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Content 1 - Content 2 - Content 3 - Content 4 - Index

Content 3

Algebra, triangle formula and elementary function ?
Elementary geometry graph computation and plot ?
Algebraic equation ?
Matrix · determinant · system of linear equations ?
Differential calculus ?
Integral calculus ?
Analytic geometry and differential geometry ?
Vector algorithms and tensor field and theory algorithms and Riemannian geometry ?
Abstract algebraic - linear space - Functional Analysis ?
Complex function ?
Integral transform ?
Special Function ?
Ordinary Differential Equations (ODE) ?
Partial Differential Equations (PDE) ?
Integral equations ?
Statistical probability and stochastic processes ?
Error theory and experimental data processing ?
Optimization methods ?
Finite element method ?
Elementary number theory ?
Set theory and general topology ?

Elementary Math
Chapter 1
Algebra, triangle formula and elementary function

§1 algebraic formula
1. number expansion, classification and basic rules of operation
2. complex
3. number of columns and simple series
4. multiplication and factorization factor( $x^2-1$ )
5. fractional
6. proportion
7. radical
8. inequality solve( 2*x-1>0 )
9. factorial, arrangement and combination
10. Triangle and polynomial theorem
11. principle of mathematical induction and drawers
§2 elementary function and numerical computation

Chapter 2
Elementary geometry, graph computation and plot

Chapter 3
Algebraic equation

§1 root expression of two, three, quartic order equation
1. Second Order Equation x^2+b*x+c=0
2. Solve equation by solve() solve( x^2+5*x+6=0 )
3. Numerical Solve equation by nsolve() nsolve( x^2+5*x+6=0 )
§2 algebraic equation nature
§3 special solution of algebraic equation
§4 approximate calculation of real roots

Higher Math
Chapter 4
Matrix · determinant · system of linear equations

Chapter 5
Differential calculus

Chapter 6
Integral calculus

Chapter 7
Analytic geometry and differential geometry

Chapter 8
Vector algorithms and tensor field and theory algorithms and Riemannian geometry

§1 vector algorithm ?
§2 field theory ?
§3 affine coordinate system ?
§4 tensor algorithm ?
§5 Riemannian geometry ?

Chapter 9
Abstract algebraic - linear space - Functional Analysis

§1 abstract algebra ?
§2 linear space and linear subspace ?
§3 linear transformation ?
§4 unitary space ?
§5 quadratic and Hermitian type ?
§6 phalanx of Standard Form ?
§7 preliminary functional analysis ?

Chapter 10
Complex function

§1 analytic function ?
- Basic concepts
- analytic function
- analytic continuation
- elementary analytic functions
- Riemann surface fulcrum and branch
§2 conformal mapping ?
§3 integral of complex function ?
§4 Taylor series Laurent series · residue theorem ?

Chapter 11
Integral transform

§1 Fourier series ?
§2 generalized Fourier series and Fourier - Bessel series ?
§3 Laplace transform ?
1. Laplace transform laplace( sin(x) )
2. Inverse Laplace transform inverselaplace( 1/(x^2+1) )
§4 Fourier transform ?
§5 fast Fourier transform ?
§6 Mellin transform ?
§7 Hankel transform ?
§8 Legendre transform and its inversion formula ?

Chapter 12
Special Function

§1 Special functions defined by integral
§2 Orthogonal polynomial
§3 Hypergeometric function
§4 Legendre function
§5 Bessel function
§6 Elliptic function
§7 Bernoulli number and Bernoulli polynomial

Bessel function
1. First kind of Bessel function besselJ(n,x)
2. Second kind of Bessel function besselY(n,x)
3. Third kind of Bessel function hankel1(n,x)
Beta function Beta(a,b)
beta function $beta(x)$
Elliptic integral
Elliptic function
error function erf(x) erfi(x)
eta function $eta(x)$
Exponential integral Ei(x) Ein(x) En(x)
Fresnel function fresnelS(x) = S(x) fresnelC(x)
Gamma function
1. Gamma function $Gamma(x)$
2. log Gamma $logGamma(x)$
3. digamma function digamma(x) $psi(x)$ = polygamma(x)
4. trigamma function trigamma(x) $psi^((1))(x)$ = polygamma(1,x)
5. polygamma function polygamma(n,x) = $psi^((n))(x)$
6. Incomplete Gamma function $Gamma(n,x)$
7. small gamma function smallgamma(n,x) $gamma(n,x)$ = $Gamma(n,0,x)$
Gaussian integral $Phi(x)$
harmonic number harmonic(x) H(x)
hypergeometric function
Legendre function
Lerch L(x,a,b)
logarithmic integral li(x)
sine and cosine integral si(x) ci(x)
Mittag-Leffler function $E_0.5(x)$
orthogonal polynomials
polylog function polylog(n,x)
zeta function $zeta(x)$

