Multivariable calculus and vector analysis

A surface that is not orientable

A surface is orientable is has two sides. Then, one can orient the surface by choosing one side to be the positive side (i.e., choose a direction for the normal vector).

Some unusual surfaces however are not orientable because they have only one side. One classical examples is called the Möbius strip. You can construct a Möbius strip by taking a strip of paper, twisting it half a turn, and then taping the ends together. A Möbius strip with a normal vector is shown below.

If you drag the red slider, the normal vector moves along the Möbius strip. Once you drag the slider half way, the normal vector has moved all the way around the surface. However, now the normal vector is pointing in the opposite direction. Since the normal vector didn’t switch sides of the surface, you can see that Möbius strip actually has only one side. For this reason, the Möbius strip is not orientable.

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