a- Let P (x, y) and Q (x, y) be continuous and have continuous first partial derivatives at each point of a simply connected region R. A necessary and sufficient condition that [P dx + Q dy] =0 around every closed path C in R is that identically in R. Similiarly we can conclude that a necessary and sufficient condition that [P dx + JP JQ ay x JP 20 Q dy] be independent of the path in R joining points A and B is that ay ax identically in R. Which theorem can be used to prove these statements? = b- According to the above mentioned statements, prove that [(6xy y)dx + (12) (6xy3x y) dy is independent of the path joining (1.2) and (3, 4). c- Find the value of Integral by using potential function.