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• 3.4 Gradient, Divergence, Curl, and the Del Operator 227

LetF: X Rn Rn be a continuous vector field. Let (a, b) bean interval inR that contains0. (Think of (a, b)asatime inter-val.) A flow of F is a differentiable function : X (a, b)Rn of n + 1 variables such that

t(x, t) = F((x, t)); (x, 0) = x.

Intuitively, we think of (x, t) as the point at time t on the flowline of F that passes through x at time 0. (See Figure 3.37.)Thus, the flow of F is, in a sense, the collection of all flow linesof F. Exercises 2631 concern flows of vector fields.

(x, 0) = x

(x, t)F( (x, t))

Figure 3.37 The flow of the vector field F.

26. Verify that

:R2 R R2,

(x, y, t) =(

x + y2

et + x y2

et ,

x + y2

et + y x2

et)

is a flow of the vector field F(x, y) = (y, x).

27. Verify that

:R2 R R2,(x, y, t) = (y sin t + x cos t, y cos t x sin t)

is a flow of the vector field F(x, y) = (y,x).28. Verify that

:R3 R R3,(x, y, z, t) = (x cos 2t y sin 2t, y cos 2t+ x sin 2t, zet )

is a flow of the vector field F(x, y, z) = 2y i+2x j z k.

29. Show that if : X (a, b) Rn is a flow of F, then,for a fixed point x0 in X , the map x: (a, b) Rn givenby x(t) = (x0, t) is a flow line of F.

30. If is a flow of the vector field F, explain why((x, t), s) = (x, s + t). (Hint: Relate the value ofthe flow at (x, t) to the flow line of F through x. Youmay assume the fact that the flow line of a continuousvector field at a given point and time is determineduniquely.)

31. Derive the equation of first variation for a flow of avector field. That is, if F is a vector field of class C1with flow of class C2, show that

tDx(x, t) = DF((x, t))Dx(x, t).

Here the expression Dx(x, t) means to differentiate with respect to the variables x1, x2, . . . , xn , that is,by holding t fixed.

3.4 Gradient, Divergence, Curl, and the DelOperator

In this section, we consider certain types of differentiation operations on vectorand scalar fields. These operations are as follows:

1. The gradient, which turns a scalar field into a vector field.2. The divergence, which turns a vector field into a scalar field.3. The curl, which turns a vector field into another vector field. (Note: The curl

will be defined only for vector fields on R3.)

We begin by defining these operations from a purely computational point of view.Gradually, we shall come to understand their geometric significance.

The Del OperatorThe del operator, denoted , is an odd creature. It leads a double life as bothdifferential operator and vector. In Cartesian coordinates on R3, del is defined by

• 228 Chapter 3 Vector-Valued Functions

the curious expression

= i x

+ j y

+ k z

. (1)

The empty partial derivatives are the components of a vector that awaits suitablescalar and vector fields on which to act. Del operates on (i.e., transforms) fieldsvia multiplication of vectors, interpreted by using partial differentiation.

For example, if f : X R3 R is a differentiable function (scalar field),the gradient of f may be considered to be the result of multiplying the vector by the scalar f , except that when we multiply each component of by f , weactually compute the appropriate partial derivative:

f (x, y, z) =(i

x+ j

y+ k

z

)f (x, y, z) = f

xi+ f

yj+ f

zk.

The del operator can also be defined in Rn , for arbitrary n. If we takex1, x2, . . . , xn to be coordinates for Rn , then del is simply

=(

x1,

x2, . . . ,

xn

)= e1

x1+ e2

x2+ + en

xn, (2)

where ei = (0, . . . , 1, . . . , 0), i = 1, . . . , n, is the standard basis vector for Rn .

The Divergence of a Vector FieldWhereas taking the gradient of a scalar field yields a vector field, the process oftaking the divergence does just the opposite: It turns a vector field into a scalarfield.

DEFINITION 4.1 Let F: X Rn Rn be a differentiable vector field.Then the divergence of F, denoted div F or F (the latter read del dotF), is the scalar field

div F = F = F1x1

+ F2x2

+ + Fnxn

,

where x1, . . . , xn are Cartesian coordinates for Rn and F1, . . . , Fn are thecomponent functions of F.

It is essential that Cartesian coordinates be used in the formula of Definition 4.1.(Later in this section we shall see what divF looks like in cylindrical and sphericalcoordinates for R3.)

EXAMPLE 1 If F = x2yi+ xzj+ xyzk, then

div F = x

(x2y)+ y

(xz)+ z

(xyz) = 2xy + 0+ xy = 3xy. !

