- #1
techmologist
- 306
- 12
On p35 of Jackson's Classical Electrodynamics 3rd Edition, the author gives the expansion of the charge density [itex]\rho(\mathbf{x'}) [/itex] around [itex]\mathbf{x'}=\mathbf{x}[/itex] as
[tex]\rho(\mathbf{x'}) = \rho(\mathbf{x}) + \frac{r^2}{6}\nabla^2\rho + ...[/tex]
where [tex]r = |\mathbf{x} - \mathbf{x'}|[/tex]
My question is, where did the first order terms go, and why is the second order term not in the form of a Hessian matrix? For instance:
[tex]\rho(\mathbf{x'}) = \rho(\mathbf{x}) + \mathbf{r} \cdot \nabla \rho + \frac{1}{2}\mathbf{r^T}H_{\rho}\mathbf{r} + ... [/tex].
He says the charge density doesn't change much in the small spherical volume under consideration, so maybe that explains the absense of first order terms, but I still don't see how to get his version of the second order term. Thank you.
[tex]\rho(\mathbf{x'}) = \rho(\mathbf{x}) + \frac{r^2}{6}\nabla^2\rho + ...[/tex]
where [tex]r = |\mathbf{x} - \mathbf{x'}|[/tex]
My question is, where did the first order terms go, and why is the second order term not in the form of a Hessian matrix? For instance:
[tex]\rho(\mathbf{x'}) = \rho(\mathbf{x}) + \mathbf{r} \cdot \nabla \rho + \frac{1}{2}\mathbf{r^T}H_{\rho}\mathbf{r} + ... [/tex].
He says the charge density doesn't change much in the small spherical volume under consideration, so maybe that explains the absense of first order terms, but I still don't see how to get his version of the second order term. Thank you.