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18.02 Multivariable Calculus
Fall 2007
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Stokes' Theorem
1.
Introduction; statement of the theorem.
The normal form of Green's theorem generalizes in 3space to the divergence theorem.
What is the generalization to space of the tangential form of Green's theorem? It says
where C is a simple closed curve enclosing the plane region R.
Since the left side represents work done going around a closed curve in the plane, its
natural generalization to space would be the integral
$
F dr representing work done going
around a closed curve in 3space.
In trying to generalize the righthand side of (I), the space curve C can only be the
boundary of some piece of surface
S

which of course will no longer be a piece of a plane.
So it is natural to look for a generalization of the form
gjc
F
.
dr
=
/L(something derived from F)dS
The surface integral on the right should have these properties:
a) If curl
F
=
0 in Bspace, then the surface integral should be 0; (for
F is then
a gradient field, by V12,
(4),
so the line integral is 0, by Vll, (12)).
b) If C is in the xyplane with
S as its interior, and the field
F does not depend
on
z
and has only a kcomponent, the righthand side should be
These
JL
curl
F dS
.
things suggest that the theorem we are looking for in space is
Stokes' theorem
For the hypotheses, first of all C should be a closed curve, since it is the boundary of S,
and it should be oriented, since we have to calculate a line integral over it.
S is an oriented surface, since we have to calculate the flux of curl
F
through it. This
means that
S is twosided, and one of the sides designated as positive; then the unit normal
n is the one whose base is on the positive side. (There is no "standard" choice for positive
side, since the surface
S is not closed.)
cubical surface:
no boundary
It is important that C and
Sbe compatibly oriented. By this we mean that the righthand
rule applies: when you walk in the positive direction on C, keeping
S to your left, then your
head should point in the direction of
n. The pictures give some examples.
2
V. VECTOR INTEGRAL CALCLUS
The field
F
=
Mi
+
N j
+
Pk should have continuous first partial derivatives, so that
we will be able to integrate curl
F. For the same reason, the piece of surface
S should be
piecewise smooth and should be finite
i.e., not go off to infinity in any direction, and have
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This note was uploaded on 05/04/2011 for the course MATH 18.02 taught by Professor Auroux during the Spring '08 term at MIT.
 Spring '08
 Auroux
 Multivariable Calculus

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