Multivariable calculus and vector analysis

Normal vector of parametrized surfaces

In our calculation of surface area, we approximated each small segment of the surface as a parallelogram spanned by $\frac{\partial \mathbf{\Phi }}{\partial u}\left(u,v\right)\Delta u$ and $\frac{\partial \mathbf{\Phi }}{\partial v}\left(u,v\right)\Delta v$. To find the area of the parallelogram, we took the magnitude of their cross product.

One feature of the cross product is that its magnitude is the area of the parallelogram spanned by the two vectors. The other feature of the cross product is that it is perpendicular to both of the vectors. It turns out that the vectors $\frac{\partial \mathbf{\Phi }}{\partial u}\left(u,v\right)$ and $\frac{\partial \mathbf{\Phi }}{\partial v}\left(u,v\right)$ are both tangent to the surface. Consequently, their cross product

$$\begin{array}{lll}\hfill \frac{\partial \mathbf{\Phi }}{\partial u}\left(u,v\right)\times \frac{\partial \mathbf{\Phi }}{\partial v}\left(u,v\right)& & \hfill \end{array}$$

is perpendicular to the surface and therefore is a normal vector to the surface. We frequently want a unit normal vector, meaning a normal vector with length one. To obtain a unit normal vector, we just divide by its magnitude:

$$\begin{array}{lll}\hfill \mathbf{n}=\frac{\frac{\partial \mathbf{\Phi }}{\partial u}\left(u,v\right)\times \frac{\partial \mathbf{\Phi }}{\partial v}\left(u,v\right)}{∥\frac{\partial \mathbf{\Phi }}{\partial u}\left(u,v\right)\times \frac{\partial \mathbf{\Phi }}{\partial v}\left(u,v\right)∥}.& & \hfill \end{array}$$

Contents by topic

1. Geometry in higher dimensions

2. Differential Calculus

3. Integral calculus

4. Vector calculus

4a. Vector field basics

4b. Parametrized curves

4c. Parametrized surfaces

• Parametrized surfaces
• Parametrized surface examples
• Surface area of parametrized surfaces
• Surface area calculation
• Normal vector of parametrized surfaces
• Orienting surfaces
• A surface that is not orientable
• Surface area example

4d. Path integrals and line integrals

4e. Surface integrals

5. The fundamental theorems of vector calculus

6. Review material