analysis

Introduction & Top Questions
Historical background
- Bridging the gap between arithmetic and geometry
- Discovery of the calculus and the search for foundations
Technical preliminaries
- Numbers and functions
  - Number systems
  - Functions
- The problem of continuity
  - Approximations in geometry
  - Infinite series
  - The limit of a sequence
  - Continuity of functions
- Properties of the real numbers
Calculus
- Differentiation
  - Average rates of change
  - Instantaneous rates of change
  - Formal definition of the derivative
  - Graphical interpretation
  - Higher-order derivatives
- Integration
  - The fundamental theorem of calculus
  - Antidifferentiation
  - The Riemann integral
Ordinary differential equations
- Newton and differential equations
  - Newton’s laws of motion
  - Exponential growth and decay
- Dynamical systems theory and chaos
Partial differential equations
- Musical origins
  - Harmony
  - Normal modes
  - Partial derivatives
    - D’Alembert’s wave equation
    - Trigonometric series solutions
- Fourier analysis
Complex analysis
- Formal definition of complex numbers
- Extension of analytic concepts to complex numbers
- Some key ideas of complex analysis
Measure theory
Other areas of analysis
- Functional analysis
- Variational principles and global analysis
- Constructive analysis
- Nonstandard analysis
History of analysis
- The Greeks encounter continuous magnitudes
  - The Pythagoreans and irrational numbers
  - Zeno’s paradoxes and the concept of motion
  - The method of exhaustion
- Models of motion in medieval Europe
- Analytic geometry
- The fundamental theorem of calculus
  - Differentials and integrals
  - Discovery of the theorem
  - Calculus flourishes
- Elaboration and generalization
  - Euler and infinite series
  - Complex exponentials
  - Functions
  - Fluid flow
- Rebuilding the foundations
  - Arithmetization of analysis
  - Analysis in higher dimensions

References & Edit History Related Topics

Images

transformation of a circular region into an approximately rectangular region

an infinite geometrical series

graph of distance traveled versus time elapsed for the motion of an automobile

Two panels showing function graphing concepts: Panel A illustrates plotting an arbitrary function f by selecting values of t and marking corresponding f(t) points; Panel B graphs the specific parabola f(t) equals t squared, with plotted points showing how t values map to t squared. Both graphs use dashed lines to connect points to the axes for clarity.

difference between average and instantaneous rates of change

A graph of the function f(x) = x cubed minus 3x plus 2 is shown on a blue background, with a red curve passing through key points. Local maximum and minimum points are marked with black dots at x equals negative 1 and x equals 1, illustrating important features of the curve.

integral region graph

A graph illustrating a calculus concept shows the curve of a function f, with shaded regions under the curve representing the area A(t) from a to t, and an additional striped area labeled hf(t) from t to t+h.

Riemann integral

Poincaré section, or map

At a Glance

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analysis summary

analysis: Media

mathematics

Images

transformation of a circular region into an approximately rectangular region

transformation of a circular region into an approximately rectangular region

\This suggests that the same constant...

Encyclopædia Britannica, Inc.

an infinite geometrical series

an infinite geometrical series

Clearly, the sum of the square's parts (

Encyclopædia Britannica, Inc.

graph of distance traveled versus time elapsed for the motion of an automobile

graph of distance traveled versus time elapsed for the motion of an automobile

Because the speed of the automobile is constant in this example (50 kilometres...

Encyclopædia Britannica, Inc.

Two panels showing function graphing concepts: Panel A illustrates plotting an arbitrary function f by selecting values of t and marking corresponding f(t) points; Panel B graphs the specific parabola f(t) equals t squared, with plotted points showing how t values map to t squared. Both graphs use dashed lines to connect points to the axes for clarity.

graph of a function

Part A illustrates the general idea of graphing any function: choose a value...

Encyclopædia Britannica, Inc.

difference between average and instantaneous rates of change

difference between average and instantaneous rates of change

The graph of f(t) shows the secant between (t, f(t))...

Encyclopædia Britannica, Inc.

A graph of the function f(x) = x cubed minus 3x plus 2 is shown on a blue background, with a red curve passing through key points. Local maximum and minimum points are marked with black dots at x equals negative 1 and x equals 1, illustrating important features of the curve.

curve sketched with the help of calculus

This graph of f(x) = x³ − 3x...

Encyclopædia Britannica, Inc.

integral region graph

integral region graph

The shaded region, bounded by the vertical lines t = a, t...

Encyclopædia Britannica, Inc.

A graph illustrating a calculus concept shows the curve of a function f, with shaded regions under the curve representing the area A(t) from a to t, and an additional striped area labeled hf(t) from t to t+h.

fundamental theorem of calculus

Graphical illustration of the fundamental theorem of calculus:

Encyclopædia Britannica, Inc.

Riemann integral

Riemann integral

If the shaded areas in B (using inscribed rectangles) and C (using circumscribed...

Encyclopædia Britannica, Inc.

Poincaré section, or map

Poincaré section, or map

The trajectory, or orbit, of an object x is sampled periodically, as...

Encyclopædia Britannica, Inc.

vibrating violin string

vibrating violin string

A violin string, with rest length l, is plucked and its displacement,...

Encyclopædia Britannica, Inc.

A diagram of the complex plane with horizontal real axis and vertical imaginary axis, showing a point labeled x + iy and dashed lines connecting it to coordinates x on the real axis and y on the imaginary axis.

point in the complex plane

A point in the complex plane. Unlike real numbers, which can be located by a single...

Encyclopædia Britannica, Inc.

multiple paths in the complex plane

multiple paths in the complex plane

Multiple paths in the complex plane. The graph illustrates that two distinct points...

Encyclopædia Britannica, Inc.

Lebesgue integral

Lebesgue integral

Note that the areas, or slices, to be summed are horizontal rather than vertical....

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Pythagorean theorem

Pythagorean theorem

Visual demonstration of the Pythagorean theorem. This may be the original proof of...

Encyclopædia Britannica, Inc.

Eudoxus's method of exhaustion

Eudoxus's method of exhaustion

Eudoxus calculated the volume of a pyramid with successively smaller prisms that...

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Archimedes' parabolic segment calculation

Archimedes' parabolic segment calculation

Employing Eudoxus's method of exhaustion, Archimedes first showed how to calculate...

Encyclopædia Britannica, Inc.

Merton acceleration theorem

Merton acceleration theorem

Discovered in the 1330s by mathematicians at Merton College, Oxford, the Merton acceleration...

Encyclopædia Britannica, Inc.

A green circle rolls along a straight horizontal line, and a point on its circumference traces a looping red cycloid curve above the line.

cycloid

A cycloid is produced by a point on the circumference of a circle as the circle rolls...

Encyclopædia Britannica, Inc.

Leibniz's model of integration

Leibniz's model of integration

Gottfried Wilhelm Leibniz expressed integration as the summing of the areas of thin...

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intermediate value theorem

intermediate value theorem

The intermediate value theorem proves the intuitively obvious assertion that, for...

Encyclopædia Britannica, Inc.

Riemann sphere

Riemann sphere

This model of the Riemann sphere has its south pole resting on the origin of the...

Encyclopædia Britannica, Inc.

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