9.7 Curl and Divergence

INTRODUCTION

In Section 9.1 we introduced the concept of vector function of one variable. In this section we examine vector functions of two and three variables.

Vector Fields

Vector functions of two and three variables,

are also called vector fields. For example, the motion of a wind or a fluid can be described by means of a velocity field because a vector can be assigned at each point representing the velocity of a particle at the point. See FIGURE 9.7.1(a) and 9.7.1(b). The concept of a force field plays an important role in mechanics, electricity, and magnetism. See Figure 9.7.1(c) and 9.7.1(d).

4 visual representations. Visual representation a. Caption. Airflow around an airplane wing. Figure. The airplane wing is slightly angled downward. Several arrows are drawn moving above and under the wing. The arrows representing the airflow moving above the wing, are labeled V subscript a, and arrows, representing the airflow moving under the wing, are labeled V subscript b. Visual representation b. Caption. Laminar flow of blood in an artery; cylindrical layers of blood flow faster near the center of the artery. Figure. 4 cylinders representing cylindrical layers of blood flow, of decreasing sizes, are fitted one inside the other. Several right facing arrows indicate the direction of the blood flow. Visual representation c. Caption. Inverse square force field; magnitude of the attractive force is large near the particle. Figure. Several arrows, arranged in several circular layers, point toward a point at the center. Visual representation d. Caption. Lines of force around two equal positive charges. Figure. Several arrows, arranged in a circular pattern around 2 positive charges placed side by side, point outward. The arrows pointing away from one positive charge also move away from the arrows pointing away from the other positive charge.

FIGURE 9.7.1 Various vector fields

EXAMPLE 1 Two-Dimensional Vector Field

Graph the two-dimensional vector field F(x, y) = –yi + xj.

SOLUTION

One manner of proceeding is simply to choose points in the xy-plane and then graph the vector F at that point. For example, at (1, 1) we would draw the vector F(1, 1) = –i +j.

For the given vector field it is possible to systematically draw vectors of the same length. Observe that F = , and so vectors of the same length k must lie along the curve defined by = k; that is, at any point on the circle x2 + y2 = k2 a vector would have length k. For simplicity let us choose circles that have some points on them with integer coordinates. For example, for k = 1, k = , and k = 2, we have:

x2 + y2 = 1: At the points (1, 0), (0, 1), (–1, 0), (0, –1) the corresponding vectors j, –i, –j, i have the same length 1.

x2 + y2 = 2: At the points (1, 1), (–1, 1), (–1, –1), (1, –1) the corresponding vectors –i + j, –i – j, i – j, i + j have the same length .

x2 + y2 = 4: At the points (2, 0), (0, 2), (–2, 0), (0, –2) the corresponding vectors 2j, –2i, –2j, 2i have the same length 2.

The vectors at these points are shown in FIGURE 9.7.2. ≡

A graph. 3 concentric circles and twelve vectors are graphed on an x y coordinate plane. The first circle passes through the following marked points: (negative 2, 0), (0, 2), (2, 0), (0, negative 2). The second circle passes through the following marked points: (negative 1, 1), (1, 1), (1, negative 1), (negative 1, negative 1). The third circle passes through the following marked points: (negative 1, 0), (0, 1), (1, 0), (0, negative 1). The first vector begins at the point (0, 1) and ends at the point (1, 1). The second vector begins at the point (1, 1) and ends at the point (negative 2, 0). The third vector begins at the point (negative 2, 0) and points vertically down. The fourth vector begins at the point (negative 1, 0) and ends at the point (negative 1, negative 1). The fifth vector begins at the point (negative 1, negative 1) and ends at the point (0, negative 2). The sixth vector begins at the point (0, negative 2) and points horizontally to the right. The seventh vector begins at the point (0, negative 1) and ends at the point (1, negative 1). The eighth vector begins at the point (1, negative 1) and ends at the point (2, 0). The ninth vector begins at the point (2, 0) and points vertically up. The tenth vector begins at the point (1, 0) and ends at the point (1, 1). The eleventh vector begins at the point (1, 1) and ends at the point (0, 2). The twelfth vector begins at the point (0, 2) and points horizontally to the left.

