I am starting to getting used to compution of line and contour integrals. Recently I came accros this problem:
I am supposed to calculate Work done by force $$\overset{\to}{F}=(-\frac{\sin x}{y}-\frac{\sin y}{x^2}-\frac{1}{y^3}-\frac{2y}{x^3},-\frac{\cos x}{y^2}+\frac{\cos y}{x}+\frac{3x}{y^4}+\frac{1}{x^2}),$$when its moving an object from $A=\lbrack\pi;\frac{\pi}{2}\rbrack$ to $B=\lbrack\frac{\pi}{2};\pi\rbrack$.
We should probably use independce of line integral on integral path and calculate desired work using potential, but I am new at this. Could somebody please help me out? I would be grateful.
The line integral is independent on integral path if the field is conservative, that is if it is the gradient of a scalar potential.
In your case the field is a $2$-D vector: $\vec F=(F_x,F_y)$ and we know that a necessary condition for such a field to be conservative is that $$ \frac{\partial F_x}{\partial y}=\frac{\partial F_y}{\partial x} $$
For your filed you can test that this is not the case, so the integral is path dependent and we need some path to evaluate the work done by the field.
After the change , the field is conservative and the scalar potential is $$ f(x,y)=\frac{\cos x}{y}+\frac{\sin y}{x}-\frac{x}{y^3}+\frac{y}{x^2} $$ (do you know how to find this result?) and you can calculate the work done by the force between the points $A=[\pi, \pi/2]$ and $B=[\pi/2,\pi]$ as: $$ W= f(\pi/2,\pi)-f(\pi,\pi/2) $$