HKAL Pure Mathematics

HKAL Pure Mathematics

1. Sets

Sets (maths online) In this applet the understanding of simple definitions of the form { x Î N | property } may be checked.

2. Mathematical Proofs

Pythagoras' Constant (Favorite Mathematical Constants) Proof of irrationality of Ö(2) by contradiction. Finding Ö(2) by iterative methods

Rational and Irrational Numbers    Includes a proof of irrationality of Ö(5).
Notes on Methods of Proof    Various types of proof
Mathematical Proofs     A couple of interesting "mathematical proofs"

3. The Summation Sign

The Summation Sign Definition and some examples
Sum (Eric Weisstein's World of Mathematics) Many identities (including SUM(k^p,k=1..n), arithmetico-geometric series)

4. Mathematical Induction

Mathematical Induction (Interactive Mathematics Miscellany and Puzzles)    with different examples
Mathematical Induction (The Technique of Proof by Induction)    including strong induction
math-history-list: mathematical induction    newsgroup articles
Chapter 5: Mathematical Induction    Simple examples and the proof of Euclid's algorithm
Mathematical Induction (by Jan van Eijck)    with various examples

5. Binomial Theorem

Binomial Formulas (S.O.S. Math) Formulas related to the Binomial coefficients

6. Linear Algebra

Linear Algebra WebNotes    Lecture notes by Dr. Mark V. Sapir. Although the notes are for university students, they are not too difficult.????
Cayley-Hamilton Theorem (Eric Weisstein's World of Mathematics)    The theorem in general and the special case for a 2 by 2 matrix.
Matrices (Web Study Guide for Vector Calculus)    Introduction; basic operations and properties

Matrix Calculator Applet ver 1.2    Calculator for matrix addition, subtraction, multiplication, transpose, determinant, inverse, adjoint and upper triangular matrix.
Determinants (Web Study Guide for Vector Calculus )    Determinants of order 2 and 3
Testing for Linear Dependence of Vectors (Web Study Guide for Vector Calculus)    Definition of linear dependence; test by determinant and Gaussian Elimination
Systems of Linear Equations (Web Study Guide for Vector Calculus)    Solving a system (with 2 unknowns) by matrices; determining the number of solutions of nonhomogeneous and homogeneous systems of equations
Gaussian Elimination (Web Study Guide for Vector Calculus)    Solving system of linear equations by Gaussian Elimination
Eigenvalues and Eigenvectors (Web Study Guide for Vector Calculus)    Eigenvalues, eigenvectors and symmetric matrices

7. Polynomials

Excel HSC Lectures: Polynomials    Definition, graphs, division, Remainder Theorem, Factor Theorem, multiplicity of roots, roots and coefficients.????
Binary Euclid's Algorithm    Not about polynomial. A special case of the generalization of the Euclidean Algorithm of number which can be performed easily by computers.
Review of Algebra Techniques - Factoring Quadratics    Resolve fractions with only non-repeated factors in denominators to partial fractions
Factoring a Polynomial (Tools for Algebraic Manipulation)   Factorize polynomials entered
Partial Fraction Expansion of a Rational Function (Tools for Algebraic Manipulation)    Resolve fractions entered into partial fractions
Solving a Polynomial Equation (Tools for Algebraic Manipulation)    Solve polynomial equation entered

8. Inequalities

Arithmetic and Geometric Means Prove the well known inequality but just emphasize a fact that was used by Cauchy in his proof.

(Corollaries from) Pythagoras' Theorem Includes a proof of the Arithmetic-Geometric Means Inequality.

Cauchy's Inequality (Eric Weisstein's World of Mathematics) Proof of the inequality by discriminant; vector approach of the inequality.

9. Two-Dimensional Coordinate Geometry

Conic Section (Eric Weisstein's World of Mathematics)     Figures showing that conic sections are the cross-sections of a cone; properties of conic sections.
Conic Sections     Photos showing that conic sections are the cross-sections of a cone (degenerated cases included); some properties of conic sections
Historical View of the Conic Sections (An Introduction To Conic Sections)     Manuscripts
An Introduction To Conic Sections     Examples and exercises.

Mark's Mathematics Applets (Conic Sections) This applet allows you to specify a plane in three-dimensional space. The applet plots the intersection of the plane with a cone whose sides have unit slope.

Jalacy Software (Conic Sections) This applet shows the intersection of a plane and a cone. Users are allowed to manipulate the plane and change the view.

Cardioid (Eric Weisstein's World of Mathematics) Polar and Cartesian equations; cardioid as locus of a point on a rolling circle (animation) the envelope of circles (animation); arc length and area of cardioids.

A Visual Dictionary of Special Plane Curves A comprehensive website about plane curves, with a lot of (dynamic) pictures for Mathematica, QuickTime, LiveGraphics3D, Sketchpad and Cabri.

Famous Curves Applet Index Experiment with some curves with java applets.

Analytic geometry 3 (maths online) Contains an applet showing cycloids as loci of points on rolling circles.

10. Vector Algebra

Vectors 1 (maths online) This applet examines vectors and their components in a 3D graphics from different viewpoints.

Vectors (Web Study Guide for Vector Calculus)    2D and 3D vectors; addition, scalar multiplication and magnitude
Dot Products and Projections (Web Study Guide for Vector Calculus)    Dot product of vectors (2D or 3D); projection vector by dot product; work done (physical application)
The Cross Product (Web Study Guide for Vector Calculus)    Definition and properties of cross product; area of triangle by cross product

11. Three-Dimensional Coordinate Geometry

Equations of Lines and Planes (Web Study Guide for Vector Calculus) Finding the equations by vectors; distance between a point and a plane

12. Complex Numbers

Complex Numbers Made Easy... (Live Mathematics on the Web) Complex numbers come alive with JAVA.

