Jim Lambers
MAT 280
Spring Semester 2009-10
Lecture 23 Notes
These notes correspond to Section 13.8 in Stewart and Section 8.2 in Marsden and Tromba.
Stokes’ Theorem
Let 𝐶 be a simple, closed, positively oriented, piecewise smooth plane curve, and let 𝐷 be the
region that it encloses. According to one of the forms of Green’s Theorem, for a vector field F with
continuous first partial derivatives on 𝐷, we have
∫
∫ ∫
F ⋅ 𝑑r =
(curl F) ⋅ k 𝑑𝐴,
𝐶
𝐷
where k = ⟨0, 0, 1⟩.
By noting that k is normal to the region 𝐷 when it is embedded in 3-D space, we can generalize
this form of Green’s Theorem to more general surfaces that are enclosed by simple, closed, piecewise
smooth, positively oriented space curves. Let 𝑆 be an oriented, piecewise smooth surface that is
enclosed by a such a curve 𝐶. If we divide 𝑆 into several small patches 𝑆𝑖𝑗 , then these patches are
approximately planar. We can apply Green’s Theorem, approximately, to each patch by rotating
it in space so that its unit normal vector is k, and using the fact that rotating two vectors u and
v in space does not change the value of u ⋅ v.
Most of the line integrals along the boundary curves of each path cancel with one another due
to the positive orientation of all such boundary curves, and we are left with the line integral over
𝐶, the boundary of 𝑆. If we take the limit as the size of the patches approches zero, we then obtain
∫
∫ ∫
∫ ∫
F ⋅ 𝑑r =
curl F ⋅ 𝑑S =
curl F ⋅ n 𝑑𝑆,
𝐶
𝑆
𝑆
where n is the unit normal vector of 𝑆. This result is known as Stokes’ Theorem.
Stokes’ Theorem can be used either to evaluate an surface integral or an integral over the curve
that encloses it, whichever is easier.
Example (Stewart, Section 13.8, Exercise 2) Let F(𝑥, 𝑦, 𝑧) = ⟨𝑦𝑧, 𝑥𝑧, 𝑥𝑦⟩ and let 𝑆 be the part of
the paraboloid 𝑧 = 9 − 𝑥2 − 𝑦 2 that lies above the plane 𝑧 = 5, with upward orientation. By Stokes’
Theorem,
∫ ∫
∫
curl F ⋅ 𝑑S =
𝑆
F ⋅ 𝑑r
𝐶
where 𝐶 is the boundary curve of 𝑆, which is a circle of radius 2 centered at (0, 0, 5), and parallel
to the 𝑥𝑦-plane. It can therefore be parameterized by
𝑥 = 2 cos 𝑡,
𝑦 = 2 sin 𝑡,
1
𝑧 = 5,
0 ≤ 𝑡 ≤ 2𝜋.
Its tangent vector is then
r′ (𝑡) = ⟨−2 sin 𝑡, 2 cos 𝑡, 0⟩.
We then have
∫ ∫
∫
2𝜋
curl F ⋅ 𝑑S =
F(r(𝑡)) ⋅ r′ (𝑡) 𝑑𝑡
0
𝑆
∫
2𝜋
⟨10 sin 𝑡, 10 cos 𝑡, 4 cos 𝑡 sin 𝑡⟩ ⋅ ⟨−2 sin 𝑡, 2 cos 𝑡, 0⟩ 𝑑𝑡
=
0
∫
2𝜋
−20 sin2 𝑡 + 20 cos2 𝑡 𝑑𝑡
=
0
∫
2𝜋
= 20
cos 2𝑡 𝑑𝑡
0
=
10 sin 2𝑡∣2𝜋
0
= 0.
This result can also be obtained by noting that because F = ∇𝑓 , where 𝑓 (𝑥, 𝑦, 𝑧) = 𝑥𝑦𝑧, it follows
that curl F = 0. □
Example (Stewart, Section 13.8, Exercise 8) We wish to evaluate the line integral of F(𝑥, 𝑦, 𝑧) =
⟨𝑥𝑦, 2𝑧, 3𝑦⟩ over the curve 𝐶 that is the intersection of the cylinder 𝑥2 + 𝑦 2 = 9 with the plane
𝑥 + 𝑧 = 5.
