Consider the torus $\Bbb T^2=\Bbb R^2/\Bbb Z^2$. And define a Morse function $f:\Bbb T^2\to \Bbb R$ by $f([(x,y)])=\cos(2\pi x)+\cos(2\pi y)$. Now the critical points are minimum $\big[\big(\frac{1}{2},\frac{1}{2}\big)\big]$, maximum $[(0,0)]$ and two critical points of index 1, namely $\big[\big(\frac{1}{2},0\big)\big]$ and $\big[\big(0,\frac{1}{2}\big)\big]$. Now, I want to compute Morse Homology w.r.t. coefficient in $\Bbb Z_2$. Note that, $C_0=\Bbb Z_2$ and $C_1=\Bbb Z_2\oplus\Bbb Z_2$ and $C_2=\Bbb Z_2$. Now I have to compute differential i.e. for $a\in C_k$ we have to find $$\partial_X(a)=\sum_{b\text{ is a critical point of index } k-1}n(a,b)b$$ where $n(a,b)\in \Bbb Z_2$ is the number of trajectory of the pseudo-gradient field $X$ corresponding to $f$ modulo $2$ going from $a$ to $b$.
My question is how to find $n(a,b)$. I saw that, this can be done by looking pseudo-gradient flow in square and counting. Is there any other way doing this without drawing? Alos I have not understand the way of drawing this pseudo-gradient flow in square. Also I want to know the stable and unstable manifolds, how do they look like. Any help will be appreciated.