higher mathematics knowledge point combing summary

Posted on Edited on In Waline:

Today, I finally got through the knowledge of higher mathematics, but because the whole process is long and there are many knowledge points, they are all fragmented and not systematic, so I thought I would take this opportunity to generally sort out all the knowledge points once.

Brief summary

In general, the core knowledge of higher data revolves around limits and continuity, so it is important to be proficient in finding limits.

The derivative of a one-dimensional function is actually defined by the limit. And when the increment of the independent variable tends to 0, the derivative of a unit function multiplied by the increment of the independent variable is the increment of a unit function, which defines the differentiation of a unit function, and the integral of a unit function is defined by differentiation.

The derivative of a multivariate function, the differentiation and the integral are defined in a similar relationship to that of a univariate function. There are some formulas in multivariate functions that need to be noted, such as the second type of curve integral, which can be changed into a double integral by Green’s formula in the plane, and the second type of curve integral in space can be changed into a second type of surface integral by Stokes’ formula, and then into a triple integral by Gauss’ formula.

The convergence criterion of the series is also mostly carried out by using the limit, such as the ratio method of the positive series, the root value method, and the Taylor series of the power series is simply the Taylor expansion of the monomial function, which seeks whether the limit of the residue term is 0.

The last is the differential equation, in fact, to you is the relationship between the function and the derivative, so that you find the expression of the function in line with this relationship

Higher Mathematics

Functions, Limits, Continuity

Definition

  • function

  • Function Concept

  • Segmented functions

  • Compound Functions

  • Inverse Functions

  • Primary functions

  • Limits

  • Limits of series

  • Limits of functions

  • Limits of a function when the independent variable tends to infinity

  • Limit of a function when the independent variable tends to a finite value

  • The concept of infinitesimal quantities

  • The concept of infinitely large quantities

  • continuity

  • The concept of continuity

  • Definition of interruption points

  • Classification of interruption points

  • Type I interruption points

  • Removable interruption points

  • Jumping interruption points

  • Second type of intermittent point

  • Infinite intermittent points

  • Oscillating intermittent point

  • Functional properties

  • Monotonicity

  • Parity

  • Periodicity

  • Boundedness

Theorem

  • A sufficient condition for the existence of the limit of a series: the limit of an odd term is equal to the limit of an even term

  • Limits of functions exist

  • The limit of a function exists when the independent variable tends to infinity

  • The limit of the function when the independent variable tends to a finite value

  • The relationship between the limit of a series and the limit of a function

  • Continuity of functions

  • The operation of continuous functions

  • Quadratic operations

  • Continuity of complex functions

  • Continuity of inverse functions

  • Continuity of elementary functions

  • Properties of continuous functions on closed intervals

  • Optimality theorem

  • Boundedness theorem

  • Median theorem

  • Zero theorem

  • Equivalent infinitesimal substitution theorem

Equation properties

  • Basic primitive functions

  • Power functions

  • Exponential functions

  • Logarithmic functions

  • Trigonometric functions

  • Inverse trigonometric functions

  • Limiting properties

  • boundedness

  • number-preserving

  • Infinitesimal property

  • The sum of finite infinitesimals is still infinitesimal

  • The product of finite infinitesimals is still infinitesimal

  • The product of an infinitesimal quantity and a bounded quantity is still infinitesimal

  • Comparison of infinitesimals

  • High-order infinitesimals

  • low order infinitesimal

  • same order infinitesimal

  • Equivalent infinitesimal

  • k-order infinitesimal

  • Relationship between extreme values and infinitesimals:limf(x) = A <=> f(x) = A + α(x)

  • The nature of infinitesimals

  • Relationship between infinitesimal quantities and unbounded variables: infinitesimal quantities require n > N when there is a constant |xn| > M, while unbounded variables do not require

  • Relationship between infinitely large and infinitesimal quantities

  • The method of finding the limit

  • The four rules of limits

  • Two important limits

  • sinx and x are equivalently infinitesimal

  • infinitesimal type of 1

  • Commonly used equivalent infinitesimals

  • Lopita’s law

  • Clipping criterion

  • Limit criterion for monotone bounded series

  • Monotone bounded function must have limit

  • Monotonically increasing series with upper bound must have limit

  • Monotonically decreasing series with lower bound must have limit

  • The nature of infinitesimals: the sum of infinitesimals is still infinitesimal

  • Functional continuity

  • Taylor’s formula

  • with pianos remainder term

  • with Lagrangian remainder term

  • Commonly used Taylor’s formula (McLaughlin’s formula)

  • can be introduced as equivalent infinitesimal

  • and also Taylor series

  • Use the definition of derivative to find the limit

  • Differential median theorem

  • Definition of definite integrals

  • Properties of convergence of series

Differentiation of Unitary Functions

The concept of derivatives and differentiation

  • Concept and geometric meaning of derivatives

  • The concept of derivative

  • Derivatives and derivative functions on an interval

  • Geometric meaning of derivatives

  • The concept and geometric meaning of differentiation

  • Definition

  • The linear principal part of the increment of a function is called the differential

