Fubini's theorem for differential forms? Why does $\int_{t_0}^{t_1}(\oint_{\partial\Omega}j)dt=\int\limits_{[t_0,t_1]\times\partial\Omega}dt\wedge j$?

Question

In an electrodynamics book I came across the following claim for the electric current density (twisted) 2-form $j$ along the boundary of some 3-dimensional volume $\Omega$:

$$\int_{t_{0}}^{t_{1}}\left(\oint_{\partial\Omega}j\right)dt=\int\limits _{\left[t_{0},t_{1}\right]\times\partial\Omega}dt\wedge j$$

It looks to me like an attempt to apply the Fubini's theorem for differential forms, but I do not see how the LHS can be reduced to a 2-argument function so that Fubini applies.

The context is the conservation of charge law $\frac{dQ}{dt}=-\mathcal J$, where $\mathcal J=\oint_{\partial\Omega}j$ is the electric current flowing out of $\Omega$ and $Q=\int_\Omega \rho$ is the total charge inside $\Omega$. Integrating it along time from $t_0$ to $t_1$, the book claims the equation above. The argument then continues with the definition of $J=-j\wedge dt+\rho$ (the $j\wedge dt$ part coming from the equation above) so that the conservation law can be written in a "super-global" form $\oint_{\partial\Omega_4}J=0$ for the 4-dimensional volume $\Omega_4=[t_0,t_1]\times\Omega$.

The book does it by an intermediate step that I omitted above: $$\int_{t_{0}}^{t_{1}}\left(\oint_{\partial\Omega}j\right)dt=\int_{t_{0}}^{t_{1}}dt\wedge\oint_{\partial\Omega}j=\int\limits _{\left[t_{0},t_{1}\right]\times\partial\Omega}dt\wedge j$$ (the middle term), but $dt\wedge\oint_{\partial\Omega}j$ seems syntactically invalid to me because $\oint_{\partial\Omega}j$ is a (time-dependent) number (and not a form) and so cannot be wedge-multiplied by $dt$.

Why does it hold? Does Fubini's theorem apply at all here?

Answer 1

But $dt\wedge\oint_{\partial V}j$ seems syntactically invalid to me because $\oint_{\partial V}j$ is a (time-dependent) number (and not a form) and so cannot be wedge-multiplied by $dt$.

I don't see a problem: Let $M$ be some smooth manifold and let $1\in\Omega^0(M)\subseteq\Omega(M)$ be the obvious section, then \begin{align} C^\infty(M)&\to\Omega^0(M)\\ f&\mapsto f1 \end{align} is invertible and given $f\in C^\infty(M)$ and $\alpha\in\Omega(M)$ we make sense of $f\wedge\alpha$ by identifying $f$ with $f1$ (it turns out that $f\wedge\alpha$ is nothing but the pointwise scalar multiplication of $f$ with $\alpha$). In your case $M=[t_0,t_1]$.

Why does $\int_{t_0}^{t_1}(\oint_{\partial V}j)dt=\int\limits_{[t_0,t_1]\times\partial V}dt\wedge j$?

• Firstly, note the abuse of notation: The $dt$ on the LHS is an element of $\Omega^1([t_0,t_1])$ and the $dt$ on the RHS is an element of $\Omega^1([t_0,t_1]\times\partial V)$. Similarly the $j$ on the LHS is a function $[t_0,t_1]\to\Omega^2(\partial V)$ and the $j$ on the RHS is an element of $\Omega^2([t_0,t_1]\times\partial V)$.
• The equation boils down to Fubini's theorem if $\partial V$ is a trivial manifold, i.e. if $\partial V$ equals the domain of some chart: Just consider such a chart and write out the definition of all the integrals. The general case is an immediate consequence: We can consider a finite cover $(U{}_\alpha)_\alpha$ of $\partial V$ by domains of charts and some partition of unity $(\psi_\alpha)_\alpha$ such that $$\int_{[t_0,t_1]}dt \int_{\partial V}j=\int_{[t_0,t_1]}dt \sum_\alpha\int_{U_\alpha}j_\alpha=\sum_\alpha\int_{[t_0,t_1]}dt \int_{U_\alpha}j_\alpha=\\ =\sum_\alpha\int_{[t_0,t_1]\times U_\alpha}dt \wedge j_\alpha=\int _{\left[t_{0},t_{1}\right]\times\partial V}dt\wedge j$$where $j_\alpha\in\Omega^2(U_\alpha)$ is defined in the obvious way, i.e. $j_\alpha=i_\alpha^*\psi_\alpha j$, where $i_\alpha:U_\alpha\to M$ is the inclusion.

Answer 2

Under a suitable cover $\{U_\alpha,\varphi_\alpha\}_\alpha$ and partition of unity $\{f_\alpha\}_\alpha$, $j=\sum_\alpha f_\alpha j=\sum_\alpha j_\alpha dx\wedge dy$. Now

\begin{align} \int_{I}\left(\iint_{\partial V}j\right)dt&\overset{\text{(PU)}}{=}\int_{I}\left(\sum_{\alpha}\iint_{U_{\alpha}}j_{\alpha}dx\wedge dy\right)dt\\&\overset{\text{(i)}}{=}\int_{I}\left(\sum_{\alpha}\iint_{\varphi_{\alpha}(U_{\alpha})}j_{\alpha}dxdy\right)dt\\&\overset{\text{(F)}}{=}\sum_{\alpha}\iiint_{I\times\varphi_{\alpha}(U_{\alpha})}j_{\alpha}dtdxdy\\&\overset{\text{(i)}}{=}\sum_{\alpha}\iiint_{I\times U_{\alpha}}j_{\alpha}dt\wedge dx\wedge dy\\&=\iiint_{I\times\partial V}dt\wedge\sum_{\alpha}\left(j_{\alpha}dx\wedge dy\right)\\&\overset{\text{(PU)}}{=}\iiint_{I\times\partial V}dt\wedge j \end{align}

where in $\overset{\text{(PU)}}{=}$ we apply the parition of unity, in $\overset{(i)}{=}$ we apply the definition of the integral for differential forms, and in $\overset{\text{(F)}}{=}$ we apply Fubini's theorem for the component functions $j_\alpha$.