Suppose the class is asked to find the line integral of F ⃗=2xyi ⃗+yzj ⃗+y^2 k ⃗ over the curve C which is the boundary of the upper hemisphere x^2+y^2+z^2=4,z≥0, oriented in a counter-clockwise direction when viewed from above. One student gives the following solution:
curl F ⃗=-2xj ⃗+2k ⃗. By Stokes’ Theorem, ∫F ⃗⋅dr ⃗=∫(-2xj ⃗+2k ⃗ )⋅dA ⃗ , where S is the hemisphere. Since div(-2xj ⃗+2k ⃗ )=0, by the Divergence Theorem, ∫(-2xj ⃗+2k ⃗ )⋅dA ⃗ =∫0 dV=0, where W is the solid hemisphere. Hence, ∫F ⃗⋅dr ⃗=0.
Critique this argument, explaining which parts are correct and find correct answer.

Question

Suppose the class is asked to find the line integral of F ⃗=2xyi ⃗+yzj ⃗+y^2 k ⃗ over the curve C which is the boundary of the upper hemisphere x^2+y^2+z^2=4,z≥0, oriented in a counter-clockwise direction when viewed from above. One student gives the following solution:
curl F ⃗=-2xj ⃗+2k ⃗. By Stokes’ Theorem, ∫F ⃗⋅dr ⃗=∫(-2xj ⃗+2k ⃗ )⋅dA ⃗ , where S is the hemisphere. Since div(-2xj ⃗+2k ⃗ )=0, by the Divergence Theorem, ∫(-2xj ⃗+2k ⃗ )⋅dA ⃗ =∫0 dV=0, where W is the solid hemisphere. Hence, ∫F ⃗⋅dr ⃗=0. 
	Critique this argument, explaining which parts are correct and find correct answer.