FIGURE 9.7.2 Vector field in Example 1

In Section 9.5 we saw that the del operator

combined with a scalar function ϕ(x, y, z) produces a vector field

called the gradient of or a gradient field. The del operator can also be combined with a vector field F(x, y, z) = P(x, y, z)i + Q(x, y, z)j + R(x, y, z)k in two different ways: in one case producing another vector field and in the other producing a scalar function. We will assume hereafter that P, Q, and R have continuous partial derivatives.

DEFINITION 9.7.1 Curl

The curl of a vector field F = Pi + Qj + Rk is the vector field

.

In practice, curl F can be computed from the cross product of the del operator and the vector F:

(1)

There is another combination of partial derivatives of the component functions of a vector field that occurs frequently in science and engineering. Before we state the next definition, consider the following motivation.

If F(x, y, z) = P(x, y, z)i + Q(x, y, z)j + R(x, y, z)k is the velocity field of a fluid, then as shown in FIGURE 9.7.3, the volume of the fluid flowing through an element of surface area ΔS per unit time—that is, the flux of the vector field F through the area ΔS—is approximated by

(height)(area of base) = (compn F) ΔS = (F · n) ΔS, (2)

where n is a unit vector normal to the surface. Now consider the rectangular parallelepiped shown in FIGURE 9.7.4. To compute the total flux of F through its six sides in the outward direction we first compute the total flux out of parallel faces. The area of face F1 is Δx Δz and its outward unit normal is –j, and so by (2) the flux of F through F1 is approximately

A rectangular parallelepiped and a left inclining parallelepiped share a common base that is shaded and labeled delta S. The height of the rectangular parallelepiped is labeled comp subscript n F. A vector labeled F begins at the center of the common base, and goes up to the left at the center of the parallelepiped and parallel to its lateral faces. A second vector labeled n, begins at the center of the common base, and goes vertically up at the center of the rectangular parallelepiped.

FIGURE 9.7.3 Fluid flow through element of area ΔS

A graph. A rectangular parallelepiped is graphed on a three dimensional x y z coordinate system. The bottom left vertex of the face at the back is at the point (x y z). The bottom edge of the face in the front is labeled delta y. The bottom edge of the face on the right is labeled delta x. The vertical edge at the back of the face on the right is labeled delta z. The front face of the rectangular parallelepiped is labeled F subscript 1, and the right face is shaded and labeled F subscript 2.

FIGURE 9.7.4 What is total flux of vector field through this element?

F · (–j) Δx Δz = –Q(x, y, z) Δx Δz.

The flux out of face F2, whose outward normal is j, is approximated by

(F · j) Δx Δz = Q(x, y + Δy, z) Δx Δz.

Consequently the total flux out of these parallel faces is

Q(x, y + Δy, z) Δx Δz + (–Q(x, y, z) Δx Δz) = [Q(x, y + Δy, z) – Q(x, y, z)] Δx Δz. (3)

By multiplying (3) by Δy/Δy and recalling the definition of a partial derivative, we get for Δy close to 0,

Arguing in exactly the same manner, we see that the contributions to the total flux out of the parallelepiped from the two faces parallel to the yz-plane, and from the two faces parallel to the xy-plane, are, in turn,

Adding the results, we see that the net flux of F out of the parallelepiped is approximately

By dividing the last expression by Δx Δy Δz, we get the outward flux of F per unit volume:

It is this combination of partial derivatives that is given a special name.

DEFINITION 9.7.2 Divergence

The divergence of a vector field F = Pi + Qj + Rk is the scalar function

.

Observe that div F can also be written in terms of the del operator as

(4)

EXAMPLE 2 Curl and Divergence

If F = (x2y3 – z4)i + 4x5y2zj – y4z6k, find (a) curl F, (b) div F, and (c) div(curl F).

SOLUTION

(a) From (1),

(b) From (4),

(c) From Definition 9.7.2 and part (a) we find

≡

We ask you to prove the following two important properties. If f is a scalar function with continuous second partial derivatives, then

curl(grad f) = ∇ × ∇f = 0. (5)

Also, if F is a vector field having continuous second partial derivatives, then

div(curl F) = ∇ ·(∇ × F) = 0. (6)

See part (c) of Example 2 and Problems 29 and 30 in Exercises 9.7.

