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=36An 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/RealNumberReal 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=37Infinite 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/ExampleOfTaylorPolynomialsForSinXExample 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