James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

1072 Chapter 16 Vector Calculus
z
an example of a vector field, called the gravitational field, because it associates a
vector [the force Fsxd] with every point x in space.
Formula 3 is a compact way of writing the gravitational field, but we can also write
it in terms of its component functions by using the facts that x − x i 1 y j 1 z k and
| x | − sx 2 1 y 2 1 z 2 :
x
y
Fsx, y, zd −
2mMGx
sx 2 1 y 2 1 z 2 d 3y2 i 1
2mMGy
sx 2 1 y 2 1 z 2 d 3y2 j 1
2mMGz
sx 2 1 y 2 1 z 2 d 3y2 k
The gravitational field F is pictured in Figure 14.
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FIGURE 14
Gravitational force field
Example 5 Suppose an electric charge Q is located at the origin. According to Coulomb’s
Law, the electric force Fsxd exerted by this charge on a charge q located at a point
sx, y, zd with position vector x − kx, y, zl is
4 Fsxd − «qQ
| x | 3 x
where « is a constant (that depends on the units used). For like charges, we have
qQ . 0 and the force is repulsive; for unlike charges, we have qQ , 0 and the force is
attractive. Notice the similarity between Formulas 3 and 4. Both vector fields are
examples of force fields.
Instead of considering the electric force F, physicists often consider the force per
unit charge:
Esxd − 1 q
Fsxd −
«Q
| x | 3 x
Then E is a vector field on R 3 called the electric field of Q.
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Gradient Fields
If f is a scalar function of two variables, recall from Section 14.6 that its gradient =f (or
grad f ) is defined by
=f sx, yd − f x sx, yd i 1 f y sx, yd j
Therefore =f is really a vector field on R 2 and is called a gradient vector field. Likewise,
if f is a scalar function of three variables, its gradient is a vector field on R 3 given by
_4 4
4
=f sx, y, zd − f x sx, y, zd i 1 f y sx, y, zd j 1 f z sx, y, zd k
Example 6 Find the gradient vector field of f sx, yd − x 2 y 2 y 3 . Plot the gradient
vector field together with a contour map of f. How are they related?
SOLUtion The gradient vector field is given by
FIGURE 15
_4
=f sx, yd − −f
−x i 1 −f
−y j − 2xy i 1 sx 2 2 3y 2 d j
Figure 15 shows a contour map of f with the gradient vector field. Notice that the
gradient vectors are perpendicular to the level curves, as we would expect from
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