Content deleted Content added
→Multiple issues: rp |
→Periodic functions: new section |
||
Line 35:
::I have considered the section "Tangent hyperplane to the graph of the function, expressed in term of the gradient" as off-topic because of the emphasis on geometry and equations, which does not belong to the subject of the article. Moreover, it supposes that the reader knows about high dimensional geometry and hypersurfaces. However a section on the best approximation of a differentiable function by a linear function would have almost the same mathematical content and fit exactly the subject of the article.
:: About the generalities, I'll propose soon a new presentation. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 14:14, 1 July 2013 (UTC)
== Periodic functions ==
Unable to find a ref for now (should be able to in time), but this would be good to add, my rough formulation would be:
{{quotation|A function of the form:
:<math>f\,:\,X\rightarrow \mathbb{R}</math>
:<math>f(\boldsymbol{x} + \boldsymbol{T}) = f(\boldsymbol{x}) </math>
where {{nowrap|'''''T''''' {{=}} (''T''<sub>1</sub>, ''T''<sub>2</sub>, ..., ''T''<sub>''n''</sub>)}}, and using [[interval notation]]:
:<math>X = \{ \boldsymbol{x}\in\mathbb{R}^n \,:\, x_i \in [a_i,a_i+T_i] \}</math>
where ''i'' can be any or all of 1, 2, ..., ''n'', is a '''[[periodic function]] of several real variables''', with periods ''T<sub>i</sub>''. Not all of the ''T<sub>i</sub>'' have to be nonzero.
For example:
:<math>f\,:\, X \rightarrow \mathbb{R}</math>
:<math>f(x,y,z) = z \sin (ax) \cos (by) </math>
with ___domain:
:<math>X = \{ (x,y,z)\in\mathbb{R}^3 \,:\, x\in[0,2\pi/a] \,, y\in[-\pi/b,\pi/b]\,,z\in(-\infty,\infty)\}</math>
is periodic in ''x'' and ''y'', but not ''z'':
:<math>f(x,y,z) = f(x + 2\pi/a,y + 2\pi/b,z+0) </math>
}}
A physical example would be the [[Bloch wave]]. [[user:Maschen|'''M∧''Ŝ''''']][[special:contributions/Maschen|''c''<sup>2</sup>''ħ''ε]][[user talk:maschen|''И<sub>τlk</sub>'']] 13:33, 8 July 2013 (UTC)
| |||