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 RuleRight Hand RuleMidpoint RuleTrapezoidal RuleSimpson’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