I had the idea of writing this post a couple of weeks ago, but I didn't feel like I had enough stuff to write here at that time. Now I do, so here goes. (Also, here's hoping that inputting LaTeX into this post works once more.)
When I took 18.03 — Differential Equations in 2010 fall, one of the topics covered was linear time-invariant systems. The general system of interest was $Lu(t) = f(t)$ where $L$ is a linear time-invariant operator. The technique of course is to find a weight function $w(t)$ where $Lw(t) = \delta(t)$, and once that is done, the solution is $u(t) = \int_{-\infty}^{\infty} f(t') w(t - t') dt'$ which is a convolution of the input $f$ with the weight $w$. The professor mentioned that it is essentially akin to inverting the operator $L$, but while I could see the general utility in this method, I never quite understood why it might be considered inversion on any deeper level.
Last semester, I took 8.07 — Electromagnetism II, and there we discussed Green's functions a little more in the context of electromagnetism & electrodynamics. In a static situation, the Green's function comes up in solving the Poisson equation $\nabla^2 \phi = -\rho$. In this case, $\nabla^2 G(\vec{x}, \vec{x}') = -\delta(\vec{x} - \vec{x}')$ is solved by the familiar potential of a unit point charge $G(\vec{x}, \vec{x}') = \frac{1}{4\pi |\vec{x} - \vec{x}'|}$. I started to see a little more clearly why this worked, because if a general charge distribution was some superposition of point charges, then a general potential distribution should be the same superposition of point charge potentials. However, it still wasn't entirely clear to me how this was "inversion" per se. Follow the jump to see what changed.
When I took 18.03 — Differential Equations in 2010 fall, one of the topics covered was linear time-invariant systems. The general system of interest was $Lu(t) = f(t)$ where $L$ is a linear time-invariant operator. The technique of course is to find a weight function $w(t)$ where $Lw(t) = \delta(t)$, and once that is done, the solution is $u(t) = \int_{-\infty}^{\infty} f(t') w(t - t') dt'$ which is a convolution of the input $f$ with the weight $w$. The professor mentioned that it is essentially akin to inverting the operator $L$, but while I could see the general utility in this method, I never quite understood why it might be considered inversion on any deeper level.
Last semester, I took 8.07 — Electromagnetism II, and there we discussed Green's functions a little more in the context of electromagnetism & electrodynamics. In a static situation, the Green's function comes up in solving the Poisson equation $\nabla^2 \phi = -\rho$. In this case, $\nabla^2 G(\vec{x}, \vec{x}') = -\delta(\vec{x} - \vec{x}')$ is solved by the familiar potential of a unit point charge $G(\vec{x}, \vec{x}') = \frac{1}{4\pi |\vec{x} - \vec{x}'|}$. I started to see a little more clearly why this worked, because if a general charge distribution was some superposition of point charges, then a general potential distribution should be the same superposition of point charge potentials. However, it still wasn't entirely clear to me how this was "inversion" per se. Follow the jump to see what changed.
