vector field using green's theorem+other integration

Question

So am I supposed to be using green's theorem for the first question, but where I'm confused is that there are three variables if I do, dx dy dz (I haven't learn how to use green's theorem for 3 variables. Can someone show me how to do a) at least and I'll see how I can approach b) and c) from there (if possible show solutions to b and c). Thanks

Answer 1

What you require essentially is the result

A line integral from $P$ to $Q$ is indeendent of path followed Iff $Curl \bar{F} =0$ Iff integral is zero on any closed path passing through $P$ and $Q$.