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