Quadratic, cubic and quartic equations History about quadratic, cubic and quartic equations.

13. Graphs of Functions

Transformation of Functions (Tools for Analyzing Functions)    This tool demonstrates how the graph of a function f(x) changes under transformations such as f(x) + c, f(x+c), f(cx) where c is a constant.
Arithmetic Operations with Functions (Tools for Analyzing Functions)    This tool is used to graph functions which are arithmetic combinations of two functions. For example, given f(x) and g(x), it graphs f(x) + g(x).
The Composition of f(x) & g(x): f(g(x)) or g(f(x)) (Tools for Analyzing Functions)    This tool graphs the composite function formed by two functions of a single variable. Given f(x) and g(x), it graphs f(g(x)) or g(f(x)).
Asymptotes of Rational Functions (Tools for Analyzing Functions)    This tool computes the vertical and horizontal asymptotes of a rational function.
Inverse Functions (Tools for Analyzing Functions)    This tool attempts to find an inverse function of f(x). Of course, this is not possible for many functions.
Tools for Plotting Functions & Equations    Tools for plotting functions and equations, 2-D or 3-D, with derivatives, in parametric and polar form.

Mark's Mathematics Applets (Cubic Polynomials) The purpose of this applet is to help users to relate the coefficients in a cubic polynomial to the shape of its graph.

14. Sequences and Series

Series, Convergence, and Series Tests Quite comprehensive. Formal definition of sequences and series; geometric series, p-series and telescoping series; series tests (out of syllabus but not too difficult)

15. Limit and Continuity

The Natural Logarithmic Base (Favorite Mathematical Constants)    Properties of e.
Continuous Function (Eric Weisstein's World of Mathematics)    Definition and some properties of continuous functions.
Limit (Eric Weisstein's World of Mathematics)       Definition of a limit of a function (epsilon-delta approach).
Given e, there exists a d.... (Tools for Computing Limits and Derivatives)    This tool gives a graphical representation of the epsilon-delta definition of a limit.
L'Hospital's Rule (Eric Weisstein's World of Mathematics)    The theorem and an example such that the rule fails.
Limits and l'Hôpital's Rule    Quite comprehensive. Definition and laws of limit; different cases of l'Hôpital's Rule.
Computing a Limit (Tools for Computing Limits and Derivatives)    This tool attempts to compute the limit of a function. The function is allowed to have parameters in which case the limit we be found (hopefully!) in terms of the parameters. There is an option that plots the graph of the function in a neighborhood of point where the limit is being taken.

16. Differential Calculus

Differentiation Tutor (by Steven Tschantz)    This is an online tutorial in how to take derivatives of common mathematical expressions. This program allows you to input an arbitrary expression and computes the derivative of your expression explaining each step in the computation.
Evaluating Derivatives (by Steven Tschantz)    This is an experimental online lesson in how to take derivatives of common mathematical expressions. This program allows you to input an arbitrary expression and computes the derivative of your expression explaining each step in the computation.
Derivatives of a Function (Tools for Using the Derivative)    This tool computes one or more derivatives of a function f(x).
Computing the Derivative Implicitly (Tools for Using the Derivative)    This tool differentiates an equation, F(x,y)=0, treating y as a function of x and then solves the differentiated equation for y'(x).
The Derivative (S.O.S. Math)    List of derivatives; higher derivatives and Leibniz Formula
Derivative (Eric Weisstein's World of Mathematics)    Definition of derivative; list of derivatives; properties of derivative.
Stationary Point (Eric Weisstein's World of Mathematics)    Diagrams of minimum, maximum and inflection points.
Minimum (Eric Weisstein's World of Mathematics)    Local minimum and global minimum.
Extremum (Eric Weisstein's World of Mathematics)    Graphs of sin(1/x), cos(1/x), sin(e^(2x+9)) (with many extrema).
First Derivative Test (Eric Weisstein's World of Mathematics)    No example.
Second Derivative Test (Eric Weisstein's World of Mathematics)    No example; Extremum test (2-D function) also.
Graphing the Tangent Line (Tools for Using the Derivative)    This tool computes the equation of the tangent line to the graph of a given function f(x) at the specified point x=a.

17. Integral Calculus

Computing Riemann Sums (Tools for Integration)    This tool calculates the Riemann sums (left-end point, right-end point, and midpoint) for a given function f(x) over an interval [a,b].
Antidifferentiation (Tools for Integration)    This tool calculates the antiderivative of the given function.
Definite Integral of a Function (Tools for Integration)    This tool calculates the definite integral of a function f(x) over an interval [a,b].
The Integrator    Give the integrals of functions entered; powered by Mathematica.
Integration Assistant (Tools for Integration)    This megatool leads the user through the computation of an integral (antiderivative). Individual tools provide assistance with integrand manipulation (e.g. partial fraction expansion), u-substitution, and integration by parts.
Fundamental Theorems of Calculus (Eric Weisstein's World of Mathematics)    The first, second and complex theorems of calculus (no proof, no example).
Integral (Eric Weisstein's World of Mathematics) List of some common indefinite integrals; some methods of integration.
Riemann Sum (Eric Weisstein's World of Mathematics)    Riemann Sum and Riemann Integral.
Cauchy Principal Value (Eric Weisstein's World of Mathematics)     Cauchy principal value of integral; definition only.
Integration by Parts (S.O.S. Math)    Integration by parts; links to typical examples
S.O.S. Mathematics - Calculus    Comprehensive
Arc Length of a Curve (Tools for Integration)    This tool computes the arc length of a curve.

First Form of Fundamental Theorem of Calculus This tool uses the Fundamental Theorem of Calculus to calculate derivatives.

Last Updated: 23rd October, 2000