>>16662774Vector Fields in R2 and R3
The vectors in Figure 1 are air velocity vectors that indicate the wind speed and direction
at points 10 m above the surface elevation in the San Francisco Bay area. We see at a
glance from the largest arrows in part (a) that the greatest wind speeds at that time
occurred as the winds entered the bay across the Golden Gate Bridge. Part (b) shows the
very different wind pattern 12 hours earlier. Associated with every point in the air we can
imagine a wind velocity vector. This is an example of a velocity vector field.
(a) 6:00 PM (b) 6:00 AM
FIGURE 1 Velocity vector fields showing San Francisco Bay wind patterns on a particular spring day
Other examples of velocity vector fields are illustrated in Figure 2: ocean currents and
flow past an airfoil.
Nova Scotia
(a) Ocean currents off the coast of Nova Scotia (b) Airflow past an inclined airfoil
ONERA photograph, Werle, 1974
FIGURE 2
Velocity vector fields Another type of vector field, called a force field, associates a force vector with each
point in a region. An example is the gravitational force field that we will look at in
Example 4.
16.1
SECTION 16.1 Vector Fields 1125
In general, a vector field is a function whose domain is a set of points in R 2 (or R 3)
and whose range is a set of vectors in V2 (or V3).
1
The best way to picture a vector field is to draw the arrow representing the vector
Fsx, yd starting at the point sx, yd. Of course, itβs impossible to do this for all points sx, yd,
but we can form a reasonable impression of F by drawing vectors for a few representative
points in D as in Figure 3. Since Fsx, yd is a two-dimensional vector, we can write it in
terms of its component functions P and Q as follows:
Fsx, yd β Psx, yd i 1 Qsx, yd j β kPsx, yd, Qsx, ydl
or, for short, F β P i 1 Q j
Notice that P and Q are scalar functions of two variables and are sometimes called scalar
fields to distinguish them from vector fields.
2
A vector field F on R 3 is pictured in Figure 4. We can express it in terms of its com-
ponent functions P, Q, and R as
Fsx, y, zd β Psx, y, zd i 1 Qsx, y, zd j 1 Rsx, y, zd k
As with the vector functions in Section 13.1, we can define continuity of vector fields
and show that F is continuous if and only if its component functions P, Q, and R are
continuous.
We sometimes identify a point sx, y, zd with its position vector x β kx, y, zl and write
Fsxd instead of Fsx, y, zd. Then F becomes a function that assigns a vector Fsxd to a vec-
tor x.
EXAMPLE 1 A vector field on R 2 is defined by Fsx, yd β 2y i 1 x j. Describe F by
sketching some of the vectors Fsx, yd as in Figure 3.
SOLUTION Since Fs1, 0d β j, we draw the vector j β k0, 1l starting at the point s1, 0d
in Figure 5. Since Fs0, 1d β 2i, we draw the vector k21, 0l with starting point s0, 1d.
Continuing in this way, we calculate several other representative values of Fsx, yd in the
table and draw the corresponding vectors to represent the vector field in Figure 5.
sx, yd Fsx, yd sx, yd Fsx, yd
s1, 0d k0, 1l s21, 0d k0, 21l
s2, 2d k22, 2l s22, 22d k2, 22l
s3, 0d k0, 3l s23, 0d k0, 23l
s0, 1d k21, 0l s0, 21d k1, 0l
s22, 2d k22, 22l s2, 22d k2, 2l
s0, 3d k23, 0l s0, 23d k3, 0l
FIGURE 3
Vector field on R@
0
(x, y)
F(x, y)
x
y
FIGURE 4
Vector field on R#
y
0
z
x
(x, y, z)
F (x, y, z)
FIGURE 5
F(x, y)=_y i+x j
F (1, 0)
F (0, 3) F (2, 2)
0 x
y
2
_2
_2 2
Definition Let D be a set in R 2 (a plane region). A vector field on R 2 is a
function F that assigns to each point sx, yd in D a two-dimensional vector Fsx, yd.
Definition Let E be a subset of R 3. A vector field on R 3 is a function F that
assigns to each point sx, y, zd in E a three-dimensional vector Fsx, y, zd.
1126 CHAPTER 16 Vector Calculus
It appears from Figure 5 that each arrow is tangent to a circle with center the origin.
To confirm this, we take the dot product of the position vector x β x i 1 y j with the
vector Fsxd β Fsx, yd:
x οΏ½ Fsxd β sx i 1 y jd οΏ½ s2y i 1 x jd β 2xy 1 yx β 0
This shows that Fsx, yd is perpendicular to the position vector kx, yl and is therefore
tangent to a circle with center the origin and radius | x | β sx 2 1 y 2 . Notice also that
| Fsx, yd | β ss2yd2 1 x 2 β sx 2 1 y 2 β | x |
so the magnitude of the vector Fsx, yd is equal to the radius of the circle. β
Some graphing software is capable of plotting vector fields in two or three dimen-
sions. The results give a better impression