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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
Chapter 2
Elementary geometry, graph computation and plot
- §1 triangle and quadrangle
- §2 circle and regular polygon
- §3 practical geometry mapping
- §4 solid figure volume, surface area, side area
Chapter 3
Algebraic equation
- §1 root expression of two, three, quartic order equation
- Second Order Equation
x^2+b*x+c=0
- Solve equation by solve()
solve( x^2+5*x+6=0 )
- 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
- §1 matrix and determinant
- §2 matrix operation
- §3 system of linear equations
Chapter 5
Differential calculus
- sequence and function limit
- Convergence and operation of series
- differential
ddx sin(x)
- Fractional differential calculus
d1.5dx1.5 cos(x)
Chapter 6
Integral calculus
- single variable function integral
- definite integral
∫10 sin(x) dx
- multiple integrals, curvilinear integral and surface integral
- integral application
- region function δ(x)
- Integral table
∫ sin(x) dx
- Fractional integral calculus
∫ cos(x) (dx)0.5
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
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
- First kind of Bessel function besselJ(n,x)
- Second kind of Bessel function besselY(n,x)
- Third kind of Bessel function hankel1(n,x)
- Beta function Beta(a,b)
- beta function
β(x)
- Elliptic integral
- Elliptic function
- error function erf(x)
erfi(x)
- eta function
η(x)
- Exponential integral
Ei(x)
Ein(x)
En(x)
- Fresnel function
fresnelS(x) = S(x)
fresnelC(x)
- Gamma function
- Gamma function Γ(x)
- log Gamma logΓ(x)
- digamma function
digamma(x) ψ(x) = polygamma(x)
- trigamma function
trigamma(x) ψ(1)(x) = polygamma(1,x)
- polygamma function
polygamma(n,x) = ψ(n)(x)
- Incomplete Gamma function Γ(n,x)
- small gamma function smallgamma(n,x)
γ(n,x) = Γ(n,0,x)
- Gaussian integral
Φ(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
E0.5(x)
- orthogonal polynomials
- polylog function
polylog(n,x)
- zeta function
ζ(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
d2dx2 y - 4y =0
dxdt=x,dydt=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( d0.5dx0.5y=2y )
- Test solution for (fractional) differential equation by test()
test( exp(4x), d0.5dx0.5y=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( dydt-d0.5dx0.5y=2y )
- Test solution for (fractional) differential equation by test()
Chapter 15
Integral equations
- The general concept
∫ 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( ∫ y dx = 2y )
dsolve( d-0.5dx-0.5y=2y )
- Test solution for (fractional) integral equation by test()
test( exp(1/4 x), d-0.5dx-0.5y=2y )
Chapter 16
Statistical probability and stochastic processes
- §1 probability theory
?
- standard normal probability density function
Guassian function
gauss(x)
- standard normal distribution function
Gaussian integral
gaussi(x) ϕ(x)
- §2 Mathematical Statistics
- Estimation of overall parameters ?
- Statistical Hypothesis Experiment ?
- Regression Analysis ?
- 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
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
- Digital Library of Mathematical Functions
- math handbook 2016
- math handbook (Russian)
- Handbook of Special Functions
See Also