Use the Divergence Theorem to calculate the flux of F across

Use the Divergence Theorem to calculate the flux of F across S, where F = zi + yj + zxk and S is the surface of the tetrahedron enclosed by the coordinate planes and the plane x/4 + y/4 + z = 1 integral integral_S F. dS =

Solution

(x_/4)+(y_/4)+z=1

0<=z<=1-(x_/4)-(y_/4), 0<=y<=4-x, 0<=x<=4

given F=zi +yj +zxk

divF=0+1+x

divF=1+x

_S F.dS

= divF dV

=_{[0 to 4] [0 to 4-x] [0 to 1-(x/4)-(y/4)]} (1+x) dz dy dx

=_{[0 to 4] [0 to 4-x][0 to 1-(x/4)-(y/4)]} (1+x) z dy dx

=_{[0 to 4] [0 to 4-x]}(1+x)(1-(x/4)-(y/4) -0) dy dx

=_{[0 to 4] [0 to 4-x]}(1+x)(1-(x/4)-(y/4)) dy dx

=_{[0 to 4][0 to 4-x]}(1+x)(y-(xy/4)-(y²/8)) dx

=_{[0 to 4]}  [(1+x)((4-x)-(x(4-x)_/4)-((4-x)²_/8))-(1+x)(0-(0)-(0))] dx

=_{[0 to 4]}  [(1+x)((4-x)-(x(4-x)_/4)-((4-x)²_/8))] dx

=_{[0 to 4]}  [(x³_/8)-(7x²_/8)+x+2] dx

=_{[0 to 4]}  [(x⁴_/32)-(7x³_/24)+(1_/2)x²+2x]

=[(4⁴_/32)-(7*4³_/24)+(1_/2)4²+(2*4)] -[0-0+0+0]

=8-(56_/3)+8+8

=16_/3

_S F.dS =16_/3