Subdivisions
Mathematical analysis includes the following subfields.
- Differential equations
- Real analysis, a rigorous study of derivatives and integrals of functions of real variables. This includes the study of sequences and their limits, series.
- Multivariable calculus
- Real analysis on time scales – a unification of real analysis with calculus of finite differences
- Measure theory – given a set, the study of how to assign to each suitable subset a number, intuitively interpreted as the size of the subset.
- Vector calculus
- Functional analysis studies spaces of functions and introduces concepts such as Banach spaces and Hilbert spaces.
- Calculus of variations deals with extremizing functionals, as opposed to ordinary calculus which deals with functions.
- Harmonic analysis deals with Fourier series and their abstractions.
- Geometric analysis involves the use of geometrical methods in the study of partial differential equations and the application of the theory of partial differential equations to geometry.
- Complex analysis, the study of functions from the complex plane to itself which are complex differentiable (that is, holomorphic).
- Several complex variables
- Clifford analysis, the study of Clifford valued functions that are annihilated by Dirac or Dirac-like operators, termed in general as monogenic or Clifford analytic functions.
- p-adic analysis, the study of analysis within the context of p-adic numbers, which differs in some interesting and surprising ways from its real and complex counterparts.
- Non-standard analysis, which investigates the hyperreal numbers and their functions and gives a rigorous treatment of infinitesimals and infinitely large numbers.
- Numerical analysis, the study of algorithms for approximating the problems of continuous mathematics.
- Computable analysis, the study of which parts of analysis can be carried out in a computable manner.
- Stochastic calculus – analytical notions developed for stochastic processes.
- Set-valued analysis – applies ideas from analysis and topology to set-valued functions.
- Tropical analysis (or idempotent analysis) – analysis in the context of the semiring of the max-plus algebra where the lack of an additive inverse is compensated somewhat by the idempotent rule A + A = A. When transferred to the tropical setting, many nonlinear problems become linear.
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