Thanks very much that solved the problem! I just wanted to check my answer and happily it was correct.

Thanks for the help. Yes, I meant of course z>=0, that's right. When I enter the above the extra comma in "Surface(," produces: Error, invalid function arguments. So I've deleted it (and also entered r=2 in front of it a couple of times to try whether that'd help but it didn't). I do get rid of that error but when I enter the rest all I get is: Error, (in Flux_Surface) error with surface input I have also put the definition for "hemisphere" directly in the Flux command like this: Flux(VectorField(`<,>`(3, 2, 0)), Surface(`ϕ` = 0 .. 1/2*Pi, theta = 0 .. 2*Pi, coords = spherical[r, phi, theta])) and I've also used a very simple vector field like the one above to avoid problems with the coordinate system but I still get the error Error, (in Flux_Surface) error with surface input I've also tried setting the coordinates to spherical by typing Error, (in Flux_Surface) error with surface input but that doesn't help either.