Physical Interpretations

The word curl was introduced by James Clerk Maxwell (1831–1879), a Scottish physicist, in his studies of electromagnetic fields. However, the curl is easily understood in connection with the flow of fluids. If a paddle device, such as shown in FIGURE 9.7.5, is inserted in a flowing fluid, then the curl of the velocity field F is a measure of the tendency of the fluid to turn the device about its vertical axis w. If curl F = 0, then the flow of the fluid is said to be irrotational, which means that it is free of vortices or whirlpools that would cause the paddle to rotate.* In FIGURE 9.7.6 the axis w of the paddle points straight out of the page.

A paddle device consists of 2 vertical planes labeled A and B intersecting each other perpendicularly. The intersection of the 2 planes is a vertical line. A vector labeled w, begins at the top of the intersection line, and points vertically up.

FIGURE 9.7.5 Paddle device

2 visual representations. Visual representation a. Caption. Irrotational flow. Figure. 6 horizontal lines contain identical horizontal vectors pointing to the right. 3 paddle devices, with their axes pointing straight out of the page, are placed side by side between the third and fourth horizontal lines. 4 identical and smaller horizontal right facing vectors are placed to the left, between and to the right of the 3 paddle devices respectively. Visual representation b. Caption. Rotational flow. Figure. 3 paddle devices, with their axes pointing straight out of the page, are placed side by side between 2 horizontal lines. 3 identical horizontal right facing vectors are placed to the left and between the 3 paddle devices respectively. 3 curved lines, representing vortices or whirlpools, are placed just above and under each of the 3 paddle devices.

FIGURE 9.7.6 Irrotational flow in (a); rotational flow in (b)

In the motivational discussion leading to Definition 9.7.2 we saw that the divergence of a velocity field F near a point P(x, y, z) is the flux per unit volume. If div F(P) > 0, then P is said to be a source for F, since there is a net outward flow of fluid near P; if div F(P) < 0, then P is said to be a sink for F, since there is a net inward flow of fluid near P; if div F(P) = 0, there are no sources or sinks near P. See FIGURE 9.7.7.

2 visual representations. Visual representation a. Caption. Div F(P) > 0; P a source. Figure. The region inside a boundary is shaded and contains a point labeled P. 6 identical vectors arranged in a circular pattern around and close to the point P, point outward away from the point P. Visual representation b. Caption. Div F(P) < 0; P a sink. Figure. The region inside a boundary is shaded and contains a point labeled P. 6 identical vectors arranged in a circular pattern around and close to the point P, point inward toward the point P.

FIGURE 9.7.7 P a source in (a); P a sink in (b)

The divergence of a vector field can also be interpreted as a measure of the rate of change of the density of the fluid at a point. In other words, div F is a measure of the fluid’s compressibility. If ∇ · F = 0, the fluid is said to be incompressible. In electromagnetic theory, if ∇ · F = 0, the vector field F is said to be solenoidal.

9.7 Exercises Answers to selected odd-numbered problems begin on page ANS-25.

In Problems 1–6, graph some representative vectors in the given vector field.

  1. F(x, y) = xi + yj
  2. F(x, y) = –xi + yj
  3. F(x, y) = yi + xj
  4. F(x, y) = xi + 2yj
  5. F(x, y) = yj
  6. F(x, y) = xj

In Problems 7–16, find the curl and the divergence of the given vector field.

  1. F(x, y, z) = xzi + yzj + xyk
  2. F(x, y, z) = 10yzi + 2x2zj + 6x3k
  3. F(x, y, z) = 4xyi + (2x2 + 2yz)j + (3z2 + y2)k
  4. F(x, y, z) = (x – y)3i + e−yzj + xye2yk
  5. F(x, y, z) = 3x2yi + 2xz3j + y4k
  6. F(x, y, z) = 5y3i + (x3y2 – xy)j – (x3yz – xz)k
  7. F(x, y, z) = xe−zi + 4yz2j + 3ye−zk
  8. F(x, y, z) = yz ln xi + (2x – 3yz)j + xy2z3k
  9. F(x, y, z) = xyexi – x3yzezj + xy2eyk
  10. F(x, y, z) = x2 sin yzi + z cos xz3j + ye5xyk

In Problems 17–24, let a be a constant vector and r = xi + yj + zk. Verify the given identity.

