Branches of Mathematics
Analysis
Also Known As Mathematical Analysis
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The branch that grew out of the calculus of Newton and Leibniz: the rigorous study of limits, continuity, differentiation, integration and infinite series. Nineteenth century mathematicians (Cauchy, Weierstrass, Riemann) re-founded calculus on precise epsilon-delta definitions of limit, replacing the informal infinitesimals of the subject's founders with a fully rigorous apparatus that still underlies real and complex analysis today. Two centuries before Newton and Leibniz, the Kerala school of astronomy and mathematics in southern India, working from Madhava of Sangamagrama (c. 1340 to c. 1425) onward, derived infinite series for sine, cosine and arctangent equivalent to what are now called Taylor series, apparently without the wider theory built around them in Europe and without influencing the European tradition, so far as surviving evidence shows; whether any transmission route to Europe existed is a live historical question rather than an established fact.
Facts
Central QuestionHow do quantities behave in the limit, continuously and infinitesimally, and what does it mean, rigorously, for an infinite process to converge? 1 Key DebateWhether the nineteenth century's rigorous epsilon-delta re-founding of calculus rescued the subject from genuine paradox in Newton and Leibniz's original infinitesimals, or merely papered over an intuition that Abraham Robinson's nonstandard analysis later showed could itself be made fully rigorous. 1 Cross-Tradition Connections
Associated With
Includes
The conjecture links an arithmetic invariant (rank) to an analytic object (the L-function's behavior at a point), the same dual number-theory and analysis character the atlas already records for the Riemann Hypothesis.
Sources
1. MacTutor History of Mathematics Archive
University of St Andrews, School of Mathematics and Statisticshttps://mathshistory.st-andrews.ac.uk/HistTopics/Real_numbers_1/Quote, https://mathshistory.st-andrews.ac.uk/HistTopics/Real_numbers_1/
It seems clear that Pythagoras would have thought of 1, 2, 3, 4, ... (the natural numbers in the terminology of today) in a geometrical way, not as lengths of a line as we do, but rather in the form of discrete points.
Wikipedia: Mathematical Analysis
Wikimedia FoundationLead sectionQuote, Lead section
Mathematical analysis is the branch of mathematics that studies functions, spaces, and operators through quantitative methods of approximation and convergence.
Euler's Identity (Wikipedia)
Wikimedia FoundationIncludes: Euler's Identity, lead paragraphQuote, Includes: Euler's Identity, lead paragraph
Euler's identity is considered an exemplar of mathematical beauty, as it shows a profound connection between the most fundamental numbers in mathematics.
Intermediate Value Theorem (Wikipedia)
Wikimedia FoundationIncludes: Intermediate Value Theorem, introductory paragraphQuote, Includes: Intermediate Value Theorem, introductory paragraph
if f is a continuous function whose domain contains the interval [a, b] and s is a number such that f(a)<s<f(b), then there exists some x between a and b such that f(x)=s
Mean Value Theorem (Wikipedia)
Wikimedia FoundationIncludes: Mean Value Theorem, opening paragraphQuote, Includes: Mean Value Theorem, opening paragraph
the average rate of change of such a function over an interval is equal to the instantaneous rate of change at some point in the interval
Integral (Wikipedia)
Wikimedia FoundationIncludes: Integral, lead paragraphQuote, Includes: Integral, lead paragraph
The major advance in integration came in the 17th century with the independent discovery of the fundamental theorem of calculus by Leibniz and Newton.
Integral (Wikipedia)
Wikimedia FoundationAssociated With: Isaac Newton, lead paragraphQuote, Associated With: Isaac Newton, lead paragraph
The major advance in integration came in the 17th century with the independent discovery of the fundamental theorem of calculus by Leibniz and Newton.
Integral (Wikipedia)
Wikimedia FoundationAssociated With: Gottfried Wilhelm Leibniz, lead paragraphQuote, Associated With: Gottfried Wilhelm Leibniz, lead paragraph
The major advance in integration came in the 17th century with the independent discovery of the fundamental theorem of calculus by Leibniz and Newton.
Continuous Function (Wikipedia)
Wikimedia FoundationIncludes: Continuity, lead paragraphQuote, Includes: Continuity, lead paragraph
a small variation of its argument induces at most a small variation of its value
Sequence (Wikipedia)
Wikimedia FoundationIncludes: Sequence, Examples and notation sectionQuote, Includes: Sequence, Examples and notation section
Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences.
Series, Mathematics (Wikipedia)
Wikimedia FoundationIncludes: Series, Lead sectionQuote, Includes: Series, Lead section
The study of series is a major part of calculus and its generalization, mathematical analysis.
Logarithm (Wikipedia)
Wikimedia FoundationIncludes: Logarithm, Analytic properties sectionQuote, Includes: Logarithm, Analytic properties section
In mathematical analysis, the logarithm base e is widespread because of analytical properties explained below.
Wikipedia: Measure (Mathematics)
Karl Weierstrass, Biography (MacTutor History of Mathematics)
MacTutor History of Mathematics Archive, University of St AndrewsIncludes: Karl Weierstrass, Opening summaryQuote, Includes: Karl Weierstrass, Opening summary
Known as the father of modern analysis, Weierstrass devised tests for the convergence of series and contributed to the theory of periodic functions, functions of real variables, elliptic functions, Abelian functions, converging infinite products, and the calculus of variations.
Fundamental Theorem of Calculus (Wikipedia)
Wikimedia FoundationIncludes: Fundamental Theorem of Calculus, IntroductionQuote, Includes: Fundamental Theorem of Calculus, Introduction
Calculus as a unified theory of integration and differentiation started from the conjecture and the proof of the fundamental theorem of calculus.
Riemann Hypothesis (Wikipedia)
Wikimedia FoundationIncludes: Riemann Hypothesis, Origin sectionQuote, Includes: Riemann Hypothesis, Origin section
connecting two seemingly unrelated areas in mathematics; namely, number theory, which is the study of the discrete, and complex analysis, which deals with continuous processes
Navier-Stokes Existence and Smoothness (Wikipedia)
Wikimedia FoundationIncludes: Navier-Stokes Existence and Smoothness, The Navier-Stokes equations sectionQuote, Includes: Navier-Stokes Existence and Smoothness, The Navier-Stokes equations section
The Navier-Stokes equations are nonlinear, meaning that the terms in the equations do not have a simple linear relationship
Yang-Mills Existence and Mass Gap (Wikipedia)
Wikimedia FoundationIncludes: Yang-Mills Existence and Mass Gap, Problem Requirements sectionQuote, Includes: Yang-Mills Existence and Mass Gap, Problem Requirements section
establishing axiomatic properties at least as strong as those cited in Streater & Wightman
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