...al extension of locally finite families of sets to locally finite families of open sets.
...<TD valign="top"> A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated fro
5 KB (773 words) - 22:17, 5 June 2020
...dimensional surface oriented by a (continuous) unit normal $\nu$, the flux of the vector field $v$ through the surface $\Sigma$ is given by the integral
...is computed using the [[Area formula]]. If $\Sigma$ is given by the graph of a function $f: \mathbb R^{n-1} \supset V \to \mathbb R$ with its natural or
4 KB (589 words) - 15:27, 4 February 2014
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...near space]] over a [[Field|field]] $\mathbf{F}$. Let $L ( X )$ be the set of all linear operators with domains and ranges in $X$ and let
10 KB (1,502 words) - 22:40, 12 December 2020
...e set of points at which $f$ vanishes (cf. [[Support of a function|Support of a function]]). For example, $\mathop{\mathrm{supp}} x = \mathbf{R}^1$, $\ma
...is the set of those (and only those) points such that in any neighbourhood of them the generalized function is not equal to a $C^\infty$-function. For ex
3 KB (450 words) - 09:18, 12 August 2022
...ch can be regarded as a direct generalization of the [[Fundamental theorem of calculus]], is often referred to as:
...ield are denoted by $v_1, \ldots, v_n$, then the [[Divergence|divergence]] of $v$ is given by the function
6 KB (926 words) - 13:28, 26 February 2014
...t [[Hausdorff space|Hausdorff space]] is called a paracompactum. The class of paracompacta is very extensive — it includes all metric spaces (Stone's t
...of the requirements assumed in the theory of manifolds is the requirement of paracompactness, which is not superfluous since there exist non-paracompact
9 KB (1,401 words) - 14:04, 30 July 2014
...sly-differentiable vector function defined on the closure $ \overline{G} $ of a two-dimensional Jordan-measurable domain $ G $ lying in the plane with Ca
...$ F(x(u,v),y(u,v),z(u,v)) $, then the following defines a surface integral of the first kind (or a integral over the surface area):
11 KB (1,911 words) - 18:57, 1 December 2016
of a universal fibre bundle $ \xi = ( E _ {0} , p _ {0} , B _ {0} ) $.
The universality of the bundle $ \xi $
9 KB (1,417 words) - 20:39, 23 December 2023
$#C+1 = 179 : ~/encyclopedia/old_files/data/A011/A.0101370 Algebra of functions,
realized as an algebra of continuous functions on the space of maximal ideals $ \mathfrak M $.
11 KB (1,877 words) - 16:10, 1 April 2020
...nces|[2]]]) and in studying the singularities of the fundamental solutions of hyperbolic equations (see [[#References|[1]]], [[#References|[2]]], [[#Refe
dimensional [[Lagrangian manifold|Lagrangian manifold]] of class $ C ^ \infty $
24 KB (3,360 words) - 19:39, 5 June 2020
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...roup this reduction is carried out with respect to the field of definition of the group.
33 KB (5,081 words) - 10:26, 11 November 2023
...of a certain function algebra $ A $ (cf. [[Algebra of functions|Algebra of functions]]) into the algebra $ L ( X) $
of continuous linear operators on a topological vector space $ X $.
11 KB (1,521 words) - 02:23, 16 June 2022
...t of a [[Residue of an analytic function|residue of an analytic function]] of one complex variable to several complex variables. Let $ X $
be an analytic submanifold of complex codimension one and let $ \omega ( x) $
11 KB (1,662 words) - 08:11, 6 June 2020
...e introduction of a canonical structure. M. Sato has constructed the sheaf of micro-functions on the cotangent sphere bundle $ S ^ {*} M $
of the base space $ M $
13 KB (1,805 words) - 19:06, 9 January 2024
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...rier hyperfunctions was proposed by M. Sato at the same time as his theory of hyperfunctions, in which the transformed objects can be interpreted natural
26 KB (3,903 words) - 17:01, 1 July 2020
A partial differential equation (system) of the form
is a multi-index (a set of non-negative integers), $ | \alpha | = \sum {\alpha _ {i} } $,
18 KB (2,569 words) - 22:17, 5 June 2020
A theory studying continuous families of objects in algebraic geometry.
be a class of objects in algebraic geometry (varieties, schemes, vector bundles, etc.) on
16 KB (2,402 words) - 11:49, 16 December 2019
...m. The local classification problem is to describe the equivalence classes of local dynamical systems, providing, if possible, the simplest or most conve
...e field (map). This hope is partly justified, see [[Algebraic decidability of local classification problems]].
37 KB (5,881 words) - 19:10, 24 November 2023
...t which locally has the structure (topological, smooth, homological, etc.) of $ \mathbf R ^ {n} $
...allowing local parametrization (see below); for example, the set of planes of dimension $ k $
30 KB (4,462 words) - 07:59, 6 June 2020
...ebra, topology forms a general foundation of mathematics, and promotes its unity.
...th of the class of geometric objects that fall within the sphere of action of its laws.
38 KB (5,626 words) - 17:15, 20 March 2018