  • dy is the derivative multiplied by the differential of the independent variable dx

  • Geometric meaning: the differential represents the increment of the vertical coordinate of the tangent line of the curve at that point at that point

  • The relationship between continuous, derivable, and differentiable

Calculation of derivatives and differentiation

  • Calculation of derivatives

  • Derivative formula of basic elementary functions

  • The rule of derivative of the four operations

  • Complex function derivative rule

  • Inverse function derivative rule: the derivative of the inverse function is the reciprocal of each other

  • The method of derivatives of implicit functions

  • Logarithmic derivative

  • Parametric equation derivative method

  • Derivative of segmented functions

  • Important conclusions about the derivative

  • The derivative of an even function is an odd function

  • The derivative of a derivable odd function is an even function

  • The derivative of a derivable periodic function is still a periodic function with constant period

  • Calculation of higher order derivatives

  • direct method, respectively, to find the first-order derivative, second-order derivative, third-order derivative, etc., to find the law

  • Indirect method: using the known higher order derivative formula, the algorithm, through the function of constant deformation, variable replacement to find the higher order derivative results

  • Second-order derivatives of several types of functions

  • Abstract composite functions

  • Second-order derivative of implicit functions

  • Differential calculations

  • The four rules of differentiation

  • Invariance of first-order differential forms

Median theorem, inequalities, zero problems

  • Median theorem

  • Rolle’s theorem

  • Fermat’s theorem

  • Lagrange’s median theorem

  • Corsi’s median theorem

  • Taylor’s theorem

  • Taylor’s formula of order n for Lagrange’s remainder term

  • Taylor’s formula for Peyano’s remainder term

  • McLaughlin’s formula

  • Inequality proofs

  • Monotonicity

  • Maximum value

  • Lagrange’s median formula

  • Lagrangian Remainder Taylor Formula

  • Zero point problem

  • Median theorem or zero theorem for continuous functions

  • Rolle’s theorem

Application of derivatives

  • Monotonicity of functions

  • Extreme values of functions

  • The extreme value point

  • Stationary points

  • Necessary conditions for the existence of extremes

  • First Sufficient Condition for Extreme Values

  • Second sufficient condition for extremes

  • The most value of a function

  • The point of maximum value

  • Convexity of a curve

  • Inflection point

  • Necessary conditions for an inflection point

  • First sufficient condition for inflection point

  • Second sufficient condition for the point of inflection

  • Asymptote of the curve

  • Horizontal asymptote

  • Vertical asymptote

  • Oblique asymptote

  • Arc differentiation and curvature

Integral of Unitary Functions

Concepts of indefinite and definite integrals, properties

  • Principle functions, indefinite and definite integrals

  • Geometric meaning of definite integrals

  • The definite integral is the limit of the sum of integrals

  • Basic properties of integrals

  • Properties of definite integrals

  • Existence theorem of definite integrals

  • If the function is continuous on a closed interval, the definite integral exists

  • The integral exists if the function has only finitely many interrupted points on the closed interval

  • Variable finite integral

  • variable upper integral

  • variable lower integral

  • Variable upper bound indefinite integrals are derived from the upper bound of the integral to obtain the relationship between definite and indefinite integrals

  • Newton-Leibniz formula

Calculation of definite and indefinite integrals

  • Basic integral formula

  • Basic integral method

  • Method of integration by integration (first permutation method)

  • Commutative integration method (second permutation method)

  • Several common permutation methods

  • Definite integral reduced integration method

  • The method of partial integration

  • Several definite integral formulas

Calculation of Inverse Integrals

  • Anomalous integrals: limits of variable limit integrals

  • Inverse integrals on infinite intervals

  • Anomalous integrals of unbounded functions

  • Inverse integrals of parity functions on symmetric intervals

  • An important anomalous integral

Applications of definite integrals

  • Area of a plane figure
  • Volume of a rotating body
  • Mean value of a function
  • Volume of a three-dimensional with known area of parallel sections on an interval
  • Arc length of a plane curve
  • Area of a rotating surface
  • Work done by variable forces
  • Hydrostatic pressure of liquid
  • Gravitational force
  • Center of mass (form center) of an object

Vector algebra and spatially analytic sets

vector algebra

  • The basic concept of vectors

  • Vector operations

  • addition and subtraction

  • Number multiplication

  • Product of quantities

  • Operation rules

  • Law of commutation

  • Distributive law

  • vector product

  • Mixed products

Spatial analytic geometry

  • Spatial planes and lines

  • Plane equations

  • General equation

  • Point method equation

  • Intercept type

  • Straight line equation

  • General formula

  • Symmetric

  • Parametric

  • Plane and line relationship

  • Plane to plane relationship

  • Relationship between a straight line and a line

  • Distance from point to surface

  • Point to line distance

  • Curved surface and space curve

  • Surface equation

  • Spatial curves

  • Common surfaces

  • Common quadratic surface equations

Multifunctional Differentiation

Limits and continuity of multivariable functions

  • Concept of binary functions

  • Definition

  • Geometric meaning of a binary function

  • Limits and continuity of binary functions

  • The concept of heavy limit

  • The concept of continuous binary functions

  • Properties of multivariate continuous functions

  • sum and difference product quotient of all continuous functions

  • the most value theorem

  • Mediation theorem

  • All multi-source elementary functions are continuous everywhere in their defined regions

