In Elementary Differential Geometry 2nd ed, Barret O'niell, Section 1.6
O'niell states that since p-forms follow the alternation rule, then a repeated differential is necessarily zero, that is
$$dx \wedge dx = 0$$
Yet, differentiating position with respect to time, we get
$$a = \frac {d^2 p} {dt^2}$$
then integrating back to position
$$p = \int_0^t \int_0^t a\,dt\,dt $$
which clearly does not equal zero, yet in my tangled brain looks something like $dt\,dt$. Why is this not zero?
On
The notation $da \land db$ indicate a differential $2-$ form, that is a different thing than a ''double differential'' (whatever this means).
Intuitively you can think at $da \land db$ as the ''infinitesimal'' oriented area of a parallelogram that has sides the two ''vectors'' $da$ and $db$. So if the two vectors are parallel, as in the case $da=db=dx$, this area is zero.
The proper way of writing a second antiderivative in terms of a double integral is $$ p(t) = \int_0^t\!\!\int_0^{t_1} a(t_2)\, dt_2\,dt_1 $$ This can be seen from two applications of the fundamental theorem of calculus. We see that the two differentials are indeed of different variables.