18.02 Multivariable Calculus (Spring 2009): Lecture 34
Stokes’ Theorem
May 1
Reading Material:
From
Simmons
: 21.5. From
Lecture Notes
: V13.
Last time:
Line Integrals, conservative vector fields, Curl and potential functions in 3D
Today:
Stokes’ Theorem.
2
The theorems we know
We recall here the Green’s Theorem in Normal Form for flux, its generalization in 3D, that is the
Divergence Theorem, and the Green’s Theorem for work in 2D.
Green’s Theorem in Normal Form
: This is a theorem in 2D and it is used to compute the flux
of a vector field
F
=
M
ˆ
i
+
N
ˆ
j
through a simple, piecewise smooth and closed curve in terms of a
double integral over the enclosed region
R
, more precisely
Theorem 1.
Given a continuously differentiable vector field
F
=
M
ˆ
i
+
N
ˆ
j
and a clockwise oriented
simple, piecewise smooth and closed curve
C
enclosing a region
R
Flux across
C
=
C
F
·
ˆ
n ds
=
C
(
N
ˆ
i

M
ˆ
j
)
·
dr
=
R
[
M
x
+
N
y
]
dA.
(2.1)
As we know there is a generalization of this in 3D
Divergence Theorem
: This theorem is used to compute the flux of a 3D vector filed
F
=
M
ˆ
i
+
N
ˆ
j
+
P
ˆ
k
across an oriented closed surface
S
by a triple integral over the space region
D
enclosed
by
S
. More precisely
1
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Theorem 2.
Assume the 3D vector filed
F
=
M
ˆ
i
+
N
ˆ
j
+
P
ˆ
k
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 Spring '06
 HartleyRogers
 Integrals, Multivariable Calculus, Vector Calculus, Manifold

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