Differentiation of multifunctions

  • Partial derivatives and full differentiation of binary functions

  • Definition of partial derivative

  • Geometric meaning of partial derivative

  • Full increment

  • Full differentiation

  • Definition

  • Necessary condition for the existence of full differentiation: existence of partial derivatives

  • Sufficient condition for the existence of full differentiation: partial derivatives are continuous

  • Partial derivatives and full differentiation of complex functions

  • Rules of derivatives of composite functions

  • Compound functions and multivariate functions

  • Multifunction and multifunction composite

  • Full differential form invariance

  • Higher order partial derivatives

  • Partial derivatives and full differentiation of implicit functions

  • Derivative of a univariate implicit function determined by an equation

  • Derivative of a binary implicit function determined by an equation

  • The derivative of a one-dimensional implicit function determined by a system of equations

  • Derivative of a binary implicit function determined by a system of equations

Limits and maxima

  • Unconditional extrema

  • Extreme value points

  • Necessary conditions for the existence of extreme values

  • Sufficient conditions for the existence of extremes

  • Conditional extreme values

  • Lagrange’s multiplier method

  • Maximum value

  • Find the most value on a bounded closed region

  • find the value of the function of the extreme value point in the region

  • find the most value on the boundary of the region (conditional extrema), for simpler than that bounded function can be brought directly into

  • Compare all the extreme values in the above two steps

  • application problem, the extreme value point may only have a

Directional derivatives, gradients and geometric applications

  • Directional derivatives and gradients
  • Geometric applications

Integration of multivariate functions

Recalculus

  • double integrals

  • Definition and geometric meaning of double integral

  • Properties of double integrals

  • Comparison theorem

  • Valuation theorem

  • Median theorem

  • Calculation of double integrals

  • Calculation in Cartesian coordinates

  • Calculation in polar coordinates

  • Calculation using symmetry parity

  • Using symmetry of integral domain and parity of product function

  • Symmetry of variables

  • Definition

  • Nature: Same as double integral

  • Calculation

  • Right-angle coordinates

  • First one, then two

  • First two, then one

  • Column Coordinates

  • Spherical coordinates

  • Parity

  • Rotational symmetry

Curve integral

  • Line integrals over arc lengths (first class line integrals)

  • Definition

  • Nature

  • Calculation method

  • Direct method

  • Parity

  • Symmetry

  • Line integral over coordinates (second type of line integral)

  • Definition

  • Properties

  • Calculation method (plane)

  • Direct method

  • Green’s formula (reduced to a double integral)

  • Green’s formula for complementary lines

  • Line integral is independent of path

  • Calculation method (space)

  • Direct method

  • Stokes formula (reduced to a second class surface integral)

Surface integral

  • Area fraction over area (first class)

  • Definition

  • Nature

  • Calculation

  • Direct method

  • Parity

  • Symmetry

  • Area score for coordinates (second category)

  • Definition

  • Nature

  • Calculation

  • Direct method

  • Gauss’s formula (reduced to triple integral)

  • Gauss formula for complementary surfaces

Application of multiple integrals

Dispersion and rotation

Infinite series

Constant term series

  • Concepts and properties of series

  • Infinite series

  • Parts and series

  • Convergence, divergence

  • Criterion for convergence of positive series

  • Partial and series bounded

  • Comparative discriminant

  • Ratio discriminant

  • Root discriminant

  • Geometric series (isoperimetric series)

  • Interleaved series discriminant criterion

  • Leibniz discriminant criterion

  • Absolute convergence and properties

Power series

  • Function term series, convergence domain, function

  • Power series

  • Abel’s theorem

  • Power series properties

  • Quadratic operations

  • Analytical properties

  • Power series expansion of functions (Taylor series / McLaughlin series)

Fourier series

  • Fourier coefficients and Fourier series

  • Convergence of Fourier series (Direkley’s convergence theorem)

  • Expansion of a function with period 2l

  • Expansion on [-L. L

  • expansion of parity functions on [-L. L

  • expansion on [0. L] as sine or cosine

Differential equations

First order differential equation

  • Differential equation concepts

  • Definition

  • Order of differential equations

  • General and special solutions

  • Initial conditions

  • Several special classes of first-order differential equations and their solutions

  • Separability of variables

  • Simultaneous differential equations

  • Linear differential equations

  • Bernoulli’s equation

  • Fully differential equations

Second order and higher order

  • Linear differential equations

  • Linear chi-square differential equations of order n

  • Linearly correlated and linearly uncorrelated

  • Properties of differential equations

  • Superposition of solutions of linear equations of the chi-square

  • Generalized solution structure of quadratic linear equations

  • Structure of general solutions of non-sublinear equations

  • Principle of superposition

  • The method and formulas for solving linear chi-square equations with constant coefficients of the second order

  • The general solution method and formula for some special free term second-order linear chi-square equations with constant coefficients

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