Concept

Differential equation

An equation relating a quantity to its own rate of change, whose solutions are functions rather than numbers. Solving one means finding every function satisfying it, and the useful ones pin down a single solution once a starting value is given.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The slopes the equation demands, and the curves that obey them. A field of short segments whose slope at each point is 0.9 times the height there, with 3 solution curves integrated through it; each doubles over an interval of 0.770 wherever that interval is taken.

The equation with only one answer

A rate of change proportional to the current amount is the most common description in nature, and it pins down the function completely. There is exactly one curve through each starting point, and a half-life and a doubling time are the same measurement.

analysis · the exponential

The flow of a linear equation, and the matrix that runs it for one unit of time. Paths of points moving so that their velocity is [0.25, −1.2, 1.2, 0.25] applied to their position, with the position after time 1 marked on each; the matrix taking start to finish is e^A.

The exponential of a square

The series for e makes perfect sense with a matrix in it. What comes out solves a system of equations the way the ordinary exponential solves one, and a skew matrix exponentiates into a rotation with no trigonometry anywhere.

analysis · the exponential

Named alongside it

The objects these essays reach for when they reach for this one.

e, the number Derivative Determinant Eigenvalue Eigenvector Exponential growth Half life Initial condition Integral Limit Matrix Matrix exponential

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