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Mathematics
The integral definition of the exponential function
The exponential function to the natural base, \(e^x\), can be defined in more than one way. One of them makes use of the integral \(\int_1^{e^x} \frac{dy}{y}= x\).
Physics
Maxwell's equations for electromagnetic waves
Maxwell's equations in a vacuum describe a wave propagating at speed \(c\) as the interaction of electric field \(E\) and magnetic field \(B\): \(\nabla \cdot E = 0\), \(\nabla \times E = -\frac{\partial B}{\partial t}\), \(\nabla \cdot B = 0\), \(\nabla \times B = \frac{1}{c^2} \frac{\partial E}{\partial t}\).
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.