4. Some short-answer questions from 13.3 to prepare for the final exam:
(a) What is the definition of a conservative vector field F?
(b) In Problem 3, you used a cool little test to decide whether F was conservative. Why exactly does this test always work?
(c) Why are et F be a conservative vector field. You integrate F along a straight line connecting P and Q, while your friend integrates F along a much longer, winding curve connecting P and Q. Which of you will end up with the higher conservative vector fields easier to integrate than non-conservative vector fields? (If you want, use Problem 3 as your example — describe all the calculations you would have had to do if that F had not been conservative!)
(d) Let P and Q be two points in the xy-plane, and lanswer? Briefly explain.