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Mathematics 317T2 Assignment#7(2)

Mathematics 317T2 Assignment#7(2) - Show that the vector...

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Mathematics 317 T2 Assignment #7 due Wednesday, March 10, 2010 1. For any two twice differentiable functions ) , , ( z y x f and ), , , ( z y x g show that (a) . ) ( 2 g f g f g f + = (b) . ) ( 2 2 f g g f f g g f - = - (c) ). ( ) ( 2 fg f g g f = + (d) . ) ( g f g f × = × (e) . 0 ) ( = × f g g f (f) . 2 ) ( g f f g g f × = - × 2. For any two differentiable vector fields ) , , ( z y x F and ), , , ( z y x G show that ). ( ) ( ) ( G F G F F G × = × - × 3. A vector field F for which div F = 0 is said to be divergence-free or solenoidal.
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Unformatted text preview: Show that the vector field given by k j i F z x y y-+ + = ) ( is solenoidal and find a vector potential A , where . A F × ∇ = 4. A vector field F for which curl F = 0 is said to be conservative, curl-free or irrotational. Show that the vector field given by k j i F 3 2 z y x +-= is irrotational and find a corresponding scalar potential φ , where . ϕ ∇ = F...
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