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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This note was uploaded on 01/19/2014 for the course MATH 5002 taught by Professor Adams during the Spring '08 term at Minnesota.

Ask a homework question - tutors are online