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Answer to Question #72164 in Calculus for Lastone
Question #72164
1. Compute the divergence and curl of each of the following vector fields:
a) hx
2 + y
2
, x2 − y
2
, z2
i
b) hx + y, x − y, zi
2. For any two vector fields F, G show that:
∇.(F × G) = G.(∇ × F) − F.(∇ × G).
3. Given scalar functions u(x, y, z), v(x, y, z), w(x, y, z), φ(x, y, z) and a vector field F(x, y, z) =
hu(x, y, z), v(x, y, z), w(x, y, z)i, prove that
div(φF) = φ(div F) + (∇φ) · F.
4. Find a function f such that F = ∇f where F = hyz, xz, xy + 2zi
5. Let r = hx, y, zi and r = |r|, verify that
a) ∇ · r = 3
b) div (r
n
r) = (n + 3)r
n
6. Given u = xy2
z
2 and v = yz − 3x
2
, find ∇ · [(∇u) × (∇v)]
7. Given the vector field F = he
x
sin y, ex
cos y, zi, show that F is irrotational, hence or
otherwise, find the functions f such that ∇f = F.
a) hx
2 + y
2
, x2 − y
2
, z2
i
b) hx + y, x − y, zi
2. For any two vector fields F, G show that:
∇.(F × G) = G.(∇ × F) − F.(∇ × G).
3. Given scalar functions u(x, y, z), v(x, y, z), w(x, y, z), φ(x, y, z) and a vector field F(x, y, z) =
hu(x, y, z), v(x, y, z), w(x, y, z)i, prove that
div(φF) = φ(div F) + (∇φ) · F.
4. Find a function f such that F = ∇f where F = hyz, xz, xy + 2zi
5. Let r = hx, y, zi and r = |r|, verify that
a) ∇ · r = 3
b) div (r
n
r) = (n + 3)r
n
6. Given u = xy2
z
2 and v = yz − 3x
2
, find ∇ · [(∇u) × (∇v)]
7. Given the vector field F = he
x
sin y, ex
cos y, zi, show that F is irrotational, hence or
otherwise, find the functions f such that ∇f = F.
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