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Analysis

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Analysis is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (384–322 B.C.), though analysis as a formal concept is a relatively recent development.

The word comes from the Ancient Greek ἀνάλυσις (analysis, "a breaking up", from ana- "up, throughout" and lysis "a loosening").

As a formal concept, the method has variously been ascribed to Alhazen, René Descartes (Discourse on the Method), and Galileo Galilei. It has also been ascribed to Isaac Newton, in the form of a practical method of physical discovery (which he did not name).

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Calculus is a branch of mathematics, developed from algebra and geometry, built on two major complementary ideas.

The first idea, called differential calculus, is about a vast generalization of the slope of a line. It is a theory about rates of change[disambiguation needed], defining differentiation. It permits velocity, acceleration, and the slope of a curve at a given point all to be discussed on a common conceptual basis.

The second idea, called integral calculus, is about a vast generalization of area. It is a theory about accumulation of small, even infinitesimal, quantities, defining integration. Though originally motivated by area, it includes related concepts such as volume and even distance.

The two concepts differentiation and integration define inverse operations in a sense made precise by the fundamental theorem of calculus. Therefore, in teaching calculus either may in fact be given priority, but the usual educational approach (nowadays) is to introduce differential calculus first.

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In complex dynamics, the Julia set of a holomorphic function informally consists of those points whose long-time behavior under repeated iteration of can change drastically under arbitrarily small perturbations. Above is a 3D slice of a 4D Julia set.

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Topics in Analysis

General Calculus Differentiation Integration
Functions Harmonic/Fourier analysis Measure theory Miscellaneous

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