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Since I'm still building this page, I figured I ought to use the brilliance of MathJax to depict some of the most amazing formulae I've seen. If there are any mistakes or if my cursory explanations are hard to understand, feel free to contact me at

Elegance via Equality: Maxwell's Equations, tidied up by Riemannian Geometry

Equations discussed:
Where:
$d_{i}$ is the $i^\text{th}$ exterior derivative
$\ast$ is the Hodge Star operation, $\ast : \bigwedge^{k}(M) \rightarrow \bigwedge^{n-k}(M)$
$F$ is the electromagnetic two-form
$J$ is the Noether (conserved) four-current
$A$ is the electromagnetic (connection) one-form

Notes: The following assumes some basic physics knowledge, calculus and a (possibly incomplete) understanding of differential forms in $\mathbb{R}^{n}$. Whenever the Minkowski metric arises, I assume a signature $(+,-,-,-)$ so that time-like intervals are defined for $\Delta s = \left\Vert \vec{x} \right\Vert_{\mathbb{R}^{1,3}} > 0$. If local coordinates ${x^{\mu}}$ are specified, then $\partial_{\mu} := \frac{\partial}{\partial x^{\mu}}$. Finally, we assume that we are in free space so that $\epsilon_{0},\mu_{0}$ are constant everywhere.

Maxwell's equations describe: a) how electricity and magnetism arise out of the same fundamental force, b) the time-evolution of these (vector) fields and c) provide a basis for the wave-like nature of light. These equations are fundamental to both our understanding of electricity and magnetism as well as applications of this knowledge. Once Einstein's Special Relativity truly finished the job of proving that electricity and magnetism can be looked at as the same force (in different frames of reference), the "hidden" spacetime symmetries of Maxwell's Equations came into clear view. As Einstein, David Hilbert and others were formulating general relativity, a geometric view of the differential equations that encompass Maxwell's equations became increasingly important. This is because General Relativity treats the (Lorentzian) metric tensor (field), which determines curvature, distance and preserves angles (i.e. it is a conformal map), as the main dynamic variable of interest. Heuristically this means that instead of studying the motion of an individual particle or objects ("the dynamics of an object"), we are studying the structure of the ambient space that objects reside in, as we assume that the "shape" of the universe is constantly changing. Hence Maxwell's Equations marked one of the first forays into the geometric study of partial differential equations. If you've taken an Electricity and Magnetism course, you may have seen Maxwell's equations presented as:

In case you were wondering, I'm using CGS units to eliminate the feisty constants, $\mu_{0},\epsilon_{0}$. Now where's the geometry? Firstly, take a look at the equation $\nabla\cdot\mathbf{B} = 0$. From the Biot-Savart Law, we know that $\mathbf{B} = \frac{I}{4\pi}\int\frac{d\ell \times \hat{\mathbf{r}}}{\mathbf{r}\cdot\mathbf{r}}$, which defines the magnetic field (to physicists) as an axial vector. Intuitively, what this means is that the orientation of $\mathbf{B}$ isn't symmetry with regard to mirror reflections. For example if I reverse orientation (i.e. Line Element: $d\ell \mapsto -d\ell$, Displacement Vector : $\hat{\mathbf{r}} \mapsto -\hat{\mathbf{r}}$), then I actually end up with the same field, as per the Biot-Savart Law. Mathematically, this says that $\mathbf{B}$ doesn't transform under $\mathsf{O}(n)$ as a "standard" vector does. In fact, this was probably the first physical realization of W. Clifford's forgotten but extremely useful concept of a Clifford Algebra. This odd behavior of the magnetic field is formally characterized by saying that the magnetic field is actually a two-form, i.e. $\mathbf{B} \in \bigwedge^{2}{\mathbb{R}^3}$. Geometrically, this fact acknowledges the arbitrarily choice of a right-handed system, $\mathbf{a},\mathbf{b},\mathbf{a}\times\mathbf{b}$ — the cross product could just as easily give a resultant that yields left-handed coordinates, $\mathbf{a},\mathbf{b},-\mathbf{a}\times\mathbf{b}$.     With this key concept in hand, we can now meaningfully write $\mathbf{B} = B_{ij}\, dx^{i}\wedge dx^{j}$. From Gauss's equation on magnetic monopoles, $\nabla\cdot\mathbf{B} = 0$ and the fact that $d \mathbf{B} = \frac{\partial B_{ij}}{\partial x^{k}} \, dx^{k} \wedge dx^{i} \wedge dx^{j} = \text{div}\mathbf{B}$, we know that $\mathbf{B}$ must be a closed 2-form. One thing that has always bothered me is the rather hand-waving proof one receives in E&M classes about the existence of the Magnetic Vector potential, $\mathbf{A}$, defined by $\nabla\times\mathbf{A} = \mathbf{B}$. However, we can now recover such a proof by simply applying the Poincaré Lemma to $\mathbf{B}$. This lemma prescribes that all closed forms (on $\mathbb{R}^{n}$) are exact on any contractible, open subset of $\mathbb{R}^{n}$. This is good enough, since the various theorems for the Cauchy Problem (existence and uniqueness for partial differential equations) most likely won't give solutions defined for all space.

