way # 2 -- without Gauss

> restart : with(linalg): i:= [1,0,0] : j := [0,1,0] : k := [0,0,1]:

Warning, new definition for norm

Warning, new definition for trace

> F_1 := x;

> F_2 := y;

> F_3 := 3 ;

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

Parameterize S:

> x := sin(p) * cos(t);

> y := sin(p) * sin(t);

> z := cos(p);

> Phi := vector( [x,y,z]) ;

> T_p_cross_T_t := scalarmul( Phi , sin(p) );

> F_on_S := [ x + 3*y^5, y + 10*x*z, z - x*y ] ;

> F_on_S := [ simplify(dotprod(F,i)), simplify(dotprod(F,j)) , simplify(dotprod(F,k)) ] ;

> intgrand := dotprod( F , T_p_cross_T_t );

> intgrand := simplify(intgrand) ;

> flux := int(int(intgrand , t = 0 .. 2*Pi), p = 0 .. Pi );