Analysis

Overview

Mathematical analysis typically refers to the topics that are studied in calculus.

Limits

The original development of the calculus utilizd the notion of an infinitesimal quantity. The logical foundations of this notion was questioned and not established until the developtment of non-standard analysis. The calculus was put on a rigorous footing with the developmebt of infinite sequences/series and the notion of a limit.

• Sequences
• Series
• Limits
• Continuity

The idea of closeness was further abstracted and developed in topology.

Differentiation

• Differentiation is a process(operator) which produces a function whose value at any point is equal to the tangent of some source function at that point.
• Differential Forms: are used to extend calculus to be used on curved surfaces and manifolds.
• Functional Differentiation - gives a notion of differentiating a functional by a function.

Integrals

Integration is at the heart of analysis. However, there are several different definitions of an integral.

• Riemann Integral - basic integral studied in elementary calculus
• Riemann Stieltjes Integral - an extension of the standard Riemann integral
• Multivariable Riemann Integral - applicable in multi-variable settings.
• Integration of Forms - extension of integration that applies when working with multi-variable calculus on manifolds
• Complex Integration - integration over functions of a complex variable
• Lebesgue Integral - abstract version of integration
• Stochastic Integrals:

Additional Topics

• Differential Equations
• Measure Theory : is the most robust and abstract form of integration. It exends the notions developed in Lebesgue intragration to the most general situations.
• Vector Calculus :
• Complex Numbers and Analysis :
• Approximation Theory
• Manifolds