Advanced Mathematics

Differentiation

Differential Calculus of Single-variable Functions

Function
The concept of functions
Function definition:

If for each number $x\in D$, the variable $x$ always corresponds to a certain $y$ according to certain rules, then $x$ is called a function of $y$, denoted as $y=f(x)$; x is often called the independent variable, $y$ the dependent variable, and $D$ domain.

Composite function:

Suppose the domain of $y=f(u)$ is $D_f, u=g(x)$, the domain is $D_g$, and the value domain is $R_{s}$; if $D_f\cap R_g\neq\phi, $, then the function $y=f[g(x)]$ is called the composite function of functions $y=f(u)$ and $u=g(x)$. Its domain is $\left\{x|x\in D_g, g(x)\in D_f\right\}$

Inverse function:

Let the domain of function $y=f(x)$ be ${D}$, and the range be $\underline{R}_{\underline{\nu}}$. If for any $y\in R_y$, there exists a unique and determined $x\in D$ such that $y=f(x)$, then it is denoted as $x=f^{-1}(y)$, and called the inverse of function $y=f(x)$

The properties of functions
Monotonicity
Parity:
Common odd functions: $\sin x,\tan x,\arcsin x,\arctan x,\ln\frac{1-x}{1+x},{\ln(x+\sqrt{1+x^2})},\frac{e^x-1}{e^x+1},f(-x)=-f(x)$
Common even functions: $x^2,|x|,\cos x,f(x)=f(-x)$
Periodicity
Boundedness
Limit
The nature of limits
Properties of the limit of sequences
Boundedness: $\text{ If the sequence is }\left\{x_n\right\}\text{ Convergence, then the sequence }\left\{x_n\right\}\text{ There must be boundaries }$
Number preservation;
Limit properties of functions
Local boundedness: if $\lim_{x\to x_0}f(x)$ exists, then $f(x)$ is in a punctured neighborhood of $x_0$
Number preservation
Function limit
Functions tend toward infinite values
Functions and infinity:

$\lim_{x\to\infty}f(x)=A$ iff, for every $\varepsilon>0$, there is an $X>0$ such that $|x|>X$ implies $|f(x)-A|<\varepsilon$.

The function tends toward negative infinity:

$\lim_{x\to-\infty}f(x)=A$ iff, for every $\varepsilon>0$, there is an $X>0$ such that $x<-X$ implies $|f(x)-A|<\varepsilon$.

The function tends toward positive infinity:

$\lim_{x\to+\infty}f(x)=A$ iff, for every $\varepsilon>0$, there is an $X>0$ such that $x>X$ implies $|f(x)-A|<\varepsilon$.

The function tends toward finite values
$\lim_{x\to x_0}f(x)=A$ iff, for every $\varepsilon>0$, there is a $\delta>0$ such that $0<|x-x_0|<\delta$ implies $|f(x)-A|<\varepsilon$.
Unilateral limit
left limit: $\lim_{x \to x_{0}^{-}}f(x) = A \Leftrightarrow \begin{cases} \hspace{1em} \forall \xi >0, \exists \delta>0, x_{0}-\delta <x<x_{0} \ \mbox{whenever}, \\ \hspace{1em} \lvert f(x) - A\rvert < \xi \\ \end{cases}$
Right limit: $\lim_{x \to x_{0}^{+}}f(x) = A \Leftrightarrow \begin{cases} \hspace{1em} \forall \xi >0, \exists \delta>0, x_{0}<x<x_{0}+\delta \ \mbox{whenever}, \\ \hspace{1em} \lvert f(x) - A\rvert < \xi \\ \end{cases}$
Relationship between limit and unilateral limit:

$\lim_{x\to x_0}f(x)=A\Leftrightarrow\lim_{x\to x_0^-}f(x)=\lim_{x\to x_0^+}f(x)=A$

Limit of the sequence
Definition of sequence limits:

$\lim_{n\to\infty}x_n=a$ iff, for every $\varepsilon>0$, there is an $N$ such that $n>N$ implies $|x_n-a|<\varepsilon$.

Criteria for the existence of sequence limits:

If there is $N, $ when $n>N$, $x_n\leq y_n\leq z_n$, $\lim_{n\to\infty}x_n=\lim_{n\to\infty}z_n=a, $ then $\lim_{n\to\infty}y_n=a$

Find the limit
Eight methods for finding limits
Infinity and infinitesimal
Infinitesimal
Definition of infinitesimal: If the function $f(x)$ has zero limit at $x\to x_0($ or $x\to\infty)$, then $f(x)$ is called an infinitesimal at $x\to x_0($ or $x\to\infty)$.
Comparison of infinitesimals:

-Iso-orderinfinitesimal: $\alpha(x)\text{ and }\beta(x)$ The result of division isconstant C(C is not equal to 0);

-Equivalentinfinitesimal: $\alpha(x)\text{ and }\beta(x)$ divide byconstant 1;

