- #1
William Crawford
- 41
- 36
- 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.
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.