User:Boris Tsirelson/Sandbox1

In Newtonian mechanics, coordinates of moving bodies are functions of time. For example, the classical equation for a falling body; its height at a time t is

$h=f(t)=h_{0}-{\frac {1}{2}}gt^{2}$

(here h₀ is the initial height, and g is the acceleration due to gravity). Infinitely many corresponding values of t and h are embraced by a single function f.

PD Image
Vibrating string: a function changes in time

The instantaneous shape of a vibrating string is described by a function (the displacement y as a function of the coordinate x), and this function changes in time:

$y=f_{t}(x).$

Infinitely many functions f_t are embraced by a single function f of two variables,

$y=f(x,t).$

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Wave equation (classical physics)

Partial differential equation