Symbolic contour integral evaluation

Question

Can anyone help with the evaluation of the following contour integral :

$$\oint\limits_C \phi(x,y)\,dx+\psi(x,y)\,dy.$$

Where the contour $C$ is given by:

What I am looking for is how to split the contour integral in to normal integral.

Answer 1

I guess the start is following

$$ \oint\limits_C \phi(x,y)\,dx+\psi(x,y)\,dy = \int\limits_{C_1} \phi(x,y)\,dx+\psi(x,y)\,dy + \int\limits_{C_2} \phi(x,y)\,dx+\psi(x,y)\,dy + \int\limits_{C_3} \phi(x,y)\,dx+\psi(x,y)\,dy + \int\limits_{C_4} \phi(x,y)\,dx+\psi(x,y)\,dy. $$ Were $C_i$ - sides of the rectangle.

To calculate these line integrals we we parametrize each line. For instance lower bound $C_1(t) = (x(t), y(t))$, where $x(t)=t$, $y(t) = j-\Delta y$ and parameter $t\in[i-\Delta x/2,i+\Delta x/2]$ Note that $dy(t) = 0$. Then we write $$ \int\limits_{C_1} \phi(x,y)\,dx+\psi(x,y)\,dy = \int\limits_{i-\Delta x/2}^{i+\Delta x/2} \phi(x(t),y(t))\,dx(t)+\psi(x(t),y(t))\,dy(t) = \int\limits_{i-\Delta x/2}^{i+\Delta x/2} \phi(t,j-\Delta y)\,dt. $$

The key of this problem (I suppose) that contour is small. So, for example $$ \int\limits_{C_1} \phi(x,y)\,dx+\psi(x,y)\,dy = \int\limits_{C_1} \phi(x,y)\,dx= \int\limits_{i-\Delta x/2}^{i+\Delta x/2} \phi(t,j-\Delta y)\,dt\approx \varphi(i,j-\Delta y)\Delta x. $$