\alpha}}}$$

How can I Derive this?

I know how to find equation of normal to a surface. It is given by:

$$\hat{e_{n}}=\frac{\nabla\phi}{|\nabla\phi|}$$

However the answer is given using metric tensor which I am not able to derive. Here is the answer

$$({g^{\alpha\beta}}{\frac{\partial\phi}{\partial x^{u}}}{\frac{\partial\phi}{\partial x^{v}}})^{-1/2} {g^{r\alpha}}{\frac{\partial\phi}{\partial x^{\alpha}}}$$

How can I Derive this?