  1. div r = 3
  2. curl r = 0
  3. (a × ∇) × r = –2a
  4. ∇ × (a × r) = 2a
  5. ∇ · (a × r) = 0
  6. a × (∇ × r) = 0
  7. ∇ × [(r · r)a] = 2(r × a)
  8. ∇ · [(r · r)a] = 2(r · a)

In Problems 25–32, verify the given identity. Assume continuity of all partial derivatives.

  1. ∇ · (F + G) = ∇ · F + ∇ · G
  2. ∇ × (F + G) = ∇ × F + ∇ × G
  3. ∇ · (fF) = f(∇ · F) + F · ∇f
  4. ∇ × (fF) = f(∇ × F) + (∇f) × F
  5. curl(grad f) = 0
  6. div(curl F) = 0
  7. div(F × G) = G · curl F – F · curl G
  8. curl(curl F + grad f) = curl(curl F)
  9. Show that

    ∇ · ∇f = .

    This is known as the Laplacian and is also written ∇2f.

  10. Show that ∇ · (f ∇f) = f ∇2f + ∇f 2, where ∇2f is the Laplacian defined in Problem 33. [Hint: See Problem 27.]
  11. Find curl(curl F) for the vector field F = xyi + 4yz2j + 2xzk.
    1. Assuming continuity of all partial derivatives, show that curl(curl F) = −∇2F + grad(div F), where

      ∇2F = ∇2(Pi + Qj + Rk) = ∇2Pi + ∇2Qj + ∇2Rk.

    2. Use the identity in part (a) to obtain the result in Problem 35.
  13. Any scalar function f for which ∇2f = 0 is said to be harmonic. Verify that f(x, y, z) = (x2 + y2 + z2)−1/2 is harmonic except at the origin. ∇2f = 0 is called Laplace’s equation.
  14. Verify that

    f(x, y) = arctan , x2 + y2 ≠ 1

    satisfies Laplace’s equation in two variables

    ∇2f = = 0.

  15. Let r = xi + yj + zk be the position vector of a mass m1 and let the mass m2 be located at the origin. If the force of gravitational attraction is

    ,

    verify that curl F = 0 and div F = 0, r ≠ 0.

  16. Suppose a body rotates with a constant angular velocity ω about an axis. If r is the position vector of a point P on the body measured from the origin, then the linear velocity vector v of rotation is v = ω × r. See FIGURE 9.7.8. If r = xi + yj + zk and ω = ω1i + ω2j + ω3k, show that ω = curl v.
    A vertical line segment labeled axis, connects 2 marked points. The point at the bottom is labeled O. The marked point at the top is the center of a circle that is perpendicular to the axis. A point labeled P is on the circle. A line segment labeled r, connects the points O and P. A vector labeled w begins at the marked point at the top, and points vertically up. A vector labeled v is tangent to the circle at the point P.

    FIGURE 9.7.8 Rotating body in Problem 40

In Problems 41 and 42, assume that f and g have continuous second partial derivatives. Show that the given vector field is solenoidal. [Hint: See Problem 31.]

  1. F = ∇f × ∇g
  2. F = ∇f × (f ∇g)
  3. The velocity vector field for the two-dimensional flow of an ideal fluid around a cylinder is given by

    for some positive constant A. See FIGURE 9.7.9.

    (a) Show that when the point (x, y) is far from the origin, F(x, y) ≈ Ai.

    (b) Show that F is irrotational.

    (c) Show that F is incompressible.

    A graph. A circle is graphed on an x y coordinate plane. The center of the circle is at the origin. Several vectors, representing the vector field, enter the left side of the viewing window in the second and third quadrants, move to the right going above and under the circle, and exit the right side of the viewing window in the first and fourth quadrants.

    FIGURE 9.7.9 Vector field in Problem 43

  4. If E = E(x, y, z, t) and H = H(x, y, z, t) represent electric and magnetic fields in empty space, then Maxwell’s equations are

    where c is the speed of light. Use the identity in Problem 36(a) to show that E and H satisfy

  5. Consider the vector field F = x2yzi – xy2zj + (z + 5x)k. Explain why F is not the curl of another vector field G.

 

*In science texts the word rotation is sometimes used instead of curl. The symbol curl F is then replaced by rot F.