way # 1 - 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] );

> n := [0, -1, 1] / norm( [0, -1, 1] , 2) ;

> intgrand := dotprod( curl_F , n );

S is an ellipse with center (0,0,3) and semi-axes of length a & b where:

> center := [ 0,0,3];

> c_y := [ 0 , 1, 4 ];

> c_x := [ 1 , 0, 3 ];

> b := norm( c_y - center , 2);

> a := norm( c_x - center , 2);

So using problem #3 from this exam:

> surface_area_of_S := Pi * a* b;

and so since the integrand is constant:

> integral := intgrand * surface_area_of_S ;

>