Difference between revisions of "Index of notation"
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* Differential Geometry | * Differential Geometry | ||
* Manifolds | * Manifolds | ||
| − | | <math>C^\infty_p(A)</math> denotes the set of all [[Germ|germs]] of <math>C^\infty</math> functions on {{M|A}} at {{M|p}} | + | | <math>C^\infty_p(A)</math> denotes the set of all [[Germ|germs]] of <math>C^\infty</math> functions on {{M|A}} at {{M|p}}<br/> |
| + | [[The set of all germs of smooth functions at a point]] | ||
|- | |- | ||
| <math>C^k([a,b],\mathbb{R})</math> | | <math>C^k([a,b],\mathbb{R})</math> | ||
Revision as of 01:19, 5 April 2015
[math]\newcommand{\bigudot}{ \mathchoice{\mathop{\bigcup\mkern-15mu\cdot\mkern8mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}} }[/math][math]\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}[/math][math]\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }[/math]Ordered symbols are notations which are (likely) to appear as they are given here, for example [math]C([a,b],\mathbb{R})[/math] denotes the continuous function on the interval [ilmath][a,b][/ilmath] that map to [ilmath]\mathbb{R} [/ilmath] - this is unlikely to be given any other way because "C" is for continuous.
Ordered symbols
These are ordered by symbols, and then by LaTeX names secondly, for example [math]A[/math] comes before [math]\mathbb{A}[/math] comes before [math]\mathcal{A}[/math]
| Expression | Context | Details |
|---|---|---|
| [math]\|\cdot\|[/math] |
|
Denotes the Norm of a vector |
| [math]\|f\|_{C^k}[/math] |
|
This Norm is defined by [math]\|f\|_{C^k}=\sum^k_{i=0}\sup_{t\in[0,1]}(|f^{(i)}(t)|)[/math] - note [math]f^{(i)}[/math] is the [math]i^\text{th}[/math] derivative. |
| [math]\|f\|_{L^p}[/math] |
|
[math]\|f\|_{L^p}=\left(\int^1_0|f(t)|^pdt\right)^\frac{1}{p}[/math] - it is a Norm on [math]\mathcal{C}([0,1],\mathbb{R})[/math] |
| [math]\|f\|_\infty[/math] |
|
It is a norm on [math]C([a,b],\mathbb{R})[/math], given by [math]\|f\|_\infty=\sup_{x\in[a,b]}(|f(x)|)[/math] |
| [math]C^\infty[/math] |
|
That a function has continuous (partial) derivatives of all orders, it is a generalisation of [math]C^k[/math] functions |
| [math]C^k[/math] [at [ilmath]p[/ilmath]] |
|
A function is said to be [math]C^k[/math] [at [ilmath]p[/ilmath]] if all (partial) derivatives of all orders exist and are continuous [at [ilmath]p[/ilmath]] |
| [math]C^\infty_p[/math] |
|
[math]C^\infty_p(A)[/math] denotes the set of all germs of [math]C^\infty[/math] functions on [ilmath]A[/ilmath] at [ilmath]p[/ilmath] |
| [math]C^k([a,b],\mathbb{R})[/math] |
|
It is the set of all functions [math]:[a,b]\rightarrow\mathbb{R}[/math] that are continuous and have continuous derivatives up to (and including) order [math]k[/math] The unit interval will be assumed when missing |
| [math]\mathcal{D}_a(\mathbb{R}^n)[/math] |
|
Denotes Set of all derivations at a point - sometimes denoted [ilmath]T_a(\mathbb{R}^n)[/ilmath] (and such authors will denote the tangent space as [ilmath]\mathbb{R}^n_a[/ilmath]) |
| [math]\bigudot_i A_i[/math] | Makes it explicit that the items in the union (the [math]A_i[/math]) are pairwise disjoint, that is for any two their intersection is empty | |
| [math]\ell^p(\mathbb{F})[/math] |
|
The set of all bounded sequences, that is [math]\ell^p(\mathbb{F})=\{(x_1,x_2,...)|x_i\in\mathbb{F},\ \sum^\infty_{i=1}|x_i|^p<\infty\}[/math] |
| [math]\mathcal{L}^p[/math] |
|
[math]\mathcal{L}^p(\mu)=\{u:X\rightarrow\mathbb{R}|u\in\mathcal{M},\ \int|u|^pd\mu<\infty\},\ p\in[1,\infty)\subset\mathbb{R}[/math] [math](X,\mathcal{A},\mu)[/math] is a measure space. The class of all measurable functions for which [math]|f|^p[/math] is integrable |
| [math]L^p[/math] |
|
Same as [math]\mathcal{L}^p[/math] |
| [math]T_p(\mathbb{R}^n)[/math] |
|
The tangent space at a point [ilmath]a[/ilmath] Sometimes denoted [ilmath]\mathbb{R}^n_a[/ilmath] - Note: sometimes can mean Set of all derivations at a point which is often denoted [ilmath]\mathcal{D}_a(\mathbb{R}^n)[/ilmath] |
Unordered symbols
| Expression | Context | Details |
|---|---|---|
| [math]\mathcal{A}/\mathcal{B}[/math]-measurable |
|
There exists a Measurable map between the [ilmath]\sigma[/ilmath]-algebras |
| [ilmath]a\cdot b[/ilmath] |
|
Vector dot product |
