# Math Ideas

rename
Updated
2009-04-27 09:57

## General Math

Add ideas for alternative methods of doing arithmetic; I will try to get more info on this.

I have heard of methods will do not involve multi-tasking to the same degree as the methods traditionally taught in the US.

I have heard of methods will do not involve multi-tasking to the same degree as the methods traditionally taught in the US.

Add charts about convergence tests for infinite sequences and series.

Add charts of common Fourier Transforms (square wave, triangle wave, sawtooth, gaussian) and Laplace Transforms and their generating functions.

Add charts for what substitutions to make when integrating functions containing

sqrt(a

When do x=a sin, a cos, a sec, a csc, a tan, and a cot work best?

sqrt(a

^{2}-x^{2}), sqrt(a^{2}+x^{2}), or sqrt(x^{2}-a^{2}).When do x=a sin, a cos, a sec, a csc, a tan, and a cot work best?

Add charts about complex numbers, how to add them, how to multiply them, how to find their magnitudes, finding complex conjugates, using e

^{ix}=cos(x)+i sin(x), etc. The chart can be many sample problems and their answers.Add charts about unit vectors, vectors, dot products, cross products, gradients, curls, divergences, and Laplacians.

Add charts about Euler Angles, rotation operators, and quaternions.

Add charts about numerical integration methods and error estimates for them (Simpson's rule, Trapezoidal Rule, etc.). Also list numerical quadrature rules.

See Physics Ideas for more.

## Series Expansions

Question | Answer |
---|---|

1 | 1/2 + 1/4 + 1/8 + 1/16 + ... + 1/2^{2n} + ... |

1/(1-x) | 1 + x + x^{2} + x^{3} + ... |

e^{x} | 1 + x + (x^{2})/2! + (x^{3})/3! + ... |

-ln(1-x) | x + (x^{2})/2 + (x^{3})/3 + (x^{4})/4 + ... |

ln(1+x) | x - (x^{2})/2 + (x^{3})/3 - (x^{4})/4 + ... |

ln[(1+x)/(1-x)] | 2 [x + (x^{3})/3 + (x^{5})/5 + (x^{7})/7 + ... ] |

2 tanh^{-1}(x) | 2 [x + (x^{3})/3 + (x^{5})/5 + (x^{7})/7 + ... ] |

tan^{-1}(x) | x - (x^{3})/3 + (x^{5})/5 - (x^{7})/7 + ... |

f(x)=f(-x) | even powers of x only |

f(x)=-f(-x) | odd powers of x only |

cos(x) | even powers of x only |

sin(x) | odd powers of x only |

tan(x) | odd powers of x only |

cos(x) | 1 - (x^{2})/2! + (x^{4})/4! - (x^{6})/6! + ... |

sin(x) | x - (x^{3})/3! + (x^{5})/5! - (x^{7})/7! + ... |

sin(x+y) | |

cos(x+y) | |

source: http://en.wikipedia.org/wiki/Taylor_series

## Equations with Special Names

Equation | Name |
---|---|

AX+XB=C | |

AX=B | |

Sylvester Equation | |

Lyapunov Equation | |

Ricati Equation | |

Schrodinger Equation | |

Maxwell's Equations | |

Heat/Diffusion Equation | |

Wave Equation | |

Fredholm Equation | |

Volterra Equation | |

## Equations with Special Functions as their Solutions

In the following, y=f(x), and y' means dy/dx=df/dx.Equation | Solutions for f(x) |
---|---|

y'' = -k^{2} y | cos(k x), sin(k x), exp(i k x) |

y' = -k y | exp(-k x) |

Neumann functions | |

Bessel functions | |

Spherical Harmonics | |

Legendre Polynomials | |

Airy functions | |

Error functions | |

y=e^{-x2} | Gaussian |

y=1/(1+x^{2}) | Lorentzian |

## Areas of Common 2-D Objects

Question | Answer |
---|---|

rectangle | L W |

circle | π r^{2}=(π d^{2})/4 |

triangle | (1/2) b h |

## Perimeters of Common 2-D Objects

Question | Answer |
---|---|

rectangle | 2 L + 2 W |

circle | 2 π r = π d |

triangle | a + b + c |