Calculus & Analysis on

The following is a list of entries on the Words page in the general area ofand ANALYSIS. Mathematical analysis is a subject of enormous scope. In the course of the 19th century became established as a major field and the 20th century saw the rise of

Sy next to an item is a link to related material on the Symbols pages, principally Earliest Uses of Symbols of Calculus, Earliest Uses of Function Symbols

You may also want to consult the indexes for entries on Set Theory and LogicFor a general perspective on word-formation in mathematics see my Mathematical Words: Origins and Sources.

## A – B

Abelian integral

Absolute continuity

Absolute convergence

Absolute differential calculus

Analysis

Analytic function

Anti-derivative

Anti-differential

Axiom

Axiom of choice

## B

Baire classes

Banach space

Banach-Steinhaus theorem

Bernoulli’s equation

Bessel function Sy

Beta & gamma functions Sy

Bijection

Brachistochrone

## C

Calculus

Calculus of variations

Cantor set

# Cauchy-Schwarz inequality

Cauchy-Riemann equations

# Circular functions

Clairaut equation

Clairaut’s theorem

# Closed set

Compact

Complementary function

Completeness

Complex analysis

Complex number

Conditionally convergent

Conformal mapping

Conjugate

Constant of integration

Continuous

Convergent

Convolution

Cotangent space

Covariant differentiation

Covering

Curl

## D – E

Daniell integral

Definite integral Sy

Del Sy

Dense

Dependent variable

Derivative Sy

Differential

Differential calculus

Differential equation

Differential geometry

Differential topology

Differentiation

Dirac δ function

Directional derivative

Dirichlet integral

Dirichlet’s principle

Distribution

Divergence Sy

Divergence theorem

Divergent

Domain

Double integration

Dummy variable

## E

Element Sy

Elliptic function

Empty set Sy

Euler’s equation

Eulerian integral

Euler-Maclaurin summation

Exact differential

Explicit function

Exponential function

Extremal

Extremum

# First derivative

Fluxion & fluent

Fourier series

Fourier’s theorem

Fourier transform

Fractal

Fréchet differential

Fubini’s theorem

Function Sy

Functional

Functional analysis

Functional calculus

Fundamental theorem of calculus

## G

Gamma function

Gauss’s theorem

Gaussian curvature

General integral

General solution

Generalized function

Geodesic

Geometric series

Gibbs phenomenon

Gram-Schmidt orthogonalization

Green’s theorem

Gregory’s series

Gregory-Newton formula

## H – J

Haar measure

Hahn-Banach theorem

Hamiltonian

Hard and Soft Analysis

Hardy classes and spaces

Hardy-Littlewood theorem

Harmonic analysis

Harmonic function

Hausdforff measure

Hausdorff space

Heine-Borel theorem

Hermite polynomial

Hessian

Hilbert space

Hölder mean

Holomorphic function

Homogeneous differential equation

Hyperset

Imaginary

Implicit differentiation

Implicit function

Improper integral

Indefinite integral

Infinitesimal

Infinity Sy

Integral  Sy

Integral calculus

Integral equation

Inverse

Jacobian

Jordan curve

Julia set

K  - N

## K

Kernel

Kolmogorov extension theorem

Kuhn-Tucker theorem

## L

Lagrange multiplier

Laplace transform

Laplacian

Laurent expansion

Lebesgue integral

L’Hospital’s rule

Limit Sy

Limit point

Linear differential equation

Logarithm Sy

Lower semicontiuity

Lorenz attractor

Maclaurin’s series

Manifold

Mapping

Measurable function

Measure

Method of exhaustion

Metric space

Modulus

Monotonic

Morse theory

Nabla  Sy

Neighborhood

Newton’s method

Non-standard analysis

Norm

Numerical differentiation

Numerical integration

# Oscillating series

## P

Partial derivative Sy

Partial solution

Pfaffian

Point-set

Point of accumulation

# Pole

Potential function

Power series

Primitive function

Projection

Real analysis

Riccati equation

Riemann integral

# Riemann zeta function

Riesz-Fischer theorem

# Runge-Kutta method

## S

Schlicht

Schwarz inequality

Schwarz’ theorem

Sequence

Series

Set and Set theory

Simpson’s rule

Single-values function

Singular integral

Singular point

Slope field

Space

Spectrum

Stieltjes integral

Stirling’s formula

Stokes’ theorem

Strange attractor

Subset

Summable

Surface integral

Surjection

Tensor

# Theorem

Transfinite

## U – V

Uniform continuity

Uniform convergence

Univalent

Variable Sy

## W – Z

Wallis’ formula

Weierstrass approximation

# Wiener-Hopf

Witch of Agnesi

Wronskian

#### Y

Young’s criterion

Young’s theorem

## Z

Zorn’s lemma

History

Although the mathematicians of antiquity calculated areas by the method of METHOD OF EXHAUSTION, it is true nonetheless that “analysis as an independent subject was created in the 17th century during the scientific revolution.” (Jahnke). For three hundred year it has been one of the most important branches of higher mathematics and has attracted the attention of many of the greatest mathematicians including Newton and Leibniz in the 17th century, Euler and Lagrange in the 18th century and Cauchy and Weierstrass in the 19th century. The entry on DIFFERENTIAL CALCULUS shows how the scope of the subject broadened and how the original differential calculus interacted with other branches of mathematics, such as geometry and topology.

Analysis is an important subject and many accounts of its history are available. Its history is a prominent topic in general works on the history of mathematics, such as Katz and Kline, and there are many more specialised works, such as Jahnke and Hawkins and Grabiner. The chapters of Jahnke contain extensive bibliographies.

References

• Morris Kline Mathematical Thought from Ancient to Modern Times. Oxford University Press, 1972. Reprinted as 3 volumes. Amazon, Amazon, Amazon.
• Victor J. Katz, A History of Mathematics: An Introduction. Addison Wesley, 1998. Amazon.
• Hans Niels Jahnke (ed.) A History of Analysis. American Mathematical Society 2003. Amazon
• Thomas Hawkins Lebesgue’s Theory of Integration: Its Origins and Development, AMS Chelsea Publishing Series, 1979/2001
• Judith V. Grabiner The Origins of Cauchy's Rigorous Calculus, Dover, 1981/2005. Amazon

# On the web

A lot of material is available but the following make a useful start.

John Aldrich, University of Southampton, Southampton, UK. (home). Most recent changes October 2009. 