How to approximate $\int f(x,y,z) dx\wedge dy \wedge dz$?

Question

For Lebesgue integrals, I know how to approximate $\int_{A} f(x)dx$ for some measurable set $A\subset \mathbb{R}^{n}$, I just decompose $A$ into measurable subsets $A_{i}$ and then make some finite "Riemann sum" $\sum f(x_{i})\mu (A_{i})$ where $x_{i}$ is a well-chosen point in $A_i$.

For integration over forms, I guess that I cannot anymore do this, since now I need to deal with orientation and stuff. Basically, I cannot just treat $dx\wedge dy \wedge dz$ as $dxdydz$. How do I proceed then to obtain a numerical approximation of $\int_{A} f(x,y,z) dx\wedge dy \wedge dz$ for some $A\subset \mathbb{R}^{3}$ ?

Well in your specific case, $dx\wedge dy\wedge dz$ and any permutation thereof, as a 3-form in a 3-space, can only be one specific measure (mod a minus sign of course). If you had say a 2-form or a 1-form instead, you'd have to consider the specific embedding of $A$ to find the Jacobian, then treat it just like your other example. — Ninad Munshi, Jul 09 '23 at 01:59

score 1 · Accepted Answer · answered Jul 09 '23 at 09:33

Underlying an integral of a differential form is the Lebesgue integral. In fact, using the standard orientation of $\mathbb{R}^3$, $$ \int_A f(x,y,z)\,dx\wedge dy\wedge dz = \int_A f(x,y,z)\,dV, $$ where $dV$ is the standard Lebesgue measure on $\mathbb{R}^3$.

Differential forms are needed for defining integrals whose value does not depend on the coordinates used. They were developed in order to define an integral over a manifold (on which there is no coordinate-independent way to integrate a function).

But to calculate an integral of a differential form, you have to write the differential form with respect to coordinates. For example, if $\omega$ is a $3$-form on $A$, then with respect to standard coordinates, it can always be written as $$ \omega = f(x,y,z)\,dx\wedge dy\wedge dz.$$ Therefore, $$ \int_A \omega = \int_A f(x,y,z)\,dx\wedge dy\wedge dz = \int_A f(x,y,z)\,dV. $$ Note that the orientation plays a role when writing $\omega$ in this form. For example, if $$ \omega = dy\wedge dx \wedge dz, $$ then $$ \int_A \omega = \int_A dy\wedge dx \wedge dz = -\int_A dx\wedge dy\wedge dz = -\int_A dV. $$

Thank's! I think I get it now. So if I want for e.g. to approximate $\int_{A}f(x_{1},...,x_{n})dx_{i_{1}}\wedge ... \wedge dx_{i_{n}}$, I just need to rearrange the $dx_{i}'s$ in order, put the good amount of minus signs, and then I have the good old Lebesgue integral. — Galois, Jul 09 '23 at 21:43
There is a twist, however. If you are using coordinates on an oriented manifold, you have to check if the orientation you used to do the integral agrees with the orientation of the manifold. If not, you should multiply your answer by $-1$. — Deane, Jul 10 '23 at 10:01

How to approximate $\int f(x,y,z) dx\wedge dy \wedge dz$?

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