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366 CHAPTER 11 Vector Differential Calculus

SECTION 11.5 PROBLEMS

In each of Problems 1 through 6, compute ∇· F and ∇× F In each of Problems 7 through 12, compute ∇ϕ and verify
and verify explicitly that ∇· (∇× F) = 0. explicitly that ∇× (∇ϕ) = O.

7. ϕ(x, y, z) = x − y + 2z 2
1. F = xi + yj + 2zk 8. ϕ(x, y, z) = 18xyz + e x
3
2. F = sinh(xyz)j 9. ϕ(x, y, z) =−2x yz 2
10. ϕ(x, y, z) = sin(xz)
y
3. F = 2xyi + xe j + 2zk
11. ϕ(x, y, z) = x cos(x + y + z)
4. F = xi + yj + 2zk 12. ϕ(x, y, z) = e x+y+z
5. F = sinh(x)i + cosh(xyz)j − (x + y + z)k 13. Let ϕ be a scalar field and F a vector field. Derive
expressions for ∇· (ϕF) and ∇× (ϕF) in terms of
6. F = sinh(x − z)i + 2yj + (z − y )k operations applied to ϕ(x, y, z) and to F(x, y, z).
2

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October 15, 2010 16:11 THM/NEIL Page-366 27410_11_ch11_p343-366