Chapter 13
Ordinary Differential Equations ( ODE )

General concept of differential equations
Differential equations y' -2y = 0
linear differential equation
higher order and a system of differential equations
$d^2/dx^2$ y - 4y =0
$dx/dt=x, dy/dt=x-y$
stability theory
Numerical Solution of Ordinary Differential Equations
Solve Differential equation for y by dsolve()
dsolve( y'-2y=0 )
Test solution for (fractional) differential equation by test()

test( exp(2x), dy/dx = 2y )

Solve Fractional differential equation by dsolve()
dsolve( $d^0.5/dx^0.5 y=2y$ )
Test solution for (fractional) differential equation by test()
test( exp(4x), $d^0.5/dx^0.5 y=2y$ )

Chapter 14
Partial Differential Equations ( PDE )

general concept of partial differential equations and problems
first order partial differential equations
second order partial differential equations
numerical solution of partial differential equations
Solve Fractional Partial differential equation by pdsolve()
pdsolve( $dy/dt - d^0.5/dx^0.5 y = 2y$ )
Test solution for (fractional) differential equation by test()

Chapter 15
Integral equations

The general concept
$int$ y dx = 2y
singular integral equation
Volterra integral equations
Approximate Solution of Integral Equation
nonlinear integral equation
Solve Fractional integral equation for y by dsolve()
dsolve( $int$ y dx = 2y )
dsolve( $d^-0.5/dx^-0.5 y=2y$ )
Test solution for (fractional) integral equation by test()
test( exp(1/4 x), $d^-0.5/dx^-0.5 y=2y$ )

Chapter 16
Statistical probability and stochastic processes

§1 probability theory ?
1. standard normal probability density function
  Guassian function gauss(x)
2. standard normal distribution function
  Gaussian integral gaussi(x) $phi(x)$
§2 Mathematical Statistics
1. Estimation of overall parameters ?
2. Statistical Hypothesis Experiment ?
3. Regression Analysis ?
4. Orthogonal experimental design ?
§3 random process ?

Chapter 17
Error theory and experimental data processing

§1 error theory ?
§2 interpolation formula ?
§3 arc curve fitting method and average method ?
§4 experimental curve smoothing method ?
§5 filter ?

Chapter 18
Optimization methods

§1 univariate function extreme problem solution (direct method) ?
§2 multivariable function extreme problem solution (direct method) ?
§3 unconditional extreme value problem solution ?
§4 conditional extremum problem solution ?
§5 variational method ?
§6 minimum (maximum) value principle ?
§7 Dynamic Programming ?

Chapter 19
Finite element method

§1 General Theory and solution step ?
§2 basic unit with linear interpolation ?
§3 parameters unit and high-order interpolation ?
§4 proposed coordination unit ?
§5 Elastic theory and Finite Element Method ?
Appendix Numerical Integration in Finite Element Method ?

Chapter 20
Elementary number theory

§1 integer ?
§2 continued fraction ?
§3 congruence 3x=2*(mod 4) ?
§4 number theoretic function ?
§5 polynomial solve( 3x-4y-2=0, x,y ) ?
§6 algebraic number ?

Chapter 21
Set theory and general topology

§1 Set (collection) ?
§2 ordinal and cardinal number ?
§3 topological space ?
§4 scale space with uniform space ?
§5 compact set of points and junction point set ?
§6 manifold ?

Reference

See Also

Math - Symbol - Handbook - math word
Elementary Math - Higher Math
Calculator - mathHand - mathHandbook
Function - Formula - equation - graphics
Fractional Calculus - differential equation
Tests - Example :

