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Note that at each p, the differential df p is not a real number, but a linear functional on tangent vectors, and a prototypical example of a differential 1-form. Since any vector v is a linear combination Σ v j e j of its components, df is uniquely determined by df p (e j) for each j and each p ∈ U, which are just the partial derivatives of ...
In differential geometry, a one-form (or covector field) on a differentiable manifold is a differential form of degree one, that is, a smooth section of the cotangent bundle. [1] Equivalently, a one-form on a manifold is a smooth mapping of the total space of the tangent bundle of to whose restriction to each fibre is a linear functional on the ...
The apparent plural form in English goes back to the Latin neuter plural mathematica , based on the Greek plural ta mathēmatiká (τὰ μαθηματικά) and means roughly "all things mathematical", although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after the pattern of ...
Mathematics, Form and Function, a book published in 1986 by Springer-Verlag, is a survey of the whole of mathematics, including its origins and deep structure, by the American mathematician Saunders Mac Lane.
If f: M → N is a submersion at p and f(p) = q ∈ N, then there exists an open neighborhood U of p in M, an open neighborhood V of q in N, and local coordinates (x 1, …, x m) at p and (x 1, …, x n) at q such that f(U) = V, and the map f in these local coordinates is the standard projection
Mathematical Notes is a peer-reviewed mathematical journal published by Springer Science+Business Media on behalf of the Russian Academy of Sciences that covers all aspects of mathematics. It is an English language translation of the Russian-language journal Matematicheskie Zametki ( Russian : Математические заметки ) and ...
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Mathematical notation is widely used in mathematics, science, and engineering for representing complex concepts and properties in a concise, unambiguous, and accurate way. For example, the physicist Albert Einstein's formula = is the quantitative representation in mathematical notation of mass–energy equivalence. [1]