Introduction to differential equations
First order ordinary differential equation
Variable separation method
homogeneous equation
Exact equation
Combined integral method
First-order linear ordinary differential equation
Bernhard differential equation and Riccati differential equation.
Parameter changing method
Singular solutions and general solutions of higher order nonlinear differential equations
Existence and uniqueness of solution
Picard iteration method
Second (High) Order Linear Differential Equation with Constant Coefficients
Linear independence and Volynski determinant
Second (High) Order Linear Differential Equation with Constant Coefficients
Second (high) order linear differential equations with variable coefficients
Cauchy equidimensional equation
Observed homogeneous solution (parameter change method)
Higher order exact equation
Dependent variable change (parameter change)
Independent variable change
Nonlinear differential equation
Synchronous linear differential equation
Series solutions of ordinary differential equations
Basic definition
Power Series Solution of Taylor Series
Fabenyi series solution of differential equation
Specially defined function
"The First Theorem of Calculus" and "Leibniz Law"
Unit step function
function
Beta function
Laplace transform (Laplace transform)
Labrador transform and its inverse transform
Basic operation theorem
Tension of periodic function
Brass transformer
Solving differential equations by laplace transform.
Solving O.D.E by Laplace Transform
Laplace transform is used to solve O.D.E without boundary, and the boundary conditions are independent of distance.
Solving Integral Equation by Laplace Transform
Bessel and Legendre functions
Bessel equation and Bessel function
Extended form of Bessel equation
Properties of Bessel function
Legendre equation
The Properties of legendre polynomials (Function)
Sturm-Liouville boundary value problem
basic concept
Regular Sturm-Joseph Liouville
Periodic Sturm-Joseph Liouville
Inner product and orthogonality of B.V.P function
Sturm-Liouville Theorem
Generalized Fourier series
Fourier series and integral
Fourier series
Fourier series of odd-even function
Fourier series in complex form with half-width expansion and full-width expansion
Fourier integration and Fourier transform
Basic properties of Fourier transform
Solving differential equations by Fourier analysis
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partial differential equation
Partial differential equations of heat transfer and wave motion in P.D.E(I) Cartesian coordinate system
basic concept
Separation of variables method for regular homogeneous partial differential equations
Transient and steady solutions of inhomogeneous partial differential equations
Heterogeneity, but only P.D.E is related to time.
Heterogeneous but all related to time.
Homogeneity of unbounded region
Laplace equation of partial differential equation (Ⅱ) Cartesian coordinates
Homogeneity rule
Homogeneous infinite partial differential equation
Heterogeneous Laplace partial differential equation
(3) Polar coordinates, cylindrical coordinates and spherical coordinates.
Laplace equation in polar coordinates.
Heat conduction partial differential equation and polar coordinates fluctuation
Laplacian of cylindrical coordinates of partial differential equations
Laplace equation in spherical coordinates.
(IV) First-order Lagrangian equation and second-order partial differential equation
First-order Lagrangian equation
Constant coefficient differential equation
The Solution of D'Alembert Wave Equation
Classification and solution of linear second-order partial differential equations
Variable combination method
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vector analysis
Basic operations of vectors
vector algebra
Vector calculus
Differential and arc length of curve
Differential of multivariate function
Directional derivative and gradient
Geometry of vectors (geometry of vectors)
Vector integral
double integral
Line Integral and Green's Theorem
Surface integral
Divergence, curl and operator
Gauss divergence theorem (Gauss divergence theorem)
Stock theorem
Green's identity (green's identity)
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Complex variable analysis
Complex variable function and complex variable function
plural
Complex plane and polar coordinates
function of a complex variable
Bifurcation and branch cutting of multivariate functions
Limit and differential of complex numbers
limit
Differential sum analysis
Cauchy-riemann equations
Complex integral
Complex integral
Cauchy integral theorem
cauchy integral formula
Complex series
Complex series
Power series and Taylor series
Laurent series
Types of isolated singularities
Residual theorem
Residue (residue)
Residue theorem (residue theorem)
Surplus at infinity
Definite integral of trigonometric function
Loss integral of rational function
Fourier integration (transformation)
Loss integral of multivalued function
Take a special road.
conformal mapping
Mapping (mapping)
conformal mapping
Bilinear transformation
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linear algebra
Matrix and linear simultaneous equations
Matrix sum basic operation
Square matrix and square matrix function
Linear Equations and gauss elimination
Inverse Matrix and gauss elimination
Gauss elimination sum basic matrix
decisive factor
decisive factor
Determinant of block matrix
Adjoint matrix and cofactor
Kramer's law
Pedestal and dimensions
Linear independence and linear correlation
rank of matrix
The relationship between linear simultaneous equations and bases
Eigenvalue problem
fundamental principle
Eigenvalues and eigenvectors
Eigenvalues and eigenvectors of square matrix function f(A)
Four operations of eigenvalues
Kelley-Hamilton Theorem and Its Application
Diagonalization theory and its application
Similarity of matrices
Diagonalization of matrix
Algebraic multiplicity, geometric multiplicity and diagonalizable conditions
Application of Diagonalization Theory
Solving Linear Simultaneous Differential Equations with Constant Coefficients
Jordan standard form
Orthogonal, Normal Matrix and Quadratic Application
Matrix inner product and Gram-Schmidt orthogonalization method
Orthogonal matrix and orthogonal diagonalization
Unitary Diagonalization and Normal Matrix Sets
Application of orthogonal matrix in quadratic form
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calculus
Limit and continuity
limit
Limit of trigonometric function
Limit of Gaussian function
continuously
Theorem related to "continuity"
Asymptote
difference
Derivative (derivative)
Differential of special points
Basic differentiable functions and basic properties of differentiation
Implicit differential (implicit differential)
Inverse function differential
Differential of Exponential Function and Logarithmic Function
Hyperbolic trigonometric function
Higher derivative function
Application of differentiation
L'Hospital's rule
Differential theorem
Increase and decrease, bump and extreme value
The Application of Differential in Drawing
Approximation and Newton approximation
integration
Using formula method
Rational functions of the first kind (denominator contains only one factor)
variable transformation
Integral chain law
Rational Functions of the Second Kind (Denominator with Quadratic Factor)
Partial integral (partial integral)
Trigonometric function integration method
Triangular substitution method of irrational function
Half-angle substitution method
Integral method always reviews exercises.