$(d^0.5y)/dx^0.5 = sin(x-1)sin(y-1)$ == ? $(d^0.5y)/dx^0.5 -cosh(y)-sinh(y)=0$ == ? $(d^1.6y)/(dx^1.6)-int y(x) (dx)^(0.8)-y-exp(x)=0$ == ? $int y(x) (dx)^0.5 -y-exp(x)$ =0 == ? $(d^0.5y)/dx^0.5-exp(y)x=0$ == ? $(d^0.5y)/dx^0.5-exp(y)y=0$ == ? $(d^0.5y)/dx^0.5=cos(x)/xy$ == ? $y(dy^0.5)/dx^0.5-sqrt(x)-1=0$ == ? $(d^1.2y)/(dx^1.2)-2(d^0.6y)/dx^0.6+y-exp(x)=0$ == ? $(d^0.5y)/dx^0.5=cos(y)exp(x)x$ == ? $(d^1.6y)/(dx^1.6)-2(d^0.8y)/dx^0.8+y-exp(x)=0$ == ? $(d^0.5y)/dx^0.5-exp(y)sqrt(x)=0$ == ? $(d^1.6y)/(dx^1.6)-3 (d^0.8y)/dx^0.8+2y-exp(x)=0$ == ? $(d^0.5y)/dx^0.5$ +log(y-1)-exp(x)-x=0 == ? $(d^0.5y)/dx^0.5-exp(y)sin(x)=0$ == ? $(d^0.5y)/dx^0.5 = ysin(x)/x$ == ? $y^((0.5))(x) -4 exp(x)y-exp(x)=0$ == ? $(dy^0.5)/dx^0.5 = 1/(x-y)$ == ? $dy/dx-(d^0.5y)/dx^0.5$ - y - exp(x)=0 == ? $(dy)/dx -exp(y-1)-x-x^2=0$ == ? $(d^1.2y)/(dx^1.2)-3dy^0.6/dx^0.6+2y-exp(x)=0$ == ? $dy/dx-(d^0.5y)/dx^0.5-y-1$ =0 == ? $(d^0.5y)/dx^0.5-cos(y)sin(x)=0$ == ? $(d^1.6y)/(dx^1.6)-(d^0.8y)/dx^0.8-y-exp(4x)=0$ == ? $dy/dx-exp(y-1)-exp(x)=0$ == ? $(dy)/dx - 2(d^0.5y)/dx^0.5-y-exp(x)=0$ == ? $(d^1.6y)/(dx^1.6)-(d^0.8y)/dx^0.8-y-exp(x)=0$ == ? $(d^0.5y)/dx^0.5 -e^(4x)-y$ =0 == ? $y^((0.5))(x) - exp(x)y-exp(x)=0$ == ? $y^((0.5))(x) - exp(x)y-4exp(x)=0$ == ? $(dy)/dx -3(d^0.5y)/dx^0.5 +2y-exp(x)=0$ == ? $y(d^0.5y)/dx^0.5-sqrt(x)-1=0$ == ? $y^((1))(x)-exp(y-1)-x=0$ == ? $(d^1.6y)/(dx^1.6)-(d^0.8y)/dx^0.8-2y-exp(x)=0$ == ? $(d^1.6y)/(dx^1.6)-(d^0.8y)/dx^0.8-y-exp(4x)=0$ == ? $(d^0.5y)/dx^0.5 - log(y-1) - exp(x) + x=0$ == ? $(dy)/dx +asin(y-1) - cos(x)-x=0$ == ? $(d^1.6y)/(dx^1.6)-3(d^0.8y)/dx^0.8+2y-exp(x)=0$ == ? $(dy)/(dx) -sqrt(y-1)-x-1 =0$ == ? $(dy)/(dx) -exp(y-1)-exp(x) = 0$ == ? $(dy)/dx$ +asinh(y-1)-cosh(x)-x =0 == ? $((d^(1/2)y)/dx^(1/2))^2 -3y (dy^0.5)/dx^0.5 + 2y^2 = 0$ == ? $(dy^0.5)/dx^0.5 = cos(x)cos(y-1)$ == ? $(d^0.5y)/dx^0.5 +log(y-1)-exp(x)-x=0$ == ? $(dy^0.5)/dx^0.5 = sin(x-1)exp(y-1)$ == ? $y(d^2y)/dx^2-(dy/dx)^2+1=0$ == ? $y^((1))(x)-exp(y-1)-log(x)=0$ == ? $(d^2y)/dx^2 exp(x)- exp(y-1)=0$ == ? $(d^1.6y)/(dx^1.6)-2 (d^0.8y)/dx^0.8-y-exp(x)=0$ == ? $(d^1.6y)/(dx^1.6)-2 (d^0.8y)/dx^0.8+y-exp(x)=0$ == ? $(dy)/dx -3 (d^0.5y)/dx^0.5+2y-exp(x)=0$ == ? $y^((0.5))(x) - xy-x=0$ == ? $y(dy^3)/dx^3-x^3-3x^2-3x-1=0$ == ? $y^((1.8))(x)-2y^((0.9))(x) +y-1=0$ == ? $y^((0.5))(x)=1/(xy-1)$ == ? $y^((2))(x)y^2-x^2-2x-1=0$ == ? $((d^0.5y)/dx^0.5)^2 -5(d^0.5y)/dx^0.5 +6=0$ == ? $y^((0.5))(x) -2 exp(x)y-4exp(x)=0$ == ? $(d^1.6y)/(dx^1.6)-(d^0.8y)/dx^0.8-y-exp(x)=0$ == ? $y^(0.5)(x)=2yexp(x)$ == ? $y^((0.5))(x)-exp(x)y^2=0$ == ? $(d^1.6y)/(dx^1.6)-2(d^0.8y)/dx^0.8+y-exp(x)=0$ == ? $y^((1))(x)-y^2-xy=0$ == ? $y^((1))(x)-y^((0.5))(x) -y-1=0$ == ? $y^((2))(x) -y^2-x^2=0$ == ? $y^((2))(x) -y^2-x^2-2xy=0$ == ? $y^((0.5))(x) -int y(x) (dx)^0.5-y-exp(x)=0$ == ? $d^0.5/dx^0.5 y -2cos(y)exp(x)=0$ == ? $d^0.5/dx^0.5 y -4sin(y)exp(x)=0$ == ? $(d^0.5y)/dx^0.5=sin(x^2)y$ == ? $(d^0.5y)/dx^0.5-sin(x)sin(y)=0$ == ? $(d^0.5y)/dx^0.5-sinh(x)sinh(y)=0$ == ? $y^((1))(x)=exp(x-y)-x$ == ? $x(d^0.5y)/dx^0.5-y-2x=0$ == ? $(d^0.5y)/dx^0.5=sinh(x-1)sinh(y-1)$ == ? $y^((0.5))(x)-exp(-x)y^2=0$ == ? $(d^0.5y)/dx^0.5=y/xsin(x)$ == ? $(dy)/dx-sin(x-y)-1=0$ == ? $(d^2.5y)/dx^2.5=y(d^0.5y)/dx^0.5$ == ? $(d^0.5y)/dx^0.5=y(dy)/dx$ == ? $(d^(2-i)y)/dx^(2-i)- y+x=0$ == ? $(d^2y)/dx^2=y^3x^2$ == ? $y(d^2y)/dx^2-x^2-3x-1=0$ == ? $y(d^2y)/dx^2-2x^2-3x-1=0$ == ? $(y-x-1)(d^2y)/dx^2-3x-1=0$ == ? $y^2(d^2y)/dx^2-x^2-4x-4=0$ == ? $(y-x-1)(d^2y)/dx^2-x^2-4x-4=0$ == ? $y(d^2y)/dx^2-2x^2-2x-1=0$ == ? $y(d^3y)/dx^3-6x^3-3x^2-3x-1=0$ == ? $y^((0))(x)y^((1))(x)y^((2))(x)=x^2$ == ? $y^((3))(x)y^((2))(x)=y^((1/2))(x)$ == ? $y^((3))(x)=exp(x)y^((1))(x)y^((1/2))(x)$ == ? $y^((1/2))(x)y^((3))(x)=exp(x)$ == ? $y^((1/2))(x)y^((2))(x)=exp(x)$ == ? $(d^0.5y)/dx^0.5-2xy-1=0$ == ? $y^2(d^0.5y)/dx^0.5-x^2-4x-4=0$ == ? $exp(y-1)(d^0.5y)/dx^0.5-x=0$ == ? $y(d^2y)/dx^2-(x-2)(2x-4)=0$ == ? $y(d^3y)/dx^3-6x^3-4x^2-4x-1=0$ == ? $exp(y-1)(d^2y)/dx^2-exp(x)=0$ == ? $y^2(d^2y)/dx^2-x^2-1=0$ == ? $1/y^2(d^2y)/dx^2-x^2-1=0$ == ? $(y-x-1)(d^3y)/dx^3-(x-2)(2x-4)(3x-1)=0$ == ? $(d^0.5y)/dx^0.5-2x^2y^2-8x^2=0$ == ? $(d^0.5y)/dx^0.5-2xy^2-8x=0$ == ? $(d^0.5y)/dx^0.5-y^2-2y-2=0$ == ? $(d^0.5y)/dx^0.5-log(y-1)exp(x)=0$ == ? $y(d^2y)/dx^2-(dy/dx)^2-1=0$ == ? $(d^2y)/dx^2-asin(y-1)-sin(x)-x=0$ == ? $dy/dx(x--y)-x--y-1 = 0$ == ?

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