#### The relation between δ(*x*) and d*x*

After introducing the Dirac delta "function" \(\delta(x)\) in the previous post, I'll try now to explain the relation between it and the differential element \(\mathrm{d}x\). In the process, I'll through all mathematical rigour out the window and aim for an intuitive understanding.

The "integral" of the Dirac delta equals 1, as we can see by taking a constant field \(\mathbf{E}(\mathbf{r}) = \mathbf{E}_0\) in Eq. (4) of post I:

\begin{equation}\int \delta(\mathbf{r} - \mathbf{r}_0) \mathrm{d}^3 \mathbf{r} = 1

\label{eq:normDirac}

\end{equation}

keeping in mind that the integration sign in \eqref{eq:normDirac} is just a convenient symbol, and not a true Riemann (or Lebesgue) integral. We could even say that the integration is superfluous, since the Dirac delta is only different from zero in \(\mathbf{r}_0\), so why not write directly:

\begin{equation}\delta(\mathbf{r} - \mathbf{r}_0) \mathrm{d}^3 \mathbf{r} |_{\mathbf{r}_0} = 1

\label{eq:normDirac2}

\end{equation}

or even:

\begin{equation}\delta(\mathbf{r} - \mathbf{r}_0) = \dfrac{1}{\mathrm{d}^3 \mathbf{r} |_{\mathbf{r}_0}}

\label{eq:normDirac3}

\end{equation}

where the bar indicates that the volume element is to be evaluated at \(\mathbf{r}_0\): we'll soon see what this means.

While Equation \eqref{eq:normDirac} can be seen as shorthand notation for a more complicated concept, \eqref{eq:normDirac2} and \eqref{eq:normDirac3} are mathematically meaningless. They can still remind us of some important properties of the Dirac delta:

While Equation \eqref{eq:normDirac} can be seen as shorthand notation for a more complicated concept, \eqref{eq:normDirac2} and \eqref{eq:normDirac3} are mathematically meaningless. They can still remind us of some important properties of the Dirac delta:

- It has units of reciprocal volume, as we have already seen.
- It transforms as the reciprocal of the volume element under a change of coordinates.

#### Change of coordinates

When switching coordinates from \(\mathbf{r}\) to \(\mathbf{u}\), the volume element changes according to:

\begin{equation}\mathrm{d}^3 \mathbf{r} |_{\mathbf{r}_0} = \left | J \left [\mathbf{r}(\mathbf{u}) \right ]_{\mathbf{u}_0} \right | \mathrm{d}^3 \mathbf{u} |_{\mathbf{u}_0}

\label{eq:Jacob}

\end{equation}

where \(\mathbf{r}_0 = \mathbf{r}(\mathbf{u}_0)\) and \(J\) is the determinant of the Jacobian matrix that converts from \(\mathbf{u}\) to \(\mathbf{r}\) (to be evaluated in \(\mathbf{u}_0\)). We assume that \(\mathbf{u}(\mathbf{r})\) is a one-to-one correspondence and that \(J\) is always nonzero.

Under the same transformation, the Dirac delta behaves as:

\begin{equation}Under the same transformation, the Dirac delta behaves as:

\delta(\mathbf{r} - \mathbf{r}_0) = \dfrac{1}{\left | J \left [\mathbf{r}(\mathbf{u}) \right ]_{\mathbf{u}_0} \right |} \delta(\mathbf{u} - \mathbf{u}_0)

\label{eq:Jacobinv}

\end{equation}

and we can conclude that

in the sense discussed above. Finally, note that \eqref{eq:Jacobinv} is an example of the composition of the Dirac delta with a function \(f(x)\) (for simplicity, we will work in one dimension), given by:

\begin{equation}**The Dirac delta behaves as one over the element volume**in the sense discussed above. Finally, note that \eqref{eq:Jacobinv} is an example of the composition of the Dirac delta with a function \(f(x)\) (for simplicity, we will work in one dimension), given by:

\delta(f(x)) = \sum_{i} \dfrac{1}{\left | f'(x_i)\right |} \delta (x - x_i)

\label{eq:comp}

\end{equation}

where \(x_i\) are the solutions of \(f(x) = 0\). They can be multiple, since \(f\) does not generally fulfill the conditions we imposed on the Jacobian transformation.

## No comments:

## Post a Comment