MANIFOLDS Meaning and
Definition
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Manifolds, in mathematics, refer to topological spaces that are locally Euclidean, meaning they resemble Euclidean space near each point. Specifically, a manifold is a set equipped with a collection of coordinate charts that enable the local description of the space. These coordinate charts are homeomorphisms between open subsets of the manifold and open subsets of Euclidean space.
Manifolds can be classified based on their dimensionality, with the most common being one-dimensional (curves), two-dimensional (surfaces), and three-dimensional (spatial). Manifolds also have different properties, such as being orientable (having an assigned consistent orientation) or non-orientable (lacking a consistent orientation).
In differential geometry, manifolds are further studied with the consideration of smooth structures. Smooth manifolds are equipped with smooth functions, allowing for the discussion of derivatives and tangent spaces. Smooth manifolds are often used to model physical spaces or describe solutions to differential equations.
Aside from their mathematical significance, manifolds also find applications in various scientific and engineering fields. For instance, in physics, manifolds are employed to describe the space-time continuum in general relativity. In computer science and machine learning, manifolds are used for data analysis and dimensionality reduction, aiming to extract meaningful information from complex datasets.
Overall, manifolds play a fundamental role in abstract mathematics, providing a framework for studying spaces with local Euclidean properties and finding application in diverse areas of research.
Common Misspellings for MANIFOLDS
Etymology of MANIFOLDS
The term "manifolds" originated from the combination of two words: "manifold" and "fold".
The word "manifold" can be traced back to the Old English word "manigfeald" (manig = many, feald = fold), meaning "many-fold" or "multi-fold". This Old English term evolved into the Middle English word "manifold", which retained the same meaning of something that has multiple folds or layers.
On the other hand, the word "fold" derives from the Old English word "fealdan", meaning "to fold". Its Indo-European root is "pel-", which also gave rise to words like "pile" and "pleat".
Similar spelling words for MANIFOLDS
Conjugate verb Manifolds
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