16
Vector Calculus
This chapter is concerned with applying calculus in the context of
vector fields
.
A
two-dimensional vector field is a function
f
that maps each point (
x, y
) in
R
2
to a two-
dimensional vector
(
u, v
)
, and similarly a three-dimensional vector field maps (
x, y, z
) to
(
u, v, w
)
. Since a vector has no position, we typically indicate a vector field in graphical
form by placing the vector
f
(
x, y
) with its tail at (
x, y
). Figure 16.1 shows a representation
of the vector field
f
(
x, y
) =
(
x/
radicalbig
x
2
+
y
2
+ 4
,
−
y/
radicalbig
x
2
+
y
2
+ 4
)
. For such a graph to be
readable, the vectors must be fairly short, which is accomplished by using a different scale
for the vectors than for the axes. Such graphs are thus useful for understanding the sizes
of the vectors relative to each other but not their absolute size.
Vector fields have many important applications, as they can be used to represent many
physical quantities: the vector at a point may represent the strength of some force (gravity,
electricity, magnetism) or a velocity (wind speed or the velocity of some other fluid).
We have already seen a particularly important kind of vector field—the gradient. Given
a function
f
(
x, y
), recall that the gradient is
(
f
x
(
x, y
)
, f
y
(
x, y
)
)
, a vector that depends on
(is a function of)
x
and
y
. We usually picture the gradient vector with its tail at (
x, y
),
pointing in the direction of maximum increase. Vector fields that are gradients have some
particularly nice properties, as we will see. An important example is
F
=
(bigg
−
x
(
x
2
+
y
2
+
z
2
)
3
/
2
,
−
y
(
x
2
+
y
2
+
z
2
)
3
/
2
,
−
z
(
x
2
+
y
2
+
z
2
)
3
/
2
)bigg
,
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