User:Boris Tsirelson/Sandbox1: Difference between revisions
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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 | 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 | ||
:<math> h(t) = h_0 - \frac12 g t^2 </math> | :<math> h = f(t) = h_0 - \frac12 g t^2 </math> | ||
(here ''h''<sub>0</sub> is the initial height, and ''g'' is the [[acceleration due to gravity]]). | (here ''h''<sub>0</sub> 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''. | ||
{{Image|Moving wave.gif|right||<small> | {{Image|Moving wave.gif|right||<small>Vibrating string: a function changes in time</small>}} | ||
The instantaneous shape of a vibrating string is described by a function (the displacement | 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: | ||
:<math> y = f_t (x). </math> | |||
Infinitely many functions ''f''<sub>''t''</sub> are embraced by a single function ''f'' of two variables, | |||
:<math> y = f(x,t). </math> | |||
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Revision as of 03:41, 19 November 2010
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
(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.
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:
Infinitely many functions ft are embraced by a single function f of two variables,
