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Hi everyone,

I have to prove this problem but I have no idea how to approach this problem. I tried something but it seems not working...

Suppose F is a vector field in R3 whose components have continuous partial derivatives. (So F satisfies the hypotheses of Stoke's Theorem.)

(a) Explain why any two surfaces S and S' oriented upward and with boundary the

unit circle x^2 + y^2 = 1 satisfy

Int Int_s (curl F dot d sigma) = Int Int_s' (curl F dot d sigma)

(b) Explain why any two surfaces S and S'', one oriented upward, the other oriented

downward, and both with boundary the unit circle x^2 + y^2 = 1 satisfy

Int Int_s (curl F dot d sigma) = - Int Int_s'' (curl F dot d sigma)

Since I don't know how to insert Greek characters, I attach 2 files here; one is the problem and another one is my attempt.

Thank you for help!

I have to prove this problem but I have no idea how to approach this problem. I tried something but it seems not working...

Suppose F is a vector field in R3 whose components have continuous partial derivatives. (So F satisfies the hypotheses of Stoke's Theorem.)

(a) Explain why any two surfaces S and S' oriented upward and with boundary the

unit circle x^2 + y^2 = 1 satisfy

Int Int_s (curl F dot d sigma) = Int Int_s' (curl F dot d sigma)

(b) Explain why any two surfaces S and S'', one oriented upward, the other oriented

downward, and both with boundary the unit circle x^2 + y^2 = 1 satisfy

Int Int_s (curl F dot d sigma) = - Int Int_s'' (curl F dot d sigma)

Since I don't know how to insert Greek characters, I attach 2 files here; one is the problem and another one is my attempt.

Thank you for help!