Higher-order infinitesimal: $\alpha(x)\text{ and }\beta(x)$ divide by 0; can be denoted as: $\alpha(x)=o(\beta(x))$
Low-order infinitesimals: $\alpha(x)\text{ and }\beta(x)$ divide by the opposite order is infinite;
$\text{ If }\lim\frac{\alpha (x)^{\color{red}{}}}{\left[\beta (x)\right]^{k}\color{red}}=C\neq 0,\text{ Said }$ α is the k-th order infinitesimal of β;
Properties of infinitesimals:

-Property 1:The sum of infinitesimalsis still infinitesimal;

-Property 2:The product of finiteinfinitesimals is still infinitesimal;

-Property 3:The product of infinitesimals andbounded quantitiesis infinitesimal;

Infinity
Definition of infinity:

If the function $f(x)$ infinite at $x\to x_0($ or $x\to\infty)$, then $f(x)$ is called an infinite quantity at $x\to x_0($ or $x\to\infty)$.

Properties of infinity:

-Property 1:The sum of finite positive infinity is infinity;

-Property 2:Theproductof finite infinity is still infinite;

-Property 3:The sum of infinite quantities and bounded variables remains infinitely large;

Continuous
Definition of continuity
Left continuous
Right continuous
Definition:

Let function $y=f(x)$ be defined in a neighborhood of point $x_{0}$. If when $x\to x_0$, the limit of function $y=f(x)$ exists and equals the value of function $f(x_0)$ at $x_{0}$, i.e., $\lim_{x\to x_0}f(x)=f(x_0)$, then function $y=f(x)$ is said to be continuous at point $x_{0}$.

The nature of continuity
Properties of continuity functions on closed intervals:
Boundedness: $\text{ If f(x) is continuous on [a,b], then f(x) is bounded on [a,b]. }$
Maximum value theorem: If $f(x)$ is continuous on $[a,b]$, then $f(x)$ must have a maximum and minimum value on $[a,b]$;
Intermediate Value Theorem: If $f(x)$ is continuity on $[a, b]$, and $f(a)\neq f(b)$, then for any number $\mathbf{C}, $ between $f(a)$ and $f(b)$ there is at least one $\xi\in(a, b), $ such that $f(\xi){=}C.$
Zero point theorem
The computational properties of continuity
Breakpoints
Definition of discontinuity points:
Several types of discontinuities
Derivative
Definition of derivatives
What is a derivative?
Left derivative and right derivative
The geometric meaning of the derivative
The rule for derivative differentiation
Differentiation rule for sum and product quotient
The rule for differentiating inverse functions
Derivative rule of composite functions
Derivative of commonly used conclusions
Logarithmic differentiation method
Higher-order derivatives
Definition of higher-order derivatives
Common formulas for finding higher-order derivatives
Implicit function
Parametric equations
Derivative application
Interpreting the monotonicity of functions
Judgment of convexity
Inflection points
Extrema and extrema of a function
Asymptote
Horizontal asymptote
Vertical asymptote
Differential
Definition of differential:
$f(x)$ is said to be differentiable at $x _0$ if $f(x_0+\Delta x)-f(x_0)=A\Delta x+o(\Delta x); $
Relationships among differentiable, differentiable, and continuity
Mean value theorem for differential calculation
Rolle's Theorem:
If three conditions are met:
1) $f$ continuous on $[a,b]$;
2)$f$ is differentiable within $(a, b)$;
3）f (a)=f (b)
So, it can be known: $\text{ then }\exists\xi\in(a,b)\text{ , to use }f^{\prime}(\xi)=0$
Derivation conclusion: There is a tangent line at a point parallel to the line connecting points ab -> Lagrange theorem;
Lagrange mean value theorem:
If the following conditions are met: 1) $f$ is continuously on $[a,b]$ 2) $f$ is differentiable within ($a,b)$;
Therefore, there exists $\xi\in(a,b)$ such that $f(b)-f(a)=f^{\prime}(\xi)(b-a)$;
Cauchy Mean Value Theorem:
If the following conditions are met:
1) $f, F$ on $[a, b]$ continuity $; $
2) $f, F$ is differentiable within $(a, b)$, and $\forall x\in(a, b), F^{\prime}(x)\neq0$
Therefore, there exists $\xi\in(a,b)$ such that $\frac{f(b)-f(a)}{F(b)-F(a)}=\frac{f^{\prime}(\xi)}{F^{\prime}(\xi)}$;

