Consider the vector field F(x, y, z) = (2yz, y sin z, 1 + cos z). (a) Find a vector field G whose curl F. (b) Let S be the half-ellipsoid 4x² + 4y² + z² = 4, z ≥ 0, oriented by the upward normal. Use Stokes's theorem to find ff, F. ds. (c) Find ff F. dS if 5 is the portion of the surface z = 1 - x² - y² above the xy-plane, oriented by the upward normal. (Hint: Take advantage of what you've already done.)

Compute the curl of F = (3y², xy, 22).

Exercise 1. A student is asked to prove If m is even and n is odd, then m² +n² is odd. The student provides the following proof: Proof. If m is even and n is odd, then m² + n² is odd. Since m is even, m = = 2k for any integer k. Since n is odd, n = 2k + 1. Note that m² + n² = (2k)² + (2k + 1)² = 8k² + 4k +1 So m² + n² is odd because it is an even number plus 1. (a) Identify the errors in the proof above. There are multiple errors. Explain each error. Your explanation should be written to the student who made the error and should try to help the student understand why what they wrote is incorrect. (b) In addition to errors, provide suggestions to improve the student's proof writing. That is, explain what parts of the proof are unclear or unjustified. Explain how such parts of the proof prevent some students from fully following the ment made. (c) Finally, provide a correct proof of the theorem above.