Physicist Sean Carroll started this. My favorite seven equations have an engineering bent, which probably seems trite to mathematicians and scientists.

$$ \begin{equation} \label{eq:ln}
\ln(a)=\int_1^a \frac{1}{x}\,dx
\end{equation} $$

$$ \begin{equation} \label{eq:fourier}
\hat{f}(\xi) = \int_{-\infty}^{+\infty} f(x)\ e^{- 2\pi i x \xi}\,dx
\end{equation} $$

$$ \begin{equation} \label{eq:maxwell}
d * F = J
\end{equation} $$

$$ \begin{equation}
dS \ge \frac{\partial Q}{T}
\end{equation} $$

$$ \begin{equation}
\label{eq:euler-lagrange}
\frac{\partial L}{\partial f} = \frac{d}{dx}\frac{\partial L}{\partial f^\prime}
\end{equation} $$

$$ \begin{equation} \label{eq:navier-stokes}
\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u}= -\frac 1 \rho \nabla \bar{p} + \nu \nabla^2 \mathbf u + \tfrac13 \, \nu \nabla (\nabla\cdot\mathbf{u}) + \mathbf{g}
\end{equation} $$

$$ \begin{equation} \label{eq:quaternion}
i^2 = j^2 = k^2 = ijk = -1
\end{equation} $$

- Natural Log: An undergrad math professor said we first define \(ln(a)\). What follows is \(e\), which enables everywhere \(e\) shows up, such as…
- Fourier transform: The foundation of signal processing, state-space controls, compression, etc.
- Let there be light, and radios, and X-rays, and electricity.
- Second Law: Helps explain how thermodynamic devices work — from refrigerators to jet engines. Entropy might also be responsible for the arrow of time and gravity.
- Euler–Lagrange equation: Necessary but insufficient in the Calculus of Variations, which is used in trajectory optimization and Finite Element Analysis. Testing was once the main way to verify that things work. Now tests anchor models and models verify designs.
- Navier-Stokes Equations: We are surrounded by air, a compressible, viscous fluid.
- Quaternions: Not as fundamentally useful as 1 – 6, given the multitude of ways to represent an attitude or rotation, but they have a beauty and a special place in the history of satellite attitude determination and controls.