FM5002-HW5-2.22.12

Compute x dx x y 2 x dx dy a b x c d x2 dy z 5 0 a

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Unformatted text preview: 2 t . Then x dx x y 8, x2 dy z 1 dz 7 ,8,9 27 2t 27 2t 2 9 2t dt 16 101 7 65 9 . 0 0042− . Let R = (2, 4) x (6, 9). 9 2x 3y Compute dy dx. R 2x 3y 9 4 dy dx R 6 2x 3y 1 2 6 6 yx 3 12 1 6 6 6 1 2. 0042− . Let Ω = f (u,x, y, z, t) be an y 0 − form in u,x, y, z, t. Show that d (dΩ) = 0. 10 We compute this for Ω n n f x1, x2, ..., xn . By defin ition dΩ xi f x1, x2, ..., xn dxi . Then d dΩ i1 d xi xj xi f x1, x2, ..., xn dxi n f x1, x2, ..., xn dxi xj xi f x1, x2, ..., xn dxj dxi . But for an y term xj xi f x1, x2, ..., xn dxj dxi , i1j1 i1 we kn ow that xi i1 n n d xj xi f x1, x2, ..., xn xi xj f x1, x2, ..., xn dxj dxi can cels with f x1, x2, ..., xn , and also that dxj dxi xi xj dxi dxj . Thus, f x1, x2, ..., xn dxi dxj , an d so the sum is 0. 0042− . Let A, B, C, P, Q, R, E, F, G, and H be expression s in u, x, y, z, t. 11 Let Ω = A dx dy + B dx dz + C dy dz + P dx dt + Q dy dt + R dz dt+E du dx+F du dy+G du dz+H du dt, so Ω is a 2 − form in u, x, y, z, t. Show that d (dΩ) = 0. 6 FM5002−HW5−2.22.12.nb By symmetry, it is en ough to show that d d A dx dy 0. Now, d A dx dy Adz dx dy Adt dx dy Adu dx dy. Therefore, z t u d d A dx d...
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This note was uploaded on 01/19/2014 for the course MATH 5002 taught by Professor Adams during the Spring '08 term at Minnesota.

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