Undergrad Partial derivative of Dirac delta of a composite argument

[William Crawford]

TL;DR

I wish to derive a formula for the partial derivative w.r.t. time of Dirac delta of a composite argument

I'm trying to prove the following statement: $$ D\partial_t\left(\delta\circ\mathbf{v}\right) = J^i\partial_i\left(\delta\circ\mathbf{v}\right), $$ where $\mathbf{v}$ is some function of time and $n$-dimensional space, $D$ is the Jacobian determinant associated with $\mathbf{v}$, that is $$ D = \frac{1}{n!}\epsilon^{i_1i_2\ldots i_n}\epsilon_{j_1j_2\ldots j_n}\partial_{i_1}v^{j_1}\partial_{i_2}v^{j_2}\cdots\partial_{i_n}v^{j_n} $$ and $J^i$ is the vector $$ J^i = \frac{1}{(n-1)!}\epsilon^{ii_2\ldots i_n}\epsilon_{j_1j_2\ldots j_n}\partial_{t}v^{j_1}\partial_{i_2}v^{j_2}\cdots\partial_{i_n}v^{j_n} $$.

So far I've been able to prove that the Jacobian determinant satisfy the conservation equation: $$\partial_t D = \partial_iJ^i.$$
My attempt at proving the above is the following. Let $\varphi$ be a test function (i.e. smooth and compact support), then
$$
\begin{align}
\int_\mathbb{R}\int_{\mathbb{R}^n}D\partial_t\left(\delta\circ\mathbf{v}\right)\varphi\ d^n\mathbf{x}\,dt
&= -\int_\mathbb{R}\int_{\mathbb{R}^n}\left(\delta\circ\mathbf{v}\right)\partial_t(D\varphi)\ d^n\mathbf{x}\,dt \\
&= -\int_\mathbb{R}\int_{\mathbb{R}^n}\left(\delta\circ\mathbf{v}\right)\left(\partial_tD\varphi + D\partial_t\varphi\right)\ d^n\mathbf{x}\,dt \\
&= -\int_\mathbb{R}\int_{\mathbb{R}^n}\left(\delta\circ\mathbf{v}\right)\partial_iJ^i\varphi\ d^n\mathbf{x}\,dt - \int_\mathbb{R}\int_{\mathbb{R}^n}D\left(\delta\circ\mathbf{v}\right)\partial_t\varphi\ d^n\mathbf{x}\,dt \\
&= \int_\mathbb{R}\int_{\mathbb{R}^n}J^i\left[\partial_i\left(\delta\circ\mathbf{v}\right)\varphi + \left(\delta\circ\mathbf{v}\right)\partial_i\varphi\right]\ d^n\mathbf{x}\,dt - \int_\mathbb{R}\int_{\mathbb{R}^n}D\left(\delta\circ\mathbf{v}\right)\partial_t\varphi\ d^n\mathbf{x}\,dt \\
&= \int_\mathbb{R}\int_{\mathbb{R}^n}J^i\partial_i\left(\delta\circ\mathbf{v}\right)\varphi\ d^n\mathbf{x}\,dt - \int_\mathbb{R}\int_{\mathbb{R}^n}\left(\delta\circ\mathbf{v}\right)\left[D\partial_t - J^i\partial_i\right]\varphi\ d^n\mathbf{x}\,dt
\end{align}
$$
However, I don't see how the last term vanishes.

 

[Ssnow]

Hi, I think there is an error in the last term, this must be zero because you must be able to reconstruct the conservation condition ...
Ssnow

 

[mad mathematician]

I guess you use the definition of a weak derivative. In your third equality where you interchange between:
$D(\delta\circ v)\partial_t \varphi$ and $(\delta\circ v)\partial_t D\varphi$, you forgot to multiply by another minus sign.