definite integral
Riemann sum and Integral Limit
definite integral
Special trigonometric function integral
Fundamental theorem of integral
Incomplete integral
Gamma function and beta function
Application of integral
zone
Arc length (arc length)
The center of mass and center of gravity of a plane.
Volume (volume)
Surface area of rotating body
double integral
double integral
Dirichlet integral transformation of double integral
Coordinate transformation of multiple integrals
Multiple integrals of polar coordinates
triple integral
centroid
Surface area of non-rotating body surface
Sequence and series
Sequence (sequence)
series
Convergence and divergence of positive series
Staggered series
Convergence region of power series
Taylor Theorem and Taylor Series
Application of Taylor series in "higher derivative"
Application of Taylor series in integral
vectors
Basic operations of vectors
Directional derivative and gradient
Geometry of vectors (geometry of vectors)
Vector Integral (Doing Work) and Green's Theorem
Divergence Theorem and Stoke Theorem
Multivariate function
Limit and continuity of multivariate functions
Partial derivative (partial derivative)
Extreme value of multivariate function
differential equation
First-order variable separation method
First-order linear ordinary differential equation
Homogeneous solution of second (high) order constant coefficient
Special Solutions of Second (Higher) Order Differential Equations with Constant Coefficients
Euler-Cauchy equation
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Automobile line power generation
Geometric vector spaces (R2 and R3 spaces)
Problem 1: dot product (inner product) and projection quantity
Question 2: Cross product (outer product) and area
Question 3: scalar triple product and volume
Question 4: Lines and planes in space
Matrix and linear simultaneous equations
Basic operations of matrices and matrices
Square Matrix and Square Matrix Algebra
Linear Equations and gauss elimination
Inverse Matrix and gauss elimination
Gauss elimination and Elementary Matrix
LU decomposition of square matrix
decisive factor
decisive factor
Determinant of block matrix
Adjoint matrix and cofactor
Cramer's law
vector space
Euclidean space
vector space
Subspace and generating space
Sum space and direct sum space
Pedestal and dimensions
Linear independence and linear correlation
Pedestal and dimensions
rank of matrix
Relationship between linear equations and base
linear mapping
linear mapping
Image set and kernel space of linear mapping
Synthesis and Inverse Mapping of Linear Maps
Rank of matrices on isomorphic spaces
Coordinate transformation and bottom changing formula
Eigenvalue problem
Eigenvalues and eigenvectors
Question 1: Type 2 2
Question 2: 3 3 and the eigenvalue has no multiple roots.
Question 3: 3 3 sum eigenvalue has multiple roots.
Eigenvalues and eigenvectors of square matrix functions
Four operations of eigenvalues
Kelley-Hamilton Theorem and Its Application
Minimum (lowest) polynomial
feature space
Diagonalization theory and its application
Similarity of matrices
Diagonalization of matrix
Algebraic multiplicity, geometric multiplicity and diagonalizable conditions
Application of Diagonalization Theory
Problem 1: Find the square matrix polynomial.
Question 2: Find the square matrix function
Problem 3: Solving Matrix Equation
Question 4: Recursion and limit of solution matrix
Solving Linear Simultaneous Differential Equations with Constant Coefficients
Question 1: first order homogeneous = ax
Question 2: Second-order Homogeneous = ax
Question 3: Heterogeneity = AX+G
Jordan standard form
Question 1: Find the Jordan form directly.
Question 2: Find the square matrix polynomial.
Question 3: Find the square matrix function
Question 4: Solve linear simultaneous differential equations with constant coefficients.
Inner product space
Definition of inner product space
Matrix inner product and Gram-Schmidt orthogonalization method
QR decomposition of square matrix
rectangular projection
Orthogonal complement set
Normal orthogonal operator and normal orthogonal matrix
Adjoint operator
Normal operator and self-adjoint operator
Normal matrix set
Orthogonal operator and unitary operator
Orthogonal Diagonalization and Unitary Diagonalization
norm of matrix
Household change
Spectral decomposition and singular value decomposition
Quadratic formula and its application
Quadratic formula and positive definite and semi-positive definite characteristics of matrix
Application of Quadratic Formula (I): Principal Axis Theorem and Multiple Integrals
Application of Quadratic Formula (Ⅱ): Rayleigh Principle and Extreme Value of Quadratic Formula
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Motion probability
permutation and combination
grade
combine
Introduction to probability
Classical probability theory
set theory
probability space
Basic theorem of probability
Conditional Probability and Independent Events
Conditional Probability and Bayesian Theorem
Random variables and probability distribution
stochastic variable
probability distribution
Expected value and variance
Joint probability distribution function
Functions and transformations of random variables
Dynamic difference and dynamic difference inequality
Expectation value and dynamic difference
Dynamic difference and dynamic difference generating function
Markov inequality and Chebyshev inequality
Discrete probability model
evenly distribution
bernoulli distribution
Binomial distribution
Hypergeometric distribution
Multiple distribution
geometric distribution
negative binomial distribution
Poisson distribution
Continuous probability model
evenly distribution
normal distribution
Exponential distribution
gamma distribution
That's it. That's it.