Lance got a custom made glow in the dark “math, physics, and chemistry” shirt for his birthday a year or two ago (It was Sofia’s idea, she bought the materials, and I got to play.)
He is clearly unimpressed with the photo op, but I wanted to share this shirt, which includes some of the what I thought were a number of the most interesting or important relationships
- Gamma function, which generalizes factorial to non-integer values.
\begin{equation}\label{eqn:formulas:20}
\Gamma(z + 1) = \int_0^\infty t^z e^{-t} dt.
\end{equation}
This formula satisfies \( n! = \Gamma(n + 1) \). - Euler’s:
\begin{equation}\label{eqn:formulas:40}
e^{n i \theta} = \lr{ \cos\theta + i \sin\theta }^n.
\end{equation} - Schr\”{o}dinger’s equation
\begin{equation}\label{eqn:formulas:60}
i \, \hbar \PD{t}{} \ket{\psi} = H \ket{\psi}.
\end{equation} - Taylor series
\begin{equation}\label{eqn:formulas:80}
f(x) = \sum_{k = 0}^\infty \frac{f^{(k)}(0)}{k!} x^k.
\end{equation} - Euler-Lagrange equations:
\begin{equation}\label{eqn:formulas:100}
\PD{x_i}{\LL} = \frac{d}{dt} \PD{\dot{x}_i}{\LL}.
\end{equation}
These formulas can be used to express most dynamics relations. You can think of them as basically being the consequence that physical laws are either inherently greedy or lazy. - Stokes’ theorem, in its geoemetric algebra form
\begin{equation}\label{eqn:formulas:120}
\oint_{\partial V} d^n \Bx \cdot T = \int_V d^n \Bx \cdot \lr{ \spacegrad \cdot T }.
\end{equation} - Quantum commutator relationships between position and momentum
\begin{equation}\label{eqn:formulas:140}
\antisymmetric{X}{P} = i \, \hbar.
\end{equation} - Fourier transform
\begin{equation}\label{eqn:formulas:160}
\tilde{f}(\omega) = \int_{-\infty}^\infty f(t) e^{-i \omega t} dt.
\end{equation} - Vector product
\begin{equation}\label{eqn:formulas:180}
\Ba \Bb = \Ba \cdot \Bb + \Ba \wedge \Bb.
\end{equation}
In geometric algebra the vector product has a dot and wedge product split. In \R{3} you can write this as
\( \Ba \Bb = \Ba \cdot \Bb + I (\Ba \cross \Bb) \), where \( I = \Be_1 \Be_2 \Be_3 \). - Relativisitic energy (Einstein’s)
\begin{equation}\label{eqn:formulas:200}
E = \frac{m c^2}{\sqrt{1 – (\Bv/c)^2}}.
\end{equation} - Cauchy contour integral relationships for the nth derivative
\begin{equation}\label{eqn:formulas:220}
f^{(n)}(s) = \frac{n!}{2 \pi i} \int_C dz \frac{f(z)}{(z-s)^{n+1}}
\end{equation} - Maxwell’s equations
\begin{equation}\label{eqn:formulas:240}
\begin{aligned}
\partial_\mu F^{\mu\nu} &= J^\nu \\
\epsilon^{\mu \nu \rho \sigma} \partial_\nu F_{\rho \sigma} &= 0.
\end{aligned}
\end{equation}
I’m partial to the geometric algebra form of Maxwell’s equations \( \grad F = J \), but that wouldn’t have looked as good on the shirt. - Dirac equation
\begin{equation}\label{eqn:formulas:260}
0 = \lr{ i \gamma^\mu \partial_\mu – m } \psi.
\end{equation}
If you had to pick a set of important formulas on a tee-shirt, which ones would you pick?
