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18.02 Multivariable Calculus
Fall 2007
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View Full DocumentV4. Green's Theorem in Normal Form
1.
Green's theorem for
flux.
Let
F
=
M
i
+
N
j
represent a twodimensional flow field, and
C
a simple
closed curve, positively oriented, with interior R.
According to the previous section,
(1)
flux of
F
across
C
=
Notice that since the normal vector points outwards, away from R, the flux is positive where
the flow is out of R; flow into R counts as negative flux.
We now apply Green's theorem to the line integral in (1); first we write the integral in
standard form (dx first, then dy):
This gives us
Green's theorem in the normal form
Mathematically this is the same theorem as the tangential form of Green's theorem

all
we have done is to juggle the symbols M and N around, changing the sign of one of them.
What is different is the physical interpretation. The left side represents the flux of
F across
the closed curve
C. What does the right side represent?
2.
The twodimensional divergence.
Once again, let
F
=
Mi
+
N
j
.
We give a name to and a notation for
the integrand of the double integral on the right of (2):
dM
div
F
=

+
,
dN
the
divergence
of
F
.
dx
dy
Evidently div
F is a scalar function of two variables. To get at its physical meaning, look
at the small rectangle pictured. If
F is continuously differentiable, then div
F is a continuous
function, which is therefore approximately constant if the rectangle is small enough. We
apply (2) to the rectangle; the double integral is approximated by a product, since the
integrand is approximately constant:
(4) flux across sides of rectangle
z
(E

+

Y)
AA
,
AA
=
area of rectangle.
Because of the importance of this approximate relation, we give a more direct derivation
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 Spring '08
 Auroux
 Multivariable Calculus

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