This post contains an overview of fundamental equations that have shaped our understanding of mathematics and physics. Each equation is accompanied by a brief explanation and its author. Each equation here could have a dedicated post, and I may do that some day, but a summarized collection gives one a cursory look good for intuition building.

1. Pythagorean Theorem (c. 570-495 BCE)

\[a^2 + b^2 = c^2\]
  • Relates the lengths of sides in a right triangle
  • Attributed to Pythagoras, though likely known earlier

2. Archimedes’ Principle (c. 287-212 BCE)

\[F_b = \rho g V\]
  • Explains flotation and buoyancy
  • Discovered by Archimedes of Syracuse

3. Heron’s Formula (c. 10-70 CE)

\[A = \sqrt{s(s-a)(s-b)(s-c)}\]
  • Calculates the area of a triangle given the lengths of its sides
  • Developed by Hero of Alexandria

4. Brahmagupta’s Formula (628 CE)

\[(ac+bd)(ad+bc) = (ad-bc)^2 + (ac-bd)^2\]
  • Relates to the product of two sums of two squares
  • Discovered by the Indian mathematician Brahmagupta

5. Cardano’s Formula (1545)

\[x = \sqrt[3]{-\frac{q}{2}+\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}+\sqrt[3]{-\frac{q}{2}-\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}\]
  • Solves cubic equations of the form $x^3 + px + q = 0$
  • Published by Gerolamo Cardano

6. Euler’s Formula for Polyhedra (1752)

\[V - E + F = 2\]
  • $V$: vertices, $E$: edges, $F$: faces
  • Relates the number of vertices, edges, and faces in a polyhedron

7. Euler’s Identity (1748)

\[e^{i\pi} + 1 = 0\]
  • Connects fundamental constants $e$, $i$, $\pi$
  • Discovered by Leonhard Euler

8. Lagrange’s Equations (1788)

\[\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = 0\]
  • Fundamental in classical mechanics
  • Developed by Joseph-Louis Lagrange

9. Fourier Transform (1822)

\[F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt\]
  • Decomposes functions into frequency components
  • Introduced by Joseph Fourier

10. Gaussian Distribution (1809)

\[f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}\]
  • Describes normal distribution
  • Developed by Carl Friedrich Gauss

11. Newton’s Second Law of Motion (1687)

\[\mathbf{F} = m\mathbf{a}\]
  • Cornerstone of classical mechanics
  • Formulated by Isaac Newton

12. Newton’s Law of Universal Gravitation (1687)

\[F = G\frac{m_1m_2}{r^2}\]
  • Describes gravitational attraction between bodies
  • Also by Newton

13. Ideal Gas Law (17th-19th century)

\[PV = nRT\]
  • Describes behavior of ideal gases
  • Combination of work by Boyle, Charles, and Gay-Lussac

14. Navier-Stokes Equations (1822)

\[\rho\left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla\mathbf{v}\right) = -\nabla p + \nabla \cdot \mathbf{T} + \mathbf{f}\]
  • Describes fluid motion
  • Formulated by Claude-Louis Navier and George Gabriel Stokes

15. Faraday’s Law of Induction (1831)

\[\mathcal{E} = -\frac{d\Phi_B}{dt}\]
  • Relates changing magnetic field to induced electromotive force
  • Discovered by Michael Faraday

16. Maxwell’s Equations (1861-1862)

\[\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}\] \[\nabla \cdot \mathbf{B} = 0\] \[\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}\] \[\nabla \times \mathbf{B} = \mu_0\left(\mathbf{J} + \epsilon_0\frac{\partial \mathbf{E}}{\partial t}\right)\]
  • Unified electricity and magnetism
  • Formulated by James Clerk Maxwell

17. Boltzmann Entropy Formula (1877)

\[S = k_B \ln W\]
  • Relates entropy to number of microstates
  • Formulated by Ludwig Boltzmann

18. Riemann Zeta Function (1859)

\[\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}\]
  • Defined for complex $s$
  • Introduced by Bernhard Riemann

19. Planck’s Law (1900)

\[B_\nu(T) = \frac{2h\nu^3}{c^2}\frac{1}{e^{h\nu/kT} - 1}\]
  • Describes black body radiation
  • Formulated by Max Planck

20. Lorentz Transformation (1904)

\[\begin{align*} t' &= \gamma(t - vx/c^2) \\ x' &= \gamma(x - vt) \\ y' &= y \\ z' &= z \end{align*}\]
  • Relates space and time measurements between inertial frames
  • Developed by Hendrik Lorentz

21. Einstein’s Mass-Energy Equivalence (1905)

\[E = mc^2\]
  • Part of Einstein’s Special Relativity
  • Demonstrates equivalence of mass and energy

22. Einstein Field Equations (1915)

\[G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}\]
  • Describes gravitation as a consequence of spacetime curvature
  • Core of Einstein’s General Relativity

23. Schrödinger Equation (1926)

\[i\hbar\frac{\partial}{\partial t}\Psi = \hat{H}\Psi\]
  • Fundamental to quantum mechanics
  • Formulated by Erwin Schrödinger

24. Heisenberg Uncertainty Principle (1927)

\[\Delta x \Delta p \geq \frac{\hbar}{2}\]
  • Sets limits on precision of simultaneous measurements
  • Formulated by Werner Heisenberg

25. Dirac Equation (1928)

\[i\hbar\gamma^\mu\partial_\mu \psi - mc\psi = 0\]
  • Describes relativistic quantum mechanics
  • Developed by Paul Dirac

26. Schrödinger’s Cat Thought Experiment (1935)

\[|\psi\rangle = \frac{1}{\sqrt{2}}(|\text{alive}\rangle + |\text{dead}\rangle)\]
  • Illustrates quantum superposition
  • Proposed by Erwin Schrödinger

27. Shannon Entropy (1948)

\[H = -\sum_{i} p_i \log_2 p_i\]
  • Measures information content
  • Developed by Claude Shannon

28. Feynman Path Integral (1948)

\[\langle \phi_f | e^{-iHt/\hbar} | \phi_i \rangle = \int \mathcal{D}\phi\, e^{iS[\phi]/\hbar}\]
  • Alternative formulation of quantum mechanics
  • Developed by Richard Feynman

29. Black-Scholes Equation (1973)

\[\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0\]
  • Models financial markets
  • Developed by Fischer Black, Myron Scholes, and Robert Merton

30. Standard Model Lagrangian (20th century)

\[\mathcal{L} = \mathcal{L}_{\text{Gauge}} + \mathcal{L}_{\text{Fermion}} + \mathcal{L}_{\text{Higgs}}\]
  • Encompasses fundamental particles and forces
  • Developed by multiple physicists over decades

These equations represent pivotal moments in the development of mathematics and physics, encapsulating profound insights into the nature of reality and the structure of mathematical systems. Each has played a crucial role in advancing our understanding of the universe and continues to influence modern research and applications.