Previously I said that I am currently reading an abstact about the Lotka Volterra equations and that I am trying to show that a modified euler method leads us to solutions, which are not spiral. You can read it here. There is all the background information you need. After understanding that $dX \wedge dX = 0 $ thanks to the User LutzL ( using a property of the wedge product ), I need a computation of the wedge product: I have to compute $dX \wedge dY $. I will copy what $X$ and $Y$ are:
$$ X = \Delta tx +x - \Delta txy$$ and $$ Y = -\Delta ty+ y + \Delta tXy $$ where we set $X:=x_{n+1}, Y:=y_{n+1}$, $x:= x_n$ and $ y := y_n$ solved for the unknown $X$ and $Y$.
Taking derivatives: $$ dX = \Delta tdx +dx - \Delta tdxy - \Delta txdy $$ and $$ dY = -\Delta tdy+ dy + \Delta tdXy + \Delta tXdy $$
Like I said I never compute a wedge product and don't really understand it. This is the last detail I need ( at least for one of my proofs regarding the abstract ). According to the author the solution has to be $$dx \wedge dy(\Delta t^3(x-2xy+xy^2)+\Delta t^2(-1 + 2x+y-2xy) + \Delta t(x-y) +1) $$
The coefficients are multiplied by each other in a wedge product:
$$f \, \mathrm dx \wedge g \, \mathrm dy = fg\, \mathrm dx \wedge \mathrm dy$$
The wedge product is defined to be linear so it's defined term by term on a sum of differential forms by distributivity of $\wedge$ over $+$.
The last property you need is that:
$\mathrm dx \wedge \mathrm dy = -\mathrm dy \wedge \mathrm dx$
Which also tells you what $\mathrm dx \wedge \mathrm dx$ is.