M344: Calculus III
Section 16.8
Stokes' Theorem¶
At this point it is natural to ask whether Green's theorem extends to curved surfaces embedded in $\mathbb{R}^3$. The answer is the following theorem.
Theorem (Stokes)¶
Let $S$ be an oriented piecewise-smooth surface that is bounded by a simple, closed, positively-oriented, piecewise-smooth curve $C$ in $\mathbb{R}^3$. Let $\mathbf{F}$ be a vector field whose component functions have continuous first partial derivatives on an open region in $\mathbb{R}^3$ that contains $S$. Then
$$
\int_C \mathbf{F}\cdot d\mathbf{r} = \int\!\!\!\!\int_S \text{curl}\, \mathbf{F}\cdot d\mathbf{S}.
$$
1. Exercise¶
Show that if $S$ is a plane region and $\partial S$ is a Jordan curve, then Stokes' theorem reduces to Green's theorem.
2. Proof of a Special Case¶
In the following video, we prove Stokes' theorem for the case when the surface $S$ is given by the graph of a function, $S = \Gamma(f,D)$.
Video proof
Deeper Considerations¶
Before we continue on to examples, let's take a moment to reflect on the significance of this theorem.
First, consider two surfaces $S_1$ and $S_2$ that share a boundary curve $C$, all of which satisfy the hypotheses of Stokes' theorem. Then
$$
\int\!\!\!\!\int_{S_1} \text{curl}\,\mathbf{F}\cdot d\mathbf{S} = \int_C \mathbf{F}\cdot d\mathbf{r} = \int\!\!\!\!\int_{S_2} \text{curl}\,\mathbf{F}\cdot d\mathbf{S}.
$$
Therefore the surface integral of $\text{curl}\,\mathbf{F}$ along any two surfaces with the same boundary curve are equal. This fact can be extremely useful when it is difficult to integrate along one surface, but easier to integral along another.
Next, we use Stokes' theorem to shed some more light on the physical interpretation of the curl of a vector field.
Video lecture
Finally, we are ready to prove once and for all that if $\text{curl}\,\mathbf{F} = \mathbf{0}$ on $\mathbb{R}^3$, then $\mathbf{F}$ is conservative on $\mathbb{R}^3$. (Recall the theorem from Section 16.3.)
Video lecture
Examples¶
We conclude this section by computing some examples.
3. Example¶
Let
$$
\mathbf{F}(x,y,z) = xyz\,\mathbf{i} + xy\,\mathbf{j} + x^2yz\,\mathbf{k}
$$
and $S$ be the surface consisting of the top and the four sides (but not the bottom) of the cube with vertices $(\pm 1, \pm 1, \pm 1)$, oriented outward.
Video solution
4. Example¶
Use Stokes' theorem to evaluate $\int_C \mathbf{F}\cdot d\mathbf{r}$, where
$$
\mathbf{F}(x,y,z) = \big\langle x + y^2, y + z^2, z + x^2 \big\rangle,
$$
and $C$ is the triangle with vertices $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$.
Video solution
5. Example¶
Use Stokes' theorem to evaluate $\int_C \mathbf{F}\cdot d\mathbf{r}$, where
$$
\mathbf{F}(x,y,z) = x^2y\,\mathbf{i} + \tfrac{1}{3}x^3\,\mathbf{j} + xy\,\mathbf{k}
$$
and $C$ is the curve of intersection of the hyperbolic paraboloid $z = y^2 - x^2$ and the cylinder $x^2 + y^2 = 1$, oriented counterclockwise as viewed from above.
Video solution
6. Example¶
If $S$ is a sphere and $\mathbf{F}$ satisfies the hypotheses of Stokes' theorem, show that
$$
\int\!\!\!\!\int_S \text{curl}\,\mathbf{F}\cdot d\mathbf{S} = 0.
$$
Video solution
Discussion¶
Questions? You can ask them here.