Multivariable Differential Calculus

Basic concepts of multivariable functions
Limits of multivariable functions
Continuity of multivariable functions
Partial derivative
Definition of partial derivatives
The geometric meaning of partial derivatives
Higher-order partial derivative
Total derivative
Definition of the total differential
A necessary condition for differentiability of multivariable functions
Sufficient condition for differentiability of multivariable functions
Multivariable function analysis
Judgments of differentiable, differentiable, continuous, and partially diverging continuous
The relationship among differentiable, differentiable, continuous, and partially differentiated continuity
Multivariable function differentiation
Multivariable Composite Function Differential
Differentiation rule for multivariable composite functions
Full-differential form invariance
Multivariable implicit function differential
Multivariable function implicit function definition
The existence theorem of implicit functions in multivariable functions
Extrema and extrema of multivariable functions
Definition of extremum of multivariable functions
The necessary condition for the existence of extremals in multivariable functions
Unconditional extremum
Conditional extremums
Lagrange multiplier method
Definition of conditional extremums

Integral Calculus

Integral Calculus of Single-Variable Functions

Indefinite integral
The concept of indefinite integrals
Definition of indefinite integrals:

An antiderivative of a function $f(x)$ is a function $F(x)$ whose derivative equals $f(x)$, that is, $F'(x)=f(x)$. The indefinite integral is the family $F(x)+C$.

Or: $\int f(x)dx=F(x)+C$

Existence of antiderivatives:
Basic properties of indefinite integrals
$(\int f(x)\mathrm{d}x)^{\prime}=f(x)$
$\mathrm{d}\int f(x)\mathrm{d}x=f(x)\mathrm{d}x$
Calculation of indefinite integrals
Basic formula
First type of substitution method
The second type of substitution method
Integration by parts
Definite integral
The concept of definite integrals
Definition of definite integrals:

$f(x)$ is bounded on $[a,b]$, arbitrarily insert a node on $[a,b]$, divide it into n small intervals $\Delta x_{1}\Delta x_{2}\cdots\Delta x_{n}$, and take any point $\xi_i$, resulting in: $\int_{a}^{b}f(x)\,dx=\lim_{\lambda\to0}\sum_{i=1}^{n}f(\xi_{i})\Delta x_{i}$.

Where: $\lambda=\max\{\Delta x_{1},\cdots,\Delta x_{n}\}$.
Fundamental theorems of calculus
Calculation of definite integrals
Newton-Leibniz formula
Substitution of definite integrals
Integral by Parts of Definite Integrals
Properties of definite integrals
Limit the points
Method 1: Formula calculation
Method 2: Extract x
Method 3: Substitution method
Improper integral
Definition of improper integrals
Two types of improper integrals
Improper integrals over an infinite interval
Improper integrals on infinite functions
Determine the divergence of improper integrals
Method 1: Definition method
Method 2: Comparative Discrimination
Method 3: P integration
Application of definite integrals
Area of a plane figure
The volume of a rotating body
The arc length of a plane curve
Lateral area of a rotating body

Multivariable Function Integral Calculus

Multiintegrals
Double integral
Definition of double integrals
Properties of double integrals
Calculation of double integrals
Double integral calculation based on the Cartesian coordinate system
Double integral calculation based on polar coordinate systems
Utilizing parity and symmetry
Triple integral
Linear area division
Line integration
By area
Points application

Differential Equations

First-Order Differential Equations

Separable variables
First-order homogeneous equation
First-order linear equation

Second-Order Differential Equations

Second-order homogeneous differential equations with constant coefficients
Second-order constant coefficient nonhomogeneous differential equation

Higher-Order Differential Equations

Reduced-order linear differential equations
Concept of higher-order differential equations

Infinite Series

Series of Constant Terms

Basic concepts
Definition of a series of constant terms
Definition of convergence of series of constant terms: $\lim_{n\to+\infty}S_{n}=\sum_{n=1}^\infty u_n$
Basic properties of series
Constant-sign series
Positive term series
Convergence of a positive-term series: $\sum_{n=1}^\infty u_n\text{ converges }\Leftrightarrow \{s_n\}\text{ is bounded above}$
Comparison test
Limit comparison test
Ratio method
Root test
Integral test
Sign-changing series
Alternating series
Definition of alternating series: $\sum_{n=1}^\infty(-1)^{n-1}u_n,u_n>0$
Leibniz test
General series
Concepts of absolute convergence and conditional convergence
Convergence tests for general series

Power series

Basic concepts of power series
Definition of a power series
$\sum_{n=0}^\infty a_n(x-x_0)^n=a_0+a_1(x-x_0)+\cdots+a_n(x-x_0)^n+\cdots$
Convergence of power series
The concepts of convergence points and divergence points
Abel's theorem
Convergence interval
Convergence domain
Radius of convergence
Radius of Convergence Determination Method One: Limit Ratio
Radius of Convergence Determination Method 2: Based on radicals
Operations on power series
Properties of rational operations
Analytical nature
The function is expanded into a power series
Basic concepts
The power series expansion of a function
Convergence of Taylor's series
The function expansion is a power series
Direct expansion method
Indirect development method
Common Development

Fourier series

Basic concepts of Fourier series
Fourier coefficient
Fourier series
Convergence
Dirichlet theorem
The function expansion is a Fourier series
Expansion of periodic functions: special cases
Expansion of periodic functions: general case

Advanced Mathematics - Mind Map