Question 5: To evaluate a line integral we must choose a path of integration. The following exercise is intended to remind us how this works for a simple mathematical example. X; =y;=1 a) Calculate...

(1,0) -> (1,1) in two straight line segments. b) Show that the integral is expected to be path independent, so that we could have guessed that all the paths attempted above would yield the same result. c) Find a function F(x,y) that satisfies oF lox = 2xy and dF löy, = x. d) Show that the integral I has the value F(1,1) – F(0,0). Why does this make sense? "/>
Extracted text: Question 5: To evaluate a line integral we must choose a path of integration. The following exercise is intended to remind us how this works for a simple mathematical example. X; =y;=1 a) Calculate the integral: I= 2xydx + x'dy along the following four paths. X; =y; =0 i) y(x)=x (a straight line) ii)y(x) = x (a parabola) iii) y(x) = iv) an “L shaped" path, (0,0) -> (1,0) -> (1,1) in two straight line segments. b) Show that the integral is expected to be path independent, so that we could have guessed that all the paths attempted above would yield the same result. c) Find a function F(x,y) that satisfies oF lox = 2xy and dF löy, = x. d) Show that the integral I has the value F(1,1) – F(0,0). Why does this make sense?