topics on calculus

topics on calculus

This entry is an overview of many

calculus

related entries which can be found here, at PlanetMath.org. By calculus we

real analysis

at the high-school level or college level, and the entries in this page should be at either level. If an entry is written at a higher level, it will be indicated with a “GL” tag.

1

Expositions

The following are books or notes on calculus:

•

http://planetmath.org/?op=getobj&from=lec&id=36

An Introduction to Calculus.

2

ℝ

: The real line

Basic properties of the

real numbers

:

•

Decimal expansion

•

Positive

•

Inverse number, Opposite number

•

(GL)

http://planetmath.org/node/RealNumber

Real number

•

Topic entry on real numbers

3

Functions of one real variable

•

Concept of

function

and

real function

4

Computing limits

•

Limit of function

•

Limit of (sin

x

)/

x

as

x

tends to 0 (

http://planetmath.org/LimitOfDisplaystyleFracsinXxAsXApproaches0

)

•

Limit rules of functions

•

Limit examples

•

Improper limits

•

L’Hôpital’s rule (

http://planetmath.org/LHpitalsRule

)

•

Growth of exponential function

•

Example of computing limits using

Taylor expansion

5

Continuity

•

Continuous

and

discontinuous functions

•

Continuity of natural power

•

Continuity of sine and cosine

•

Jump discontinuity

example (

http://planetmath.org/ExampleOfJumpDiscontinuity

)

•

Intermediate value theorem

6

Differentiation in one variable

•

Derivative

•

Sum rule

•

Power rule

•

Product rule

•

Quotient rule

•

Chain rule

•

Derivative of inverse function

•

Derivatives of sine and cosine

•

Bolzano’s theorem

•

Least and greatest value of function

•

Higher order derivatives

•

Fractional Differentiation

7

Integration of functions of one real variable

•

Integral sign

•

Definition of

Riemann integral

(

http://planetmath.org/RiemannIntegral

)

•

A

lecture on integration by substitution

•

A

lecture on integration by parts

•

A

lecture on trigonometric integrals and trigonometric substitutions

•

A lecture on the partial fraction decomposition method

8

Definite integral

•

Definite integral

•

Riemann integral

•

Fundamental theorem of calculus

•

Improper integral

•

List of improper integrals

•

Approximate integration:

Left Hand Rule

Right Hand Rule

Midpoint Rule

Trapezoidal Rule

Simpson’s Rule

•

Integral equation

•

Fractional Integration

9

Integral Transforms

•

Fourier transform

•

Laplace transform

•

Fourier-Mellin integral

•

Mellin’s

transform

•

Fourier sine transform

•

Fourier cosine transform

10

Multivariable Calculus

10.1

Differentiation

•

Iterated limit

•

Jacobian matrix

•

Differentiation

under the integral sign

10.2

Integration

•

Stokes’ theorem

11

Differential Equations

•

Differential equation

•

(GL)

Existence and uniqueness of solution of ordinary differential equations

•

Index of differential equations

•

Separation of variables

•

Method of integrating factors

•

(GL) Examples of solving a PDE:
a)

Heat equation

(

http://planetmath.org/ExampleOfSolvingTheHeatEquation

),
b)

Wave equation

(

http://planetmath.org/SolvingTheWaveEquationByDBernoulli

)

12

Infinite Series

12.1

Series of Numbers

•

Series

•

Topic entry on series of complex terms

•

http://planetmath.org/?op=getobj&from=lec&id=37

Infinite Series

: Tests for

Convergence

and Examples

•

(GL?)

Non-existence of universal series convergence criterion

•

Cauchy general condition for convergence (

http://planetmath.org/CauchyCriterionForConvergence

)

•

Geometric series

,

Harmonic series

•

Sum of series depends on order

•

Manipulating convergent series

•

(GL?)

Multiplication of series

•

Leibniz’ estimate for

alternating series

(

http://planetmath.org/LeibnizEstimateForAlternatingSeries

)

12.2

Function Sequences and Series

•

Limit of function

sequence

•

The limit of a

uniformly convergent

sequence of

continuous functions

is continuous

•

Sum function of series

•

Termwise differentiation

•

Weierstrass’ criterion of

uniform convergence

•

(GL)

Fourier series

(

http://planetmath.org/FourierSineAndCosineSeries

)

12.3

Power Series and Taylor Series

•

Power series

and

Taylor series

•

Taylor’s theorem

•

Newton’s binomial series (

http://planetmath.org/BinomialFormula

)

•

http://planetmath.org/node/ExampleOfTaylorPolynomialsForSinX

Example of

Taylor polynomials

for

sin

⁡

x

•

Example of Taylor polynomials for the

exponential function

•

Examples on how to find Taylor series from other known series

•

Getting Taylor series from differential equation

13

Additional Topic

•

Non-Newtonian calculus

Title

topics on calculus

Canonical name

TopicsOnCalculus

Date of creation

2013-03-22 15:39:33

Last modified on

2013-03-22 15:39:33

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

65

Author

alozano (2414)

Entry type

Topic

Classification

msc 97A80

Related topic

CommonFormulasInCalculusOfDifferentialForms

Related topic

NumerableSet

Related topic

NonNewtonianCalculus2

Defines

Calculus

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