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Content 3

1. Algebra, triangle formula and elementary function ?
   
2. Elementary geometry graph computation and plot ?
   
3. Algebraic equation ?
   
4. Matrix · determinant · system of linear equations ?
   
5. Differential calculus ?
   
6. Integral calculus ?
   
7. Analytic geometry and differential geometry ?
   
8. Vector algorithms and tensor field and theory algorithms and Riemannian geometry ?
   
9. Abstract algebraic - linear space - Functional Analysis ?
   
10. Complex function ?
    
11. Integral transform ?
    
12. Special Function ?
    
13. Ordinary Differential Equations (ODE) ?
    
14. Partial Differential Equations (PDE) ?
    
15. Integral equations ?
    
16. Statistical probability and stochastic processes ?
    
17. Error theory and experimental data processing ?
    
18. Optimization methods ?
    
19. Finite element method ?
    
20. Elementary number theory ?
    
21. 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
  1. function concept and classification
  2. Power functions `x^2`
  3. Exponential and logarithmic functions ln(x)
  4. Trig functions sin(x)
  5. Inverse trigonometric functions asin(x)
  6. Hyperbolic functions sinh(x)
  7. Inverse hyperbolic functions asinh(x)
  8. inverse function inverse( sin(x) )
  9. function graph

1. §1 triangle and quadrangle
   
2. §2 circle and regular polygon
   
3. §3 practical geometry mapping
   
4. §4 solid figure volume, surface area, side area
   

1. §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. §2 algebraic equation nature
   
3. §3 special solution of algebraic equation
   
4. §4 approximate calculation of real roots
   

1. §1 matrix and determinant
   
2. §2 matrix operation
   
3. §3 system of linear equations
   

1. sequence and function limit
   
2. Convergence and operation of series
   
3. differential `d/dx` sin(x)
   
   • Derivative table
   • implicit function
     
   • differential application
     
4. Fractional differential calculus `d^1.5/dx^1.5` cos(x)

1. single variable function integral
   
2. definite integral `int_0^1` sin(x) dx
   
3. multiple integrals, curvilinear integral and surface integral
   
4. integral application
   
5. region function `delta(x)`
   
6. Integral table `int` sin(x) dx
   
7. Fractional integral calculus `int` cos(x) `(dx)^0.5`

Chapter 7
Analytic geometry and differential geometry

Analytic geometry
• §1 coordinate system and coordinate transformation
  
• §2 in analytic geometry basic computation
  
• §3 in plane straight line
  
• §4 in spatial straight line and plane
  
• §5 conic section
  
• §6 quadratic surface
  
• 2D surface

Differential geometry
• §7 plane curve
  
• §8 Important plane curves
• §9 space curve
  
• §10 helical surface
  
• §11 developable surface
  
• §12 General Surfaces
  
• 3D surface

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

1. 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)
2. Beta function Beta(a,b)
3. beta function `beta(x)`
4. Elliptic integral
5. Elliptic function
6. error function erf(x) erfi(x)
7. eta function `eta(x)`
8. Exponential integral Ei(x) Ein(x) En(x)
9. Fresnel function fresnelS(x) = S(x) fresnelC(x)
10. 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)`
11. Gaussian integral `Phi(x)`
12. harmonic number harmonic(x) H(x)
13. hypergeometric function
14. Legendre function
15. Lerch L(x,a,b)
16. logarithmic integral li(x)
17. sine and cosine integral si(x) ci(x)
18. Mittag-Leffler function `E_0.5(x)`
19. orthogonal polynomials
20. polylog function polylog(n,x)
21. zeta function `zeta(x)`

Chapter 13
Ordinary Differential Equations ( ODE )

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

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

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

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

Chapter 15
Integral equations

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

• §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

1. Digital Library of Mathematical Functions
2. math handbook 2016
3. math handbook (Russian)
4. Handbook of Special Functions
   

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