edit Selected article The four charts each map part of the circle to an open interval, and together cover the whole circle. A manifold is an abstract mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be more complicated. In discussing manifolds, the idea ofdimension is important. For exa mple, lines are one- dimensional, andplanes two-dimensional. In a one dimensional manifold (o r one-manifold), every point has a neighborhood that looks like a segment of a line. Examples of one-manifolds include a line, a circle, and two separate circles. In a two-manifold, every point has a neighborhood that looks like a disk. Examples include a plane, the surface of a sphere, and the surface of a torus . Manifolds are important objects in mathematics andphysics because they allow more complicated structures to be expressed and understoo d in terms of the relatively well-understood properties of simpler spaces. ...Archive Image credit: User:KSmrqRead more... edit Picture of the month
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
8/7/2019 Math Topics
http://slidepdf.com/reader/full/math-topics 1/4
edit
Selected article
The four charts each map part of the
circle to an open interval, and
together cover the whole circle.
A manifold is an abstract mathematical space in which every point has a neighborhood which
resembles Euclidean space, but in which the global structure may be more complicated. Indiscussing manifolds, the idea of dimension is important. For example, lines are one-
dimensional, and planes two-dimensional.
In a one dimensional manifold (or one-manifold), every point has a neighborhood that looks likea segment of a line. Examples of one-manifolds include a line, a circle, and two separate circles.
In a two-manifold, every point has a neighborhood that looks like a disk . Examples include a plane, the surface of a sphere, and the surface of a torus.
Manifolds are important objects in mathematics and physics because they allow morecomplicated structures to be expressed and understood in terms of the relatively well-understood
properties of simpler spaces.
...Archive
Image credit: User:KSmrq Read more...edit
Picture of the month
8/7/2019 Math Topics
http://slidepdf.com/reader/full/math-topics 2/4
Illustration for the paradoxical decomposition of F2 used in the proof of the Banach-Tarski
theory | Chaos theory | Combinatorics | Dynamic systems | Fractals | Game theory | Geometry |Algebraic geometry | Graph theory | Group theory | Linear algebra | Mathematical logic | Model
Theory | Multi-dimensional geometry | Number theory | Numerical analysis | Optimization |Order theory | Probability and statistics | Set theory | Statistics | Topology | Algebraic topology |
Trigonometry |
Mathematics ( books) | History of mathematics | Mathematicians | Awards | Education | Institutesand societies | Literature | Notation | Theorems | Proofs | Unsolved problems
More mathematics categories
editDid you know...
y ...properties of Pascal's triangle have application in many fields of mathematics including
combinatorics, algebra, calculus and geometry?
8/7/2019 Math Topics
http://slidepdf.com/reader/full/math-topics 3/4
y ...work in artificial intelligence makes use of Swarm intelligence, which has foundationsin the behavorial examples found in nature of ants, birds, bees, and fish among others?
y ...that statistical properties dictated by Benford's Law are used in auditing of financialaccounts as one means of detecting fraud?
y ...that Modular arithmetic has application in at least ten different fields of study,
including the arts, computer science, and chemistry in addition to mathematics?y ...that outstanding mathematician Grigori Perelman was offered a Fields Medal in 2006,
in part for his proof of the Poincaré conjecture, which he declined?y ...that a regular heptagon is the regular polygon with the fewest number of sides which is
not constructible with a compass and straightedge?y ...that the Gudermannian function relates the regular trigonometric functions and the
hyperbolic trigonometric functions without the use of complex numbers?y ...that the Catalan numbers solve a number of problems in combinatorics such as the
number of ways to completely parenthesize an algebraic expression with n+1 factors?y ...that a ball can be cut up and reassembled into two balls the same size as the original
(Banach-Tarski paradox)?
Showing 9 items out of 35 More did you know
edit
W ikiProjects
TheMathematics WikiProject is the center for mathematics-related editing on Wikipedia. Join
the discussion on the project's talk page.
Project pages
y Participants y Current activity y Manual of style y Conventions
Essays
y Advice on using Wikipedia for mathematics self-study y Proofs