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Math Ideas

rename
Updated 2009-04-27 09:57

Summary

Ideas for pages that could be added to the Math section.

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.
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(a2-x2), sqrt(a2+x2), or sqrt(x2-a2).
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 eix=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.

Trig Functions

Question Answer
tan(X)sin(X)/cos(X)
cot(X)1/tan(X)
sec(X)1/cos(X)
csc(X)1/sin(X)
memorize

Series Expansions

Question Answer
11/2 + 1/4 + 1/8 + 1/16 + ... + 1/22n + ...
1/(1-x)1 + x + x2 + x3 + ...
ex1 + x + (x2)/2! + (x3)/3! + ...
-ln(1-x)x + (x2)/2 + (x3)/3 + (x4)/4 + ...
ln(1+x)x - (x2)/2 + (x3)/3 - (x4)/4 + ...
ln[(1+x)/(1-x)]2 [x + (x3)/3 + (x5)/5 + (x7)/7 + ... ]
2 tanh-1(x)2 [x + (x3)/3 + (x5)/5 + (x7)/7 + ... ]
tan-1(x)x - (x3)/3 + (x5)/5 - (x7)/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 - (x2)/2! + (x4)/4! - (x6)/6! + ...
sin(x)x - (x3)/3! + (x5)/5! - (x7)/7! + ...
sin(x+y)
cos(x+y)
memorize

 

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

Sequences with Special Names

SequenceName
Fibonacci series
geometric series
memorize

Equations with Special Names

EquationName
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
memorize

Equations with Special Functions as their Solutions

In the following, y=f(x), and y' means dy/dx=df/dx.
EquationSolutions for f(x)
y'' = -k2 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-x2Gaussian
y=1/(1+x2)Lorentzian
memorize

Areas of Common 2-D Objects

Question Answer
rectangleL W
circleπ r2=(π d2)/4
triangle(1/2) b h
memorize

Perimeters of Common 2-D Objects

Question Answer
rectangle2 L + 2 W
circle2 π r = π d
trianglea + b + c
memorize

Volumes of Common 3-D Objects

Question Answer
sphere(4 π r3)/3
cone
boxL H W
cylinderπ r2 h
memorize

Surface areas of Common 3-D Objects

Question Answer
sphere4 π r2
cone
box2 (L H + L W + W H)
cylinder2 π r2 + 2 π r h
memorize

See also

See Physics Ideas for some more mathematical ideas.

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