How to Apply Stoke's Theorem on a Hemispherical Surface?

[fonseh]

Homework Statement

In the first and second photo , it's stated earlier that the C is the boundary of surface on xy plane , but in the question in the 3rd picture , it's not stated that the C is on which surface , so , how to do this question ?
For ∫F.dr , i am not sure how to get r , coz i am not sure i should to use which surface as 2d projection .
In the first photo , it's stated earlier that C is on xy-plane , so xy - plane projection is used.
In the solution provided in the 4th photo , the author use xy plane projection , is the concept correct ?

Homework Equations

The Attempt at a Solution

 

[haruspex]

in the question in the 3rd picture , it's not stated that the C is on which surface

Not an area I know, but it looks to me that you can define C as the boundary in xy plane and that allows you to apply Stokes' theorem.