To describe the surface 𝑆 enclosed by 𝐶, we use the parameterization
𝑥 = 𝑢 cos 𝑣,
𝑧 = 5 − 𝑢 cos 𝑣,
𝑦 = 𝑢 sin 𝑣,
0 ≤ 𝑢 ≤ 3,
0 ≤ 𝑣 ≤ 2𝜋.
Using
r𝑢 = ⟨cos 𝑣, sin 𝑣, − cos 𝑣⟩,
r𝑣 = ⟨−𝑢 sin 𝑣, 𝑢 cos 𝑣, 𝑢 sin 𝑣⟩,
we obtain
r𝑢 × r𝑣 = ⟨𝑢, 0, 𝑢⟩.
We then compute
〈
curl F =
∂ ∂ ∂
, ,
∂𝑥 ∂𝑦 ∂𝑧
〉
× ⟨𝑥𝑦, 2𝑧, 3𝑦⟩ = ⟨1, 0, 𝑥⟩.
Let 𝐷 be the domain of the parameters,
𝐷 = {(𝑢, 𝑣) ∣ 0 ≤ 𝑢 ≤ 3,
We then apply Green’s Theorem and obtain
∫
∫ ∫
F ⋅ 𝑑r =
curl F ⋅ 𝑑S
𝐶
𝑆
2
0 ≤ 𝑣 ≤ 2𝜋.
∫ ∫
curl F(r(𝑢, 𝑣)) ⋅ (r𝑢 × r𝑣 ) 𝑑𝐴
=
𝐷
∫
3 ∫ 2𝜋
⟨1, 0, 𝑢 cos 𝑣⟩ ⋅ ⟨𝑢, 0, 𝑢⟩ 𝑑𝐴
=
0
∫
0
3 ∫ 2𝜋
=
0
∫
𝑢 + 𝑢2 cos 𝑣 𝑑𝑣 𝑑𝑢
0
3
2𝜋
(𝑢𝑣 + 𝑢2 sin 𝑣)0 𝑑𝑣 𝑑𝑢
0
∫ 3
𝑢 𝑑𝑢 𝑑𝑢
= 2𝜋
=
0
3
𝑢2 = 2𝜋
2 0
= 9𝜋.
□
Stokes’ Theorem can also be used to provide insight into the physical interpretation of the curl
of a vector field. Let 𝑆𝑎 be a disk of radius 𝑎 centered at a point 𝑃0 , and let 𝐶𝑎 be its boundary.
Furthermore, let v be a velocity field for a fluid. Then the line integral
∫
v ⋅ 𝑑r = 𝑖𝑛𝑡𝐶𝑎 v ⋅ T 𝑑𝑠,
𝐶𝑎
where T is the unit tangent vector of 𝐶𝑎 , measures the tendency of the fluid to move around 𝐶𝑎 .
This is because this measure, called the circulation of v around 𝐶𝑎 , is greatest when the fluid
velocity vector is consistently parallel to the unit tangent vector. That is, the circulation around
𝐶𝑎 is maximized when the fluid follows the path of 𝐶𝑎 .
Now, by Stokes’ Theorem,
∫
∫ ∫
∫ ∫
v ⋅ 𝑑r =
curl v ⋅ 𝑑S =
curl v ⋅ n 𝑑𝑆 ≈ curl V(𝑃0 ) ⋅ n(𝑃0 ) 𝑑𝑆 ≈ 𝜋𝑎2 curl v(𝑃0 ) ⋅ n(𝑃0 ).
𝐶𝑎
𝑆𝑎
𝑆𝑎
As 𝑎 → 0, and 𝑆𝑎 collapses to the point 𝑃0 , this approximation improves, and we obtain
∫
1
curl v(𝑃0 ) ⋅ n(𝑃0 ) = lim
v ⋅ 𝑑r.
𝑎→0 𝜋𝑎2 𝐶𝑎
This shows that circulation is maximized when the axis around which the fluid is circulating, n(𝑃0 ),
is parallel to curl v. That is, the direction of curl v indicates the axis around which the greatest
circulation occurs.
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Practice Problems
Practice problems from the recommended textbooks are:
∙ Stewart: Section 13.8, Exercises 1-7 odd
∙ Marsden/Tromba: Section 8.2, Exercises 5-11 odd
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