Marsden-3620111 VC September 27, 2011 10:5 258
258 Vector-Valued Functions
Because ∇ · V thus represents the diminution of density or the rate at which matter is leaving a point per unit volume per unit time, it is called the divergence. Maxwell employed the term convergence to denote the rate at which fluid approaches a point per unit volume per unit time. This is the negative of the divergence. In the case that the fluid is incompressible, as much matter must leave the cube as enters it. The total change of contents must therefore be zero. For this reason, the characteristic differential equation that any incompressible fluid must satisfy is
∇ · V = 0,
whereV is the velocity of the fluid. This equation is often known as the hydrodynamic equation. It is satisfied by any flow of water, since water is practically incompressible. The great importance of the equation for work in electricity is due to the fact that according to Maxwell’s hypothesis, electric displacement obeys the same laws as an incompressible fluid. If, then,D is the electric displacement,
divD = ∇ · D = 0.
Gibbs’ interpretation of curl was much like the one we gave in Example 9 for the rotation of a rigid body. Wilson remarks that an analysis of the meaning of curl for fluid motion was “rather difficult.” It remains a bit elusive, even today, as can be seen from our discussion following Example 9. We provide another interpretation in Chapter 8.
Find the divergence of the vector fields in Exercises 1 to 4.
1. V(x, y, z)= ex yi− ex yj+ eyzk 2. V(x, y, z)= yzi + xzj + xyk
3. V(x, y, z)= xi + ( y + cos x)j + (z + ex y)k 4. V(x, y, z)= x2i+ (x + y)2j+ (x + y + z)2k
5. Figure 4.4.11 shows some flow lines and moving regions for a fluid moving in the plane-field velocity field V.
Where is div V > 0, and also where is div V < 0?
6. Let V (x, y, z)= xi be the velocity field of a fluid in space. Relate the sign of the divergence with the rate of change of volume under the flow.
7. Sketch a few flow lines for F(x, y)= yi. Calculate ∇ · F and explain why your answer is consistent with your sketch.
8. Sketch a few flow lines for F(x, y)= −3xi − yj.
Calculate∇ · F and explain why your answer is consistent with your sketch.
figure 4.4.11The flow lines of a fluid moving in the plane.
Marsden-3620111 VC September 27, 2011 10:5 259
4.4 Divergence and Curl 259
Calculate the divergence of the vector fields in Exercises 9 to 12.
9. F(x, y)= x3i− x sin (xy)j 10. F(x, y)= yi − xj
11. F(x, y)= sin (xy)i − cos (x2y)j 12. F(x, y)= xeyi− [ y/(x + y)]j
Compute the curl,∇ × F, of the vector fields in Exercises 13 to 16.
13. F(x, y, z)= xi + yj + zk 14. F(x, y, z)= yzi + xzj + xyk
15. F(x, y, z)= (x2+ y2+ z2)(3i+ 4j + 5k) 16. F(x, y, z)= yzi− xzj + xyk
x2+ y2+ z2 Calculate the scalar curl of each of the vector fields in Exercises 17 to 20.
17. F(x, y)= sin xi + cos xj 18. F(x, y)= yi − xj
19. F(x, y)= xyi + (x2− y2)j 20. F(x, y)= xi + yj
21. Let F(x, y, z)= (x2, x2y, z+ zx).
(a) Verify that∇ · (∇ × F) = 0.
(b) Can there exist a function f :R3→ R such that F= ∇ f ? Explain.
22. (a) Which of the vector fields in Exercises 13–16 could be gradient fields?
(b) Which of the vector fields in Exercises 9–12 could be the curl of some vector field V:R3→ R3?
23. Let F(x, y, z)= (ex z, sin(x y), x5y3z2).
(a) Find the divergence of F.
(b) Find the curl of F.
24. Suppose f :R3→ R is a C2scalar function. Which of the following expressions are meaningful, and which are nonsense? For those which are meaningful, decide whether the expression defines a scalar function or a vector field.
(a) curl(grad f ) (b) grad(curl f )) (c) div(grad f )
(d) grad(div f ) (e) curl (div f ) (f) div(curl f )
25. Suppose F:R3→ R3is a C2vector field. Which of the following expressions are meaningful, and which are nonsense? For those which are meaningful, decide whether the expression defines a scalar function or a vector field.
