47. F = (yz co
48. Find poten
Example 8.
49. Show that
that any constar
50. Let φ = ln
radial vector er
51. For P = (a
FIGURE 11
21. A river 200 meters wide is modeled by the region in the xy-plane given by -100 ≤ x ≤ 100. The velocity vector field on the surface of the river is given by F = (-0.05x, 20 - 0.0001x²) in meters per second. Determine the coordinates of those points that have the maximum speed.
22. The velocity vectors in kilometers per hour for the wind speed of a tornado near the ground are given by the vector field F = (-y / e^(x²+y²-1)², x / e^(x²+y²-1)²). Determine the coordinates of those points where the wind speed is the highest.
In Exercises 23-30, calculate div(F) and curl(F).
(a) Verify that
(b) Calculate er
(c) Find a poter
52. Which of
function for the
dicular to the lev
23. F = ⟨x, y, z⟩
24. F = ⟨y, z, x⟩
25. F = ⟨x - 2zx², z - xy, z²x²⟩
26. sin(x + z)i - ye^(xz)k
27. F = ⟨yz, xz, xy⟩
28. F = ⟨y/x, y/z, z/x⟩
29. F = ⟨e^y, sin x, cos x⟩
30. F = ⟨x / (x² + y²), y / (x² + y²), 0⟩
In Exercises 31-37, prove the identities assuming that the appropriate partial derivatives exist and are continuous.
31. div(F + G) = div(F) + div(G)
32. curl(F + G) = curl(F) + curl(G)
33. div curl(F) = 0
34. div(F × G) = G · curl(F) - F · curl(G)