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 $\mathbb{R}$ : The real line

Basic properties of the real numbers:

• Decimal expansion

• Positive

• Inverse number, Opposite number

• (GL) http://planetmath.org/node/RealNumberReal number

4 Computing limits

• Limit of function

• Limit of (sin x)/x as x tends to 0 (http://planetmath.org/LimitOfDisplaystyleFracsinXxAsXApproaches0)

• L’Hôpital’s rule (http://planetmath.org/LHpitalsRule)

• Example of computing limits using Taylor expansion

5 Continuity

• Jump discontinuity example (http://planetmath.org/ExampleOfJumpDiscontinuity)

6 Differentiation in one variable

• Bolzano’s theorem

• Least and greatest value of function

7 Integration of functions of one real variable

• Integral sign

• Definition of Riemann integral (http://planetmath.org/RiemannIntegral)

• A lecture on the partial fraction decomposition method

8 Definite integral

• Riemann integral

• Approximate integration:

• Integral equation

9 Integral Transforms

• Mellin’s transform

10 Multivariable Calculus

10.2 Integration

• Stokes’ theorem

11 Differential Equations

• (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

• http://planetmath.org/?op=getobj&from=lec&id=37Infinite Series: Tests for Convergence and Examples

• Cauchy general condition for convergence (http://planetmath.org/CauchyCriterionForConvergence)

• Leibniz’ estimate for alternating series (http://planetmath.org/LeibnizEstimateForAlternatingSeries)

12.3 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

• Getting Taylor series from differential equation