LESSON 24
Stokes’ Theorem
READ:
Section 18.2
NOTES:
There are two operations related to the differential calculus of vector fields: the
curl
and the
divergence
. Like
the gradient of a function of several real variables, these are vector field analogs of the derivative of a function
of one variable. The names are rooted in their physical interpretations, which will be sketched a little later.
For now, let’s just give the definition of the curl. In the next, and last, lesson, we will define the divergence.
From now on, all vector fields will be in 3space, so that
→
F
(
→
x
) =
F
1
(
→
x
)
→
ı
+
F
2
(
→
x
)
→
+
F
3
(
→
x
)
→
k
. Of course,
by taking
F
3
to be 0, what follows can be used for vector fields in 2space as well.
The
curl
of a vector field is another vector field. The definition is
curl
→
F
=
∂F
3
∂y

∂F
2
∂z
→
ı
+
∂F
1
∂z

∂F
3
∂x
→
+
∂F
2
∂x

∂F
1
∂y
→
k .
The complicated looking formula for the
curl
is most easily remembered by thinking of the formal identity
curl
→
F
=
∇ ×
→
F
, where the formal cross product is computed using a determinant just as with vectors. See
example 1, page 1001 of the text to see how to get the formula right.
The curl shares some of the features of ordinary derivatives. For it is
linear
. That is,
curl
a
→
F
+
b
→
G
=
a curl
→
F
+
b curl
→
G.
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 Winter '11
 JerryMetzger
 Calculus, Derivative, Fundamental Theorem Of Calculus, Vector Calculus, Line integral, Vector field, Stokes' theorem

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