Slice the region along lines with y=constant and z=constant, as shown. Starting from dr=dxi+dyj+dzk and using the equation of the surface, x+y+z=1, we have dr1=(j−i)dy along z=constant lines, and dr2=(k−i)dz along y=constant lines, leading to dA=dr1×dr2=(i+j+k)dydz. Dotting this with the given vector field E=zk yields zdydz. Integrating this as y goes from 0 to 1−z yields z(1−z)dz, and integrating this as z goes from 0 to 1 yields 1/6.
(Note that the result of the first integration yields the same integrand used in the geometric solution.)
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