Great! Now in special relativity, one expresses vector quantities in 4-vectors, which is simply a physicists way of saying any vector $v\in\mathbb{R}^{1,3}$. In the above treatment of the magnetic field, we expressed the field as a closed and exact form in $\mathbb{R}^{3}$. The question now is, can we do the same thing in $\mathbb{R}^{4}$, unifying Maxwell's Equations as well as the electric and magnetic fields? The answer is yes! Without the loss of generality, now assume that we choose units such that $c := 1$. Suppose $\phi$ and $\mathbf{A}$ are the electric and magnetic potentials, respectively. Now given standard coordinates (not light-cone coordinates!) $\{x^{\mu}\}$ in $\mathbb{R}^{1,3}$ define the electromagnetic four-potential by:
Note that $A$ is also a one-form, since $\mathbf{A}$ is a one-form (i.e. the above construction simply extends the one-form $\mathbf{A}$). Also note that we define $\phi$ slightly differently; from the first source equation, $\nabla\times\mathbf{E} = -\frac{1}{c}\frac{\partial\mathbf{B}}{\partial t}$, we see that $\mathbf{E}$ is not in general, curl-less. From calculus, we know that a vanishing curl is a necessary and sufficient condition for the existence of a scalar potential. However, since we have seen that the magnetic field has a vector potential, the source equation can be written as: $\nabla\times\mathbf{E}+\frac{1}{c}\frac{\partial\mathbf{B}}{\partial t}=\nabla\times\left(\mathbf{E}+\frac{1}{c}\frac{\partial\mathbf{A}}{\partial t}\right)=0$. Thus we define $\phi$ to be the scalar potential of $\mathbf{E}+\frac{1}{c}\frac{\partial\mathbf{A}}{\partial t}$.     Now we would like to define some simple differential equations in $A$ that characterize Maxwell's Equations fully, while yielding some geometric data. Since we found that $d\mathbf{B} = \mathbf{A}$, one might think that we can get a complete description. Let's check: $\begin{eqnarray*} dA & = & \frac{dA^{\nu}}{dx^{\mu}}dx^{\mu}\wedge dx^{\nu}\\ & = & (\partial_{\mu}A^{\nu}-\partial_{\nu}A^{\mu})\, dx^{\mu}\wedge dx^{\nu}\end{eqnarray*}$ If we compute $\partial_{\mu}A^{\nu}-\partial_{\nu}A^{\mu}$, we find that $\begin{equation*}F_{\mu\nu} := \partial_{\mu}A^{\nu}-\partial_{\nu}A^{\mu} \Rightarrow \left[\begin{array}{cccc} 0 & -E_{x} & -E_{y} & -E_{z}\\ E_{x} & 0 & -B_{z} & B_{y}\\ E_{y} & B_{z} & 0 & -B_{x}\\ E_{z} & -B_{y} & B_{x} & 0\end{array}\right]\end{equation*}$ Another thing that was also esoteric to me when I took E&M was the arbitrary of the electromagnetic field strength $F_{\mu\nu}$; from the above, we see precisely where it comes from: $F_{\mu\nu} = \partial_{\mu}A^{\nu} - \partial_{\nu}A^{\mu}$.

Let $X$ be a Riemannian Manifold. Suppose that $A$ be the connection 1-form electromagnetic potential. By construction, the Faraday form $F$ is a curvature form on a principle bundle with structure group $U(1) \cong SO(2)$ (See entry on "Bott Perioidity" to get a marginally better definition). Moreover let $d_{i} : \Omega^{i}(X) \rightarrow \Omega^{i+1}(X)$ be the $i^{th}$ exterior derivative. As such, the Poincaré lemma provides that (at least locally, since we can use a chart that maps a neighborhood of interest to a contractible open subset of $\mathbb{R}^{n}$), $F$ is exact. The Second Bianchi Identity holds for any connection over a principle bundle so that $d_{2}F = 0$. Using the definition of the Noether Current for Electromagnetism (I'll put in a more complete definition later!) and letting $\ast$ be the Hodge Star operator, we have $(1)$.