Learn: Math Ideas - Memorize.com - Remember and Understand

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Summary

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

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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 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.

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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)
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Series Expansions

Question	Answer
1	1/2 + 1/4 + 1/8 + 1/16 + ... + 1/2²ⁿ + ...
1/(1-x)	1 + x + x² + x³ + ...
e^x	1 + 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)
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source: http://en.wikipedia.org/wiki/Taylor_series

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Sequences with Special Names

Sequence	Name
	Fibonacci series
	geometric series
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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
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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'' = -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^-x²	Gaussian
y=1/(1+x²)	Lorentzian
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Areas of Common 2-D Objects

Question	Answer
rectangle	L W
circle	π r²=(π d²)/4
triangle	(1/2) b h
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Perimeters of Common 2-D Objects

Question	Answer
rectangle	2 L + 2 W
circle	2 π r = π d
triangle	a + b + c
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Volumes of Common 3-D Objects

Question	Answer
sphere	(4 π r³)/3
cone
box	L H W
cylinder	π r² h
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Surface areas of Common 3-D Objects

Question	Answer
sphere	4 π r²
cone
box	2 (L H + L W + W H)
cylinder	2 π r² + 2 π r h
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Math Ideas

Summary

General Math

Trig Functions

Series Expansions

Sequences with Special Names

Equations with Special Names

Equations with Special Functions as their Solutions

Areas of Common 2-D Objects

Perimeters of Common 2-D Objects

Volumes of Common 3-D Objects

Surface areas of Common 3-D Objects

See also

Pages linking here (main versions)