• 3.4 Gradient, Divergence, Curl, and the Del Operator 229

The notation for the divergence involving the dot product and the del operatoris especially apt: If we write

F = F1e1 + F2e2 + + Fnen,then,

F =(e1

x1+ e2

x2+ + en

xn

) (F1e1 + F2e2 + + Fnen)

= F1x1

+ F2x2

+ + Fnxn

,

where, once again, we interpret multiplying a function by a partial differentialoperator as performing that partial differentiation on the given function.

Intuitively, the value of the divergence of a vector field at a particular pointgives a measure of the net mass flow or flux density of the vector field inor out of that point. To understand what such a statement means, imagine thatthe vector field F represents velocity of a fluid. If F is zero at a point, thenthe rate at which fluid is flowing into that point is equal to the rate at whichfluid is flowing out. Positive divergence at a point signifies more fluid flowing outthan in, while negative divergence signifies just the opposite. We will make theseassertions more precise, even prove them, when we have some integral vectorcalculus at our disposal. For now, however, we remark that a vector field F suchthat F = 0 everywhere is called incompressible or solenoidal.EXAMPLE 2 The vector field F = x i+ yj has

F = x

(x)+ y

(y) = 2.

This vector field is shown in Figure 3.38. At any point in R2, the arrow whosetail is at that point is longer than the arrow whose head is there. Hence, there isgreater flow away from each point than into it; that is, F is diverging at everypoint. (Thus, we see the origin of the term divergence.)

x

y

Figure 3.38 The vector fieldF = x i+ yj of Example 2.

x

y

Figure 3.39 The vector fieldG = x i yj of Example 2.

The vector field G = x i yj points in the direction opposite to the vectorfield F of Figure 3.38 (see Figure 3.39), and it should be clear howGs divergenceof 2 is reflected in the diagram. !

EXAMPLE 3 The constant vector field F(x, y, z) = a shown in Figure 3.40is incompressible. Intuitively, we can see that each point of R3 has an arrowrepresenting a with its tail at that point and another arrow, also representing a,with its head there.

The vector field G = yi xj has

G = x

(y)+ y

(x) 0.

A sketch ofG reveals that it looks like the velocity field of a rotating fluid, withouteither a source or a sink. (See Figure 3.41.) !

The Curl of a Vector FieldIf the gradient is the result of performing scalar multiplication with the deloperator and a scalar field, and the divergence is the result of performing thedot product of del with a vector field, then there seems to be only one simple

• 230 Chapter 3 Vector-Valued Functions

y

x

z

Figure 3.40 The constant vectorfield F = a.

x

y

Figure 3.41 The vector fieldG = yi xj resembles thevelocity field of a rotating fluid.

differential operation left to be built from del. We call it the curl of a vector fieldand define it as follows:

DEFINITION 4.2 Let F: X R3 R3 be a differentiable vector field onR3 only. The curl of F, denoted curlF or F (the latter read del crossF), is the vector field

curl F = F =(i

x+ j

y+ k

z

) (F1i+ F2j+ F3k)

=

i j k

/x /y /zF1 F2 F3

=(F3y F2

z

)i+

(F1z F3

x

)j+

(F2x

F1y

)k.

There is no good reason to remember the formula for the components of thecurlinstead, simply compute the cross product explicitly.

EXAMPLE 4 If F = x2yi 2xzj+ (x + y z)k, then

F =

i j k

/x /y /zx2y 2xz x + y z

=(

y(x + y z)

z(2xz)

)i+

(

z(x2y)

x(x + y z)

)j

+(

x(2xz)

y(x2y)

)k

= (1+ 2x)i j (x2 + 2z)k. !

• 3.4 Gradient, Divergence, Curl, and the Del Operator 231

Figure 3.42 A twig in a pond where water moves with velocity given by a vector field F. In the left figure, the twig doesnot rotate as it travels, so curl F = 0. In the right figure, curl F != 0, since the twig rotates.

One would think that, with a name like curl, F should measure howmuch a vector field curls. Indeed, the curl does measure, in a sense, the twistingor circulation of a vector field, but in a subtle way: Imagine that F represents thevelocity of a stream or lake. Drop a small twig in the lake and watch it travel.The twig may perhaps be pushed by the current so that it travels in a large circle,but the curl will not detect this. What curl Fmeasures is how quickly and in whatorientation the twig itself rotates as it moves. (See Figure 3.42.) We prove thisassertion much later, when we know something about line and surface integrals.For now, we simply point out some terminology: A vector field F is said to beirrotational if F = 0 everywhere.

EXAMPLE 5 Let F = (3x2z + y2) i+ 2xy j+ (x3 2z)k. Then

F =

i j k

/x /y /z

3x2z + y2 2xy x3 2z

=(

y(x3 2z)

z(2xy)

)i+

(

z(3x2 z + y2)

x(x3 2z)

)j

+(

x(2xy)

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