5 August 2015

The Dirac delta "function" - part II

The relation between δ(x) and dx

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:

\int \delta(\mathbf{r} - \mathbf{r}_0) \mathrm{d}^3 \mathbf{r} = 1
\label{eq:normDirac}

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:

\delta(\mathbf{r} - \mathbf{r}_0) \mathrm{d}^3 \mathbf{r} |_{\mathbf{r}_0} = 1
\label{eq:normDirac2}

or even:

\delta(\mathbf{r} - \mathbf{r}_0) = \dfrac{1}{\mathrm{d}^3 \mathbf{r} |_{\mathbf{r}_0}}
\label{eq:normDirac3}

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:
• 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:

\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}

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:

\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}

and we can conclude that

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}
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.