Unit normal vector of a surface
What we're building to
- If a surface is parameterized by a function , the unit normal vector to this surface is given by the expression
- You always have two choices for a unit vector function. If a surface is closed, like a sphere or a torus, those choices can be interpreted as outward-facing and inward-facing vectors.
- This is useful for the idea of flux in three-dimensions, covered in the next article.
Unit normal vector
Example: How to compute a unit normal vector
Step 1: Find a (not necessarily unit) normal vector
Step 2: Make that a unit normal vector
- Given a surface parameterized by a function , to find an expression for the unit normal vector to this surface, take the following steps:
- Step 1: Get a (non necessarily unit) normal vector by taking the cross product of both partial derivatives of :
- Step 2: Turn this vector-expression into a unit vector by dividing it by its own magnitude:
- You can also multiply this expression by , and it will still give unit normal vectors.
- The main reason for learning this skill is to compute three-dimensional flux.