Math 275 Notes (Ultman)
Topic 5.10: Curl and Stokes’ Theorem
Textbook Section: 16.8
From the Toolbox (what you need from previous classes):
Computing partial derivatives.
Computing the dot and cross products.
Knowing what a vector line integral is. Setting up and evaluating vector
line integrals.
Knowing what a vector surface integral (aka flux integral) is. Setting up
and evaluating vector surface integrals.
Related ideas: Green’s Theorem and the Divergence Theorem; the gradient of a function ∇f and the divergence of a vector field divF = ∇ · F .
Learning Objectives (New Skills) & Important Concepts
Learning Objectives (New Skills):
Compute the curl of a vector field.
Apply Stokes’ Theorem to evaluate line integrals.
Recognize when Stokes’ Theorem can be used.
Important Concepts:
Suppose C is a closed curve, and S is a surface that has C as a
boundary. Stokes’ Theorem relates the line integral of a vector field
F around the closed curve C, to a vector surface integral over S:
˛
¨
F · dr =
curlF · dS.
C
S
Green’s Theorem is the special case of Stokes’ Theorem, where the
vector field F , the curve C, and the surface S all lie in a coordinate
plane.
curlF = ∇ × F measures the local rotation of a vector field F .
The Big Picture
If C is a closed curve and S is a surface bounded by C, Stokes’ Theorem states
that the line integral of a vector field F around C is equal to the surface integral
of the curl of F over S:
¨
˛
curlF · dS.
F · dr =
S
C
Curl measures the local rotation of the vector field F . curlF · dS measures the
circulation (vector line integral) of F around an infinitesimal curve enclosing a
region of area dS:
˛
F · dS.
curlF · dS =
infinitesimal curve
So Stokes’ Theorem states that adding up the circulation of F around infinitesimally small curves inside S is the same as the total counterclockwise circulation
around the boundary curve of S.
Green’s Theorem is a special case of Stokes’ Theorem, where the vector
field, the curve, and the surface bounded by the curve all lie in a coordinate
plane.
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How to “Read” Stokes’ Theorem
Given a vector field F (x, y , z) = P (x, y , z) ˆı + Q(x, y , z) ˆ
 + R(x, y , z) k̂ ,
Stokes’ Theorem equates a vector line integral and vector surface integral:
˛
¨
F · dr =
C
curlF · dS
S
Type of Integral:
Vector Line Integral
(1-d)
Vector Surface Integral
(2-d)
Domain of Integration:
C
a closed curve in R3
S
a 2-d region bounded by C
F (x, y , z)
curlF = ∇ × F
Integrand:
More Details
◦ Computing Curl The curl of a vector field in Cartesian coordinates can
be computed by taking the cross product of the nabla operator and the
field:
F (x, y , z) = P (x, y , z) ˆı + Q(x, y , z) ˆ
 + R(x, y , z) k̂
curlF = ∇ × F
ˆı
ˆ

k̂
∂ ∂ ∂ = ∂x ∂y ∂z P Q R ∂R ∂Q
∂R ∂P
∂Q ∂P
=
−
ˆı −
−
ˆ
 +
−
k̂
∂y
∂z
∂x
∂z
∂x
∂y
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◦ Gradient, Curl, and Divergence
If f (x, y , z) is a scalar-valued function, and F (x, y , z) is a vector field,
then:
Gradient
Curl
Divergence
∇f
∇×F
∇·F
scalar-valued function vector field
→ vector field → vector field
curl(∇f ) = ~0
vector field
→ scalar-valued function
div(curlF ) = 0
Since the curl of a gradient field is always zero, we can use curl as a test
for conservative vector fields.
◦ What Curl Measures Curl can be interpreted as the local maximal rotation of a vector field. Imagine an infinitesimal paddlewheel at each point
in the domain of F :
? The direction of curlF at a point gives the axis of rotation about
which the paddlewheel spins the fastest.
? The magnitude |curlF | is related to the maximum speed of rotation
of the paddlewheel.
? A field F with curlF = ~0 is sometimes called irrotational.
◦ Geometric motivation for Stokes’ Theorem
? curlF · dS measures the circulation of the field F around an infinitesimal curve in the surface S, bounding a parallelogram of area
dS:
˛
curlF · dS =
F · dr .
infinitesimal curve
? The double integral “adds up” these infinitesimal circulations.
Curves on the interior of the surface S share segments with their
neighbors. The circulation of the field F along these shared seg-
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ments have the same absolute value, but opposite signs, so they
cancel.
Along the bounding curve C = ∂S, the infinitesimal curves have segments that don’t have a neighbor to cancel with. Adding all these
up leads to the total circulation around the bounding curve C.
◦ Green’s Theorem is a special case of Stokes’ Theorem
Suppose C is a closed curve (oriented counterclockwise) in the xy -plane.
Choose S to be the region in the xy -plane that is bounded by C. Then
dS = k̂ dA, where dA is the area element in the xy -plane.
If
field of the form F = P (x, y ) ˆı + Q(x, y ) ˆ
 , then curlF =
F is a vector
∂Q
∂P
k̂ . So by Stokes’ Theorem:
∂x − ∂y
curlF ·dS =
F ·dr =
C
¨ ¨
˛
S
S
∂Q ∂P
−
∂x
∂y
¨
k̂ · k̂ dA =
S
∂Q ∂P
−
dA
∂x ∂y
◦ Stokes’ Theorem & “Surface Independence” Recall that the Fundamental Theorem of Line Integrals implies that line integrals over conservative vector fields are path independent: if C1 and C2 are both paths
from the point P to the point Q, then:
ˆ
ˆ
∇f · dr = f (Q) − f (P ) =
∇f · dr
C1
C2
So, only the boundary points matter, not the path you take to get from
one to another.
Stokes’ theorem implies an analogous thing about surface integrals over
curl fields. If S1 and S2 are both surfaces with boundary curve C (and
the proper orientation relative to C) then:
¨
˛
¨
curlF · dS =
F · dr =
curlF · dS
S1
C
S2
So, only the boundary curve matters, not the surface that has the curve
as a boundary.
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Comparison of the Integral Theorems (Divergence Theorem,
Stokes’ Theorem, Green’s Theorem, FTLI, FTC)
ˆ
f 0 (t) dt
f (b) − f (a) =
FTC:
b
ˆa
f (Q) − f (P ) =
∇f · dr
C
˛
¨
∂Q ∂P
−
dA
F · dr =
∂y
C
D ∂x
‹
¨
F · dr =
curlF · dS
C
S
‹
˚
F · dS =
divF dV
FTLI:
Green’s:
Stokes’:
Divergence:
S
General form:
´
∂Ω
ω=
´
Ω
W
dω.
Technical Conditions
Requirements for using Stokes’ Theorem:
? Surface: S is oriented. The surface element dS points in the direction
of the orientation.
? Curve: C = ∂S is the boundary of S. C is closed, and oriented counterclockwise as you look down at the surface along dS. (Computationally, choosing the wrong orientation will result in the incorrect sign in
your final answer.)
? Field: Both the surface S and the boundary curve C are contained in
an open set (a “bubble” around the surface S and it’s boundary curve
C = ∂S) throughout which the field F is defined and curlF exists and is
and continuous.
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