Most beautiful equations
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Intermediate
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Euler's identity
eiπ+1=0e^{i\pi} + 1 = 0
Einstein's mass-energy equivalence
E=mc2E = mc^2
The general wave equation
∂2u∂x2−1c2∂2u∂t2=0\frac{\partial^2 u}{\partial x^2} - \frac{1}{c^2} \frac{\partial^2 u}{\partial t^2} = 0
Maxwell's equations
Gauss’s law for electricity∇⋅E=ρε0Gauss’s law for magnetism∇⋅B=0Faraday’s law of induction∇×E=−∂B∂tAmpere’s law with Maxwell’s addition∇×B=μ0J+μ0ε0∂E∂t\begin{gather} \text{\footnotesize Gauss's law for electricity} \nonumber \\ \nabla \cdot \mathbb{E} = \frac{\rho}{\varepsilon_0} \nonumber \\ \text{\footnotesize Gauss's law for magnetism} \nonumber \\ \nabla \cdot \mathbb{B} = 0 \nonumber \\ \text{\footnotesize Faraday's law of induction} \nonumber \\ \nabla \times \mathbb{E} = -\dfrac{\partial \mathbb{B}}{\partial t} \nonumber \\ \text{\footnotesize Ampere's law with Maxwell's addition} \nonumber \\ \nabla \times \mathbb{B} = \mu_0 \mathbb{J} + \mu_0 \varepsilon_0 \dfrac{\partial \mathbb{E}}{\partial t} \nonumber \\ \end{gather}
Shrodinger' wave equation
−ℏ22m∂2ψ∂x2=iℏ∂ψ∂t-\dfrac{\hbar^2}{2m}\dfrac{\partial^2\psi}{\partial x^2 }= i\hbar\dfrac{\partial \psi}{\partial t}