Surface Normal Vector

This note gives different methods of calculating the normal vector $\vec{n}$ of a surface in 3 dimensions. This is useful for applications in Stokes Theorem

Parametrized Surface A surface’s $r(u,v)=f(u,v)i+g(u,v)j+h(u,v)k$ normal vector is given by

$$\vec{n}=\frac{r\_u\times r\_v}{|r\_u\times r\_v|}$$

Implicit Surface A surface given by $f(x,y,z)=c$ has normal vector

$$\vec{n}=\frac{\nabla f}{|\nabla f|}$$

Explicit Surface Explicitly defined surfaces follow the form $z=f(x,y)$ and their normal vector is

$$\vec{n}=\sqrt{f\_x^2+f\_y^2+1}$$

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When using the normal vector for integration with the area differential dσ cancellation of terms will simplify some of the calculation.