$dxdy=-dydx$ using Jacobian determinant. Why?

Question

1. How do you reslove the contradiction due to the fact that $dxdy = dydx$ as per definiton of hyperreals ?

2. Is this abuse of notation and by $dxdy$ its is actually meant $dx \wedge dy$ in both statements ? That is $$dxdy = -dydx$$ $$dy_{1}...dy_{n} = \left|\frac{\partial\left(y_{1},...,y_{n}\right)}{\partial\left(x_{1},...,x_{n}\right)}\right| dx_{1}...dx_{n}$$is actually $$dx\wedge dy= - dy \wedge dx$$ $$ dy_{1} \wedge ... \wedge dy_{n} = \left|\frac{\partial\left(y_{1},...,y_{n}\right)}{\partial\left(x_{1},...,x_{n}\right)}\right| dx_{1} \wedge ... \wedge dx_{n}$$ ?

3. Does it mean there is no such as product of infinitesimal in integration of $R^{n}$ e.g. $\intop_{A\subset R^{2}} f\left( x,y \right) dxdy$ is indeterminant ? Note I dont mean this $\intop_{b}^{a}\intop_{n}^{m} f\left( x,y \right) dxdy$ which I believe is wrong and should be written as $\intop_{b}^{a}\left(\intop_{n}^{m} f\left( x,y \right) dx \right)dy$.

Answer 1

Let me try to respond to your questions one-by-one.

(1) $dxdy=dydx$ does not cause a "contradiction" in the hyperreals any more than in the reals, since the hyperreals are an elementary extension of the reals, meaning that all first order properties are preserved (there are some fine details that need to be mentioned here; I can elaborate if you are interested). Constructing the exterior algebra leads to anticommuting 1-forms over the hyperreals just as it leads to anticommuting 1-forms over the reals.

(2) in the long term it is indeed preferable to interpret $dxdy$ in integration as $dx\wedge dy$. This is because of the fact that doing otherwise suppresses useful information. Consider for example the fact that even in 1-variable, the sign of the derivative matters when you make a change of variables. Assuming this is positive forces you to limit the discussion to oriented changes of variable, which is an unnecessary restriction.

(3) you are right, there is a kind of ambiguity built into the notation. Depending on context this may or may not need to be clarified.