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18.02 Multivariable Calculus
Fall 2007
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Twodimensional Flux
In this section and the next we give a different way of looking at Green's theorem which
both shows its significance for flow fields and allows us to give an intuitive physical meaning
for this rather mysterious equality between integrals.
We have seen that if
F is a force field and
C
a directed curve, then
(1)
work done by
F along
C
=
In words, we are integrating
F
.
T, the
tangential component
of
F, along the curve
C. In
component notation, if
F
=
Mi
+
Nj
,
then the above reads
Analogously now, we may integrate
F
.
n, the
normal component
F along
C. To
describe this, suppose the curve
C
is parametrized by the arclength s, increasing in the
positive direction on
C. The position vector for this parametrization and its corresponding
tangent vector are given respectively by
where we have used
t instead of
T since it is a unit vector its length is 1, as one can see
by dividing through by ds on both sides of
ds
=
J(dx)~
+
(d~)~
.
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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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