Question

PATH INDEPENDENCE? 13-19 from Check, and if independent, integrate from (0, 0, 0) to (a, b, c) 13. 2e (x cos 2y dx - sin 2y dy)

Answer 1

The integral is path-independent if there is a scalar function $f$ whose gradient is

$\nabla f=(2e^x\cos2y,-\sin2y)$

(at least, that's what it looks like the given integrand is)

Then

$\dfrac{\partial f}{\partial x}=2e^x\cos 2y\implies f(x,y)=2e^x\cos2y+g(y)$

Differentiating both sides with respect to $y$ gives

$\dfrac{\partial f}{\partial y}=-4e^x\sin 2y\neq-\sin2y$

so the line integral *is* dependent on the path. (again, assuming what I've written above actually reflects what the question is asking)