Determine whether or not the vextor field Fxy 2xcosyix2sinyj

Solution

df/dx = 2x.cos(y)

df/dy = x²sin(y)

We have:

df/dx = 2x.cos(y)

f(x,y) = int 2x.cos(y) dx = x²cos(y) + g(y)

df/dy = -x²sin(y) + g\'(y) = x²sin(y) -> g\'(y) = 2x²sin(y)

But this is not possible because g\'(y) is a function of y, and 2x²sin(y) is a function of x and y.

Therefore F is not conservative.