Mathematics

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oo . 5 Let x 3 y 2 z dx dy be a 2-form describing a flow in space and x S 1, 0 S y S 2, let S be the rectangle {(x, y, z): 0 z = 1} oriented counterclockwise.

Y = canst. to divide R into subrectangles and by orienting each subrectangle counterclockwise in agreement with the orientation of R. An approximating sum corresponding to this 'finely divided approximation to R' is obtained by 'evaluating' A(x, y) dx dy on each subrectangle and adding over all rectangles. To form such an approximating sum it is necessary to choose: lines x = Xi, where a = x 0 < x1 < x 2 < · · · lines y = Y;. where c = Yo < Y1 < Y2 < · · · points Pi;, one in each of the mn rectangles Rii = < Xm-1 < Xm = < Yn-1 < Yn = b d {(x, y): Xi-1 :::; X :::; Xi, Yi-1 :::; Y :::; Y;} · The approximating sum is then tThis sum is a number once A.

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