(a) curl(grad F) (b) grad(curl F)) (c) div(grad F) (d) grad(div F) (e) curl (div F) (f) div(curl F)
26. Suppose f, g, h:R → R are differentiable. Show that the vector field F(x, y, z)=!
f (x), g( y), h(z)"
27. Suppose f, g, h:R2→ R are differentiable. Show that the vector field F(x, y, z)=!
f ( y, z), g(x, z), h(x, y)"
has zero divergence.
28. Prove identity 13 in the list of vector identities.
Verify that∇ × (∇ f ) = 0 for the functions in Exercises 29 to 32.
29. f (x, y, z)=#
x2+ y2+ z2 30. f (x, y, z)= xy + yz + xz 31. f (x, y, z)= 1/(x2+ y2+ z2)
32. f (x, y, z)= x2y2+ y2z2
33. Show that F= y(cos x)i + x(sin y)j is not a gradient vector field.
34. Show that F= (x2+ y2)i− 2xyj is not a gradient field.
Marsden-3620111 VC September 27, 2011 10:5 260
260 Vector-Valued Functions
35. Prove identity 10 in the list of vector identities.
36. Suppose that∇ · F = 0 and ∇ · G = 0. Which of the following necessarily have zero divergence?
(a) F+ G (b) F× G
37. Let F= 2xz2i+ j + y3zxk and f = x2y. Compute the following quantities.
(a) ∇ f (b) ∇ × F (c) F× ∇ f (d) F· (∇ f )
38. Let r(x, y, z)= (x, y, z) and r=#
x2+ y2+ z2= ∥r∥. Prove the following identities.
(a) ∇(1/r) = −r/r3, r ̸= 0; and, in general,
∇(rn)= nrn−2r and∇(log r) = r/r2. (b) ∇2(1/r )= 0, r ̸= 0; and, in general,
∇2rn= n(n + 1) rn−2.
(c) ∇ · (r/r3)= 0; and, in general,
∇ · (rnr)= (n + 3) rn.
(d) ∇ × r = 0; and, in general, ∇ × (rnr)= 0.
39. Does∇ × F have to be perpendicular to F?
40. Let F(x, y, z)= 3x2yi+ (x3+ y3)j.
(a) Verify that curl F= 0.
(b) Find a function f such that F= ∇ f . (Techniques for constructing f in general are given in Chapter 8.
The one in this problem should be sought by trial and error.)
41. Show that the real and imaginary parts of each of the following complex functions form the components of an irrotational and incompressible vector field in the plane;
(a) (x− iy)2 (b) (x− iy)3
(c) ex−iy= ex(cos y− i sin y)
review exercises for chapter 4
For Exercises 1 to 4, at the indicated point, compute the velocity vector, the acceleration vector, the speed, and the equation of the tangent line.
1. c(t)= (t3+ 1, e−t, cos (πt/2)), at t= 1 2. c(t)= (t2− 1, cos (t2), t4), at t =√
π 3. c(t)= (et, sin t, cos t), at t= 0 4. c(t)= t2
1+ t2i+ tj + k, at t = 2
5. Calculate the tangent and acceleration vectors for the helix c(t)= (cos t, sin t, t) at t = π/4.
6. Calculate the tangent and acceleration vector for the cycloid c(t)= (t −sin t, 1−cos t) at t = π/4 and sketch.
7. Let a particle of mass m move on the path
c(t)= (t2, sin t, cos t). Compute the force acting on the particle at t= 0.
8. (a) Let c(t) be a path with∥c(t)∥ = constant; that is, the curve lies on a sphere. Show that c′(t) is orthogonal to c(t).
(b) Let c be a path whose speed is never zero. Show that c has constant speed if and only if the
acceleration vector c′′is always perpendicular to the velocity vector c′.
9. Let c(t)= (cos t, sin t,√
3t) be a path inR3. (a) Find the velocity and acceleration of this path.
(b) Find a parametrization for the tangent line to this path at t= 0.
(c) Find the arc length of this path for t∈ [0, 2π].
10. Let F(x, y, z)= (sin(xz), ex y, x2y3z5).
(a) Find the divergence of F.
(b) Find the curl of F.
11. Verify that the gravitational force field F(x, y, z)= −A (x, y, z)
(x2+ y2+ z2)3/2, where A is some constant, is curl free away from the origin.