Verify Stokes Theorem where S is the part of the paraboloid

Verify Stokes\' Theorem, where S is the part of the paraboloid z = x2 + y2 below z = 4, and F(x, y, z) = yzi + 2xyj + 3xzk.

Solution

At x=4 , x^2+y^2=4 , using cylindrical coord x= 2cosT y=2sinT z=4 R= 2cosT i+2sinTj +4k dR= (dR/dT) dT =( -2sinT i +2cosT j )dT F= 8sinT i +8sinTcosT j +24cosT k F . dR = -16sin^2T +16sinT cos^2T +0 W= INT F.dR = INT (-16sin^2T +16sinT cos^2T) dT 0 = N/INI and dS= dS I . k IdS = dA ( Over XY plane , remember that z=4 k ) dS= INI dA / r Curl F = ( 0,rsinT-3r^2, 2rsinT-r^2) curl F. dS = ( 0,rsinT-3r^2, 2rsinT-r^2) .( 2r^2 cosT , 2r^2sinT, -r)/ INI = 2r^2sinT ( rsinT-3r^2) -r ( 2rsinT-r^2))/ INI = (2r^3 sin^2T -6r^4 sinT -2r^2sinT+r^3 ) /INI = (2r^3 sin^2T -6r^4 sinT -2r^2 sinT+r^3) /INI ( INI /r ) dA dA=rdrdT =(2r^3 sin^2T -6r^4 sinT -2r^2 sinT+r^3) drdT W= INT_A (2r^3 sin^2T -6r^4 sinT -2r^2 sinT+r^3) drdT W= (1/2) r^4 INT sin^2T dT -(6/5) r^5 INT sinT dT +r^4/4 INT dT) 0