Mathematical Analysis - Subdivisions

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.