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Mathematics
The Gaussian integral
Integrating the Gaussian function from minus to plus infinity yields \(\int_{-\infty}^{\infty} e^{-x^2}dx=\sqrt{\pi}\), which is known as the Gaussian integral.
Physics
A particle's rest energy in special relativity
In Einstein's special relativity, the rest energy of a mass \(m\) is \(E = mc^2\), where \(c\) is the speed of light in a vacuum.
Physics
The heat equation
The heat equation, \(\frac{\partial \varnothing}{\partial t} = D \nabla^2 \varnothing\), uses the Laplacian of \(\varnothing\) to express diffusion through a medium.