Coordinated Multivariable Calculus

In Preview Activity 4.11.1, we saw that a simple closed curve in ${\mathbb{R}}^3$ can bound many different surfaces. For now, however, we want to focus on a smooth surface $S$ in ${\mathbb{R}}^3$ that has a well-defined normal vector $\mathbf{n}$ at every point and a boundary curve $C\text{.}$ We will use the normal vector to define an orientation of $C$ so that if a person were to walk along $C$ in the direction of the orientation with the top of their head pointing in the direction of $\mathbf{n}\text{,}$ their left arm would be over the surface $S\text{.}$ Notice that this is the same convention that we used with Green’s Theorem if we assume that the normal vector being used is $\mathbf{k}\text{.}$

In Figure 4.11.4, we show the curve $C$ from Preview Activity 4.11.1 in magenta as well as a surface $S$ that has $C$ as its boundary. The chosen normal vector $\mathbf{n}$ to $S$ is shown, as is the orientation of $C$ that matches $\mathbf{n}\text{.}$

Thinking back to Green’s Theorem, our main idea was that we could calculate the circulation around a simple closed curve in ${\mathbb{R}}^2$ by taking the double integral of the circulation density over the region bounded by the curve. As we saw in Preview Activity 4.11.1, we can break up ${\oint }_C\mathbf{F}\cdot d\mathbf{r}$ into line integrals around other simple closed curves so that overlapping portions are oriented oppositely just as we did with the square grid for Green’s Theorem. To find a three-dimensional analog of Green’s Theorem, we require that a simple closed curve $C$ in three dimensions bound a smooth surface $S$ with a normal vector $\mathbf{n}\text{.}$ In doing this, we can choose our “smaller” curves similar to the squares we used in Green’s Theorem to lie on the surface $S\text{.}$ This gives us almost all the ingredients used in Green’s Theorem, but we still need to find a suitable replacement for the circulation density.

As we saw in Section 4.5, the curl of a vector field in ${\mathbb{R}}^3$ measures the rotation of the vector field. Theorem 4.5.17 says that for a unit vector $\mathbf{v}\text{,}$ the scalar $(\mathrm{curl}(\mathbf{F})(a,b,c))\cdot \mathbf{v}$ measures the rotational strength of $\mathbf{F}$ at the point $(a,b,c)$ around the axis defined by $\mathbf{v}\text{.}$ When $\mathbf{v}$ is the normal vector to the surface $S$ at the point $(a,b,c)\text{,}$ we have the appropriate analog for the circulation density of $\mathbf{F}$ on $S$ at $(a,b,c)\text{.}$ Thus, the equivalent idea to integrating the circulation density of a two-dimensional vector field over a region in the plane is calculating the flux integral ${\iint }_D\mathrm{curl}(\mathbf{F})\cdot ({\mathbf{r}}_s\times {\mathbf{r}}_t)dA\text{,}$ where $\mathbf{r}(s,t)$ on the domain $D$ gives a parametrization of the smooth surface $S\text{.}$

A rigorous proof of the following theorem is beyond the scope of this text. However, the previous subsection and our discussion of Green’s Theorem provide an intuitive description of why this theorem is true.

When we looked Green’s Theorem, it was generally most useful when we were given a line integral and we calculated it using a double integral. In fact, except in the circumstances described in exercises 4.6.6.1 and 4.6.6.3, we did not use Green’s Theorem to rewrite a double integral as a line integral because of the difficulty of finding a suitable vector field. The situation for Stokes’ Theorem will be similar, with the exception of Exercise 1 in this section. However, Stokes’ Theorem gives us an interesting additional piece of freedom: selecting the surface $S$ through which we calculate the flux of $\mathrm{curl}(\mathbf{F})$ from amongst possibly several reasonable surfaces with boundary $C\text{.}$ The next two activities focus on the relationships between surfaces and their boundary.

We close this section with a pair of activities. The first focuses on calculating both of the integrals in Stokes’ Theorem. The second asks you to calculate some line integrals along simple closed curves and gives you the discretion to choose the best method to use for this (as well as the best surface to use, if you choose Stokes’ Theorem).

In part part b of Activity 4.11.5, there are two “reasonable” choices for the surface bounded by the circle. If you did not do so while doing the activity, we encourage you to identify both of them and compare which one makes doing the flux integral easier. In general, this will vary depending on the (curl of the) vector field in question, so we cannot give a rule for determining what surface or coordinate system to use. However, we do encourage you to think about which surface will make evaluating the flux integral easiest.