way # 2 - with Stokes' ie, int_C F dot ds = int int_S curl(F) dot dS

> restart : with(linalg):

Warning, new definition for norm

Warning, new definition for trace

> F_1 := 3*z;

> F_2 := 5*x;

> F_3 := -2*y;

> F := [ F_1, F_2, F_3 ];

> curl_F := curl( F , [x,y,z] );

S can be parameterized by :

> x := x ;

> y := y ;

> z := y + 3 ;

> phi := [ x,y,z];

So:

> T_x := [ diff(x,x), diff(y,x) , diff(z,x) ] ;

> T_y := [ diff(x,y), diff(y,y) , diff(z,y) ] ;

> cross_of_T_x_and_T_y := crossprod( T_x , T_y ) ;

> integrand := dotprod( curl_F, cross_of_T_x_and_T_y ) ;

> integral := int(int( integrand, x = -sqrt(1 - y^2) .. sqrt(1 - y^2)) , y = -1 .. 1);

>