Mathematics 317 T2Assignment#9 - Mathematics 317T2...

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Mathematics 317T2 Assignment #9 due Wednesday, March 25, 2010 1. Verify the divergence theorem when k j i F z y x = and σ is the closed surface bounded by the cylindrical surface 1 2 2 = + y x and the planes z = 0, z = 1. 2. Suppose E is conservative in the domain D . Then ϕ -∇ = E in D for some potential function ). , , ( z y x ϕ Suppose ) , , ( z y x ρ and div E are both continuous in D . Suppose ∫∫∫ ∫∫ = R dV z y x d ) , , ( 4 ρ π σ S E for any region R in D bounded by a surface σ . Show that πρ ϕ 4 2 - = in D . 3. Let ) , , ( z y x F be a continuously differentiable vector field defined on a domain D . Let R be a region in D bounded by the surface σ . Use the divergence theorem to
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Unformatted text preview: show that . ) ( ) ( ∫∫∫ ∫∫ × ∇ = × R dV dS F F n r 4. (a) Show that . | | ) ( 2 2 T T T T T ∇ + ∇ = ∇ ⋅ ∇ (b) Suppose that at any time t , the temperature distribution ) , , , ( t z y x T over a domain D in 3 R satisfies the heat conduction equation T t T 2 ∇ = ∂ ∂ κ where the diffusivity κ is a constant. Suppose D is bounded by the surface σ . If either T = 0 or = ∂ ∂ = ⋅ ∇ n T T n on σ , show that ( 29 . | | 2 2 2 1 dV T dV T t D D ∫∫∫ ∫∫∫ ∇-= ∂ ∂...
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