Portal:Analysis
Analysis
Analysis is the process of breaking a complex topic or substance into smaller parts to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle, though analysis as a formal concept is a relatively recent development. As a formal concept, the method has variously been ascribed to Ibn al-Haytham, Descartes (Discourse on the Method), Galileo, and Newton, as a practical method of physical discovery.
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In calculus, a branch of mathematics, the derivative is a measurement of how a function changes when the values of its inputs change. Loosely speaking, a derivative can be thought of as how much a quantity is changing at some given point. For example, the derivative of the position or distance of a car at some point in time is the instantaneous velocity, or instantaneous speed (respectively), at which that car is traveling (conversely the integral of the velocity is the car's position).
A closely related notion is the differential of a function.
The derivative of a function at a chosen input value describes the best linear approximation of the function near that input value. For a real-valued function of a single real variable, the derivative at a point equals the slope of the tangent line to the graph of the function at that point. In higher dimensions, the derivative of a function at a point is a linear transformation called the linearization.^{[1]}.
The process of finding a derivative is called differentiation. The fundamental theorem of calculus states that differentiation is the reverse process to integration.
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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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- ...that the Gudermannian function relates the circular trigonometric functions and the hyperbolic trigonometric functions without the use of complex numbers?
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Topics in Analysis
General | Calculus | Differentiation | Integration |
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Functions | Harmonic/Fourier analysis | Measure theory | Miscellaneous |
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Algebra | Analysis | Category theory |
Computer science |
Cryptography | Discrete mathematics |
Geometry |
Logic | Mathematics | Number theory |
Physics | Science | Set theory | Statistics | Topology |
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- ^ Differential calculus, as discussed in this article, is a very well-established mathematical discipline for which there are many sources. Almost all of the material in this article can be found in Apostol 1967, Apostol 1969, and Spivak 1994.