Evaluate $xy dx+xy^2 dy$ along C by stokes theorem where C is the square in xy-plane with vertices $(1,0),(-1,0),(0,1),(0,-1)$

Question

I am confused in putting limits of the integral. In the book limit for $y$ is $-1$ to $1$ and limit for $x$ is $-1$ to $1$ given. but i think limit for $y$ should be $-(1-x)$ to $1-x$ and limit for $x$ should be $-1$ to $1$. i am getting different answer. please help.

Answer 1

you region is bounded by the lines:

$x+y = 1, x+y = -1, x-y = 1, x - y = -1$

So yes, I get limits of $\int_{-1}^0 \int_{-1-x}^{1+x} F(x,y) dy\ dx + \int_{0}^1 \int_{x-1}^{1-x} F(x,y) dy\ dx$

Unless you did a coordinate transformation, $u = x+y, v = x-y$

And then you can integrate $\int_{-1}^{1}\int_{-1}^{1} 2F(u,v)\ du\ dv$