some questions for the final

True or False. (why)

1. There is a vector field F in such that curl(F)= y i + x j +z k. why?

2. Let D1 be an oriented surface and D2 be the same surface with opposite orientation. Then, for every integrable function f, we have why?

My thought: if f is scalar function, then this statement would be false?? Does this surface has to be a closed surface?

3. For any smooth vector field F on a torus is zero

My thought: use Stoke's theorem? omega has to be no boundary?

4. Let and be the sphere oriented with outward-pointing normal. Then, the flux increases with the radius R.