# 6.7 Stokes’ theorem

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Use Stokes’ theorem to calculate line integral ${\int }_{C}\text{F}·d\text{r},$ where $\text{F}=⟨z,x,y⟩$ and C is the boundary of a triangle with vertices $\left(0,0,1\right),\left(3,0,-2\right),$ and $\left(0,1,2\right).$

$\frac{3}{2}$

## Interpretation of curl

In addition to translating between line integrals and flux integrals, Stokes’ theorem can be used to justify the physical interpretation of curl that we have learned. Here we investigate the relationship between curl and circulation, and we use Stokes’ theorem to state Faraday’s law—an important law in electricity and magnetism that relates the curl of an electric field to the rate of change of a magnetic field.

Recall that if C is a closed curve and F is a vector field defined on C , then the circulation of F around C is line integral ${\int }_{C}\text{F}·d\text{r}.$ If F represents the velocity field of a fluid in space, then the circulation measures the tendency of the fluid to move in the direction of C .

Let F be a continuous vector field and let ${D}_{r}$ be a small disk of radius r with center ${P}_{0}$ ( [link] ). If ${D}_{r}$ is small enough, then $\left(\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}\right)\left(P\right)\approx \left(\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}\right)\left({P}_{0}\right)$ for all points P in ${D}_{r}$ because the curl is continuous. Let ${C}_{r}$ be the boundary circle of ${D}_{r}.$ By Stokes’ theorem,

${\int }_{{C}_{r}}\text{F}·d\text{r}={\iint }_{{D}_{r}}\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}·\text{N}dS\approx {\iint }_{{D}_{r}}\left(\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}\right)\left({P}_{0}\right)·\text{N}\left({P}_{0}\right)dS.$ Disk D r is a small disk in a continuous vector field.

The quantity $\left(\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}\right)\left({P}_{0}\right)·\text{N}\left({P}_{0}\right)$ is constant, and therefore

${\iint }_{{D}_{r}}\left(\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}\right)\left({P}_{0}\right)·\text{N}\left({P}_{0}\right)dS=\pi {r}^{2}\left[\left(\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}\right)\left({P}_{0}\right)·\text{N}\left({P}_{0}\right)\right].$

Thus

${\int }_{{C}_{r}}\text{F}·d\text{r}\approx \pi {r}^{2}\left[\left(\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}\right)\left({P}_{0}\right)·\text{N}\left({P}_{0}\right)\right],$

and the approximation gets arbitrarily close as the radius shrinks to zero. Therefore Stokes’ theorem implies that

$\left(\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}\right)\left({P}_{0}\right)·\text{N}\left({P}_{0}\right)=\underset{r\to {0}^{+}}{\text{lim}}\frac{1}{\pi {r}^{2}}{\int }_{{C}_{r}}\text{F}·d\text{r}.$

This equation relates the curl of a vector field to the circulation. Since the area of the disk is $\pi {r}^{2},$ this equation says we can view the curl (in the limit) as the circulation per unit area. Recall that if F is the velocity field of a fluid, then circulation ${\oint }_{{C}_{r}}\text{F}·d\text{r}={\oint }_{{C}_{r}}\text{F}·\text{T}ds$ is a measure of the tendency of the fluid to move around ${C}_{r}.$ The reason for this is that $\text{F}·\text{T}$ is a component of F in the direction of T , and the closer the direction of F is to T , the larger the value of $\text{F}·\text{T}$ (remember that if a and b are vectors and b is fixed, then the dot product $\text{a}·\text{b}$ is maximal when a points in the same direction as b ). Therefore, if F is the velocity field of a fluid, then $\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}·\text{N}$ is a measure of how the fluid rotates about axis N . The effect of the curl is largest about the axis that points in the direction of N , because in this case $\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}·\text{N}$ is as large as possible.

To see this effect in a more concrete fashion, imagine placing a tiny paddlewheel at point ${P}_{0}$ ( [link] ). The paddlewheel achieves its maximum speed when the axis of the wheel points in the direction of curl F . This justifies the interpretation of the curl we have learned: curl is a measure of the rotation in the vector field about the axis that points in the direction of the normal vector N , and Stokes’ theorem justifies this interpretation. To visualize curl at a point, imagine placing a tiny paddlewheel at that point in the vector field.

Now that we have learned about Stokes’ theorem, we can discuss applications in the area of electromagnetism. In particular, we examine how we can use Stokes’ theorem to translate between two equivalent forms of Faraday’s law . Before stating the two forms of Faraday’s law, we need some background terminology.

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