Slice the region along lines with x=constant and y=constant, as shown. Starting from dr=dxi+dyj+dzk and using the equation of the surface, x+y+z=1, we have dr1=(i−k)dx along y=constant lines, and dr2=(j−k)dy along x=constant lines, leading to dA=dr1×dr2=(i+j+k)dxdy. Dotting this with the given vector field E=zk yields (1−x−y)dxdy, and integrating this as y goes from 0 to 1−x and x goes from 0 to 1 yields 1/6.
(The traditional way to determine the limits of integration is to project the surface into the xy-plane.)
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