Elementary mathematics studies constants and advanced mathematics studies variables.
Advanced mathematics (also known as calculus, which is the general name of several courses) is an important basic subject in science and engineering colleges. As a science, advanced mathematics has its inherent characteristics, namely, high abstraction, strict logic and wide application. Abstract is the most basic and remarkable feature of mathematics-high abstraction and unity, which can profoundly reveal its essential laws and make it more widely used. Strict logic means that in the induction and arrangement of mathematical theory, whether it is concept and expression, or judgment and reasoning, we must use the rules of logic and follow the laws of thinking. Therefore, mathematics is also a way of thinking, and the process of learning mathematics is the process of thinking training. The progress of human society is inseparable from the wide application of mathematics. Especially in modern times, the appearance and popularization of electronic computers have broadened the application field of mathematics. Modern mathematics is becoming a powerful driving force for the development of science and technology, and it has also penetrated into the field of social sciences extensively and deeply. Therefore, it is very important for us to learn advanced mathematics well. However, many students are confused about how to learn this course well. If you want to learn advanced mathematics well, you must at least do the following four things:
First, understand the concept. There are many concepts in mathematics. Concepts reflect the essence of things. Only by figuring out how it is defined and what its essence is can we really understand a concept.
Secondly, master the theorem. Theorem is a correct proposition, which is divided into two parts: condition and conclusion. In addition to mastering its conditions and conclusions, we should also understand its scope of application and be targeted.
Third, do some exercises on the basis of understanding the examples. Especially remind learners that the examples in the textbook are very typical, which is helpful to understand concepts and master theorems. Pay attention to the characteristics and solutions of different examples, and do appropriate exercises on the basis of understanding examples. When writing a topic, you should be good at summing up-not only the methods, but also the mistakes. You will gain something after doing this, so you can draw inferences from others.
Fourth, clear the context. We should have an overall grasp of the knowledge we have learned and summarize the knowledge system in time, which will not only deepen our understanding of knowledge, but also help us to further study.
Advanced mathematics includes calculus and solid analytic geometry, series and ordinary differential equations. Calculus is the most systematic and widely used in other courses. The theory of calculus was completed by Newton and Leibniz. (Of course, calculus has been applied before them, but it is not systematic enough. ) The basic concepts of calculus and extreme calculus are difficult to understand.
Advanced mathematics is divided into several parts:
First, the function limit continuity.
Second, the differential calculus of unary function
3. Integral calculus of unary function
4. Vector Algebra and Spatial Analytic Geometry
Verb (abbreviation of verb) Differential calculus of multivariate functions
Six, multivariate function integral calculus
Seven, infinite series
Eight, ordinary differential equations
The high figures mainly include
First, function and limit are divided into
Constants and variables
function
Simple behavior of function
inverse function
Elementary function
Sequence limit
functional limit
Infinite quantity and infinite quantity
Comparison of infinitesimal quantities
Functional continuity
Properties of continuous functions and functional continuity of elementary functions
Second, derivative and differential
The concept of derivative
Derivation rule of sum and difference of functions
Derivation rule of product sum quotient of functions
Derivation rule of compound function
Inverse function derivative rule
higher derivative
Implicit function and its derivative rule
Differential of function
Third, the application of derivatives.
Differential mean value theorem
Uncertain problem
Determination of monotonicity of function
Extreme value of function and its solution
Maximum and minimum values of functions and their applications
Concave direction and inflection point of curve
Fourth, indefinite integral
Concept and properties of indefinite integral
The method of finding indefinite integral
Examples of several special function integrals
Five, definite integral and its application
The concept of definite integral
Integral formula of calculus
Partial substitution integral method for definite integral
Generalized integral
Six, spatial analytic geometry
Space rectangular coordinate system
Direction cosine and direction number
Plane and spatial straight line
Surfaces and space curves
Eight, multivariate function differential calculus
The concept of multivariate function
Limit and continuity of binary function
partial derivative
complete differential
Derivation method of multivariate composite function
Extreme value of multivariate function
Nine, multivariate function integral calculus
The Concept and Properties of Double Integral
Calculation method of double integral
The concept of triple integral and its calculation method
Ordinary differential equation
Basic concepts of differential equations
Differential equation and homogeneous equation of separable variables
linear differential equation
Degradable higher order equation
Structure of solutions of linear differential equations
Solution of second-order homogeneous linear equation with constant coefficients
Solution of second-order non-homogeneous linear equation with constant coefficients
XI。 infinite series
The concept of derivative
Before learning the concept of number, let's discuss the instantaneous speed of linear motion with variable speed in physics.
For example, suppose a particle moves along the X axis, and its position X is a function of time T, and y=f(x), and find the instantaneous velocity of the particle at t0?
We know that when the time increases from t0 to δt, the position of particles increases.
This is the displacement of the particle in the time period △ t, so the average velocity of the particle during this period is;
If the particle moves at a uniform speed, this is the instantaneous speed at t0; If the particle moves along a non-uniform straight line, this is not the instantaneous velocity at t0.
We think that when the time period △t is infinitely close to 0, the average velocity will be infinitely close to the instantaneous velocity of the particle t0.
Namely: instantaneous velocity of particles at t0 =
To this end, the definition of derivative is produced as follows:
Definition of derivative
Let the function y=f(x) be defined in the neighborhood of point x0. When the independent variable x has an increment △ x at x0 (x+△ x is also in this neighborhood), correspondingly,
Function has increment
If the ratio of △y to △x has a limit when △x→0, it is called the derivative of y=f(x) at x0.
Remember as follows:
It can also be recorded as:
The function f(x) has a derivative at point x0. The function f(x) is simply derivable at point x0, otherwise it is not derivable.
If the function f(x) is differentiable at every point in the interval (a, b), it is said that the function f(x) is differentiable in the interval (a, b). At this time, the function y=f(x) is different for this region.
Each definite value of x in (a, b) corresponds to a definite derivative, which constitutes a new function.
Let's call this function the derivative of the original function y=f(x).
Note: the derivative is also the limit of the difference quotient.
Left and right derivatives
We have the concept of left and right limit, and the derivative is the limit of difference quotient, so we can give the concept of left and right derivative.
Ruoji
Existence, we call it the left derivative of the function y=f(x) at x=x0.
Ruoji
Existence, we call it the right derivative of the function y=f(x) at x=x0.
Introduction to advanced mathematics
Elementary mathematics studies constants and advanced mathematics studies variables.
Advanced mathematics (also known as calculus, which is the general name of several courses) is an important basic subject in science and engineering colleges. As a science, advanced mathematics has its inherent characteristics, namely, high abstraction, strict logic and wide application. Abstract is the most basic and remarkable feature of mathematics-high abstraction and unity, which can profoundly reveal its essential laws and make it more widely used. Strict logic means that in the induction and arrangement of mathematical theory, whether it is concept and expression, or judgment and reasoning, we must use the rules of logic and follow the laws of thinking. Therefore, mathematics is also a way of thinking, and the process of learning mathematics is the process of thinking training. The progress of human society is inseparable from the wide application of mathematics. Especially in modern times, the appearance and popularization of electronic computers have broadened the application field of mathematics. Modern mathematics is becoming a powerful driving force for the development of science and technology, and it has also penetrated into the field of social sciences extensively and deeply. Therefore, it is very important for us to learn advanced mathematics well. However, many students are confused about how to learn this course well. If you want to learn advanced mathematics well, you must at least do the following four things:
First, understand the concept. There are many concepts in mathematics. Concepts reflect the essence of things. Only by figuring out how it is defined and what its essence is can we really understand a concept.
Secondly, master the theorem. Theorem is a correct proposition, which is divided into two parts: condition and conclusion. In addition to mastering its conditions and conclusions, we should also understand its scope of application and be targeted.
Third, do some exercises on the basis of understanding the examples. Especially remind learners that the examples in the textbook are very typical, which is helpful to understand concepts and master theorems. Pay attention to the characteristics and solutions of different examples, and do appropriate exercises on the basis of understanding examples. When writing a topic, you should be good at summing up-not only the methods, but also the mistakes. You will gain something after doing this, so you can draw inferences from others.
Fourth, clear the context. We should have an overall grasp of the knowledge we have learned and summarize the knowledge system in time, which will not only deepen our understanding of knowledge, but also help us to further study.
Advanced mathematics includes calculus and solid analytic geometry, series and ordinary differential equations. Calculus is the most systematic and widely used in other courses. The theory of calculus was completed by Newton and Leibniz. (Of course, calculus has been applied before them, but it is not systematic enough. ) The basic concepts of calculus and extreme calculus are difficult to understand.
Advanced mathematics is divided into several parts:
First, the function limit continuity.
Second, the differential calculus of unary function
3. Integral calculus of unary function
4. Vector Algebra and Spatial Analytic Geometry
Verb (abbreviation of verb) Differential calculus of multivariate functions
Six, multivariate function integral calculus
Seven, infinite series
Eight, ordinary differential equations
The high figures mainly include
First, function and limit are divided into
Constants and variables
function
Simple behavior of function
inverse function
Elementary function
Sequence limit
functional limit
Infinite quantity and infinite quantity
Comparison of infinitesimal quantities
Functional continuity
Properties of continuous functions and functional continuity of elementary functions
Second, derivative and differential
The concept of derivative
Derivation rule of sum and difference of functions
Derivation rule of product sum quotient of functions
Derivation rule of compound function
Inverse function derivation rule
higher derivative
Implicit function and its derivative rule
Differential of function
Third, the application of derivatives.
Differential mean value theorem
Uncertain problem
Determination of monotonicity of function
Extreme value of function and its solution
Maximum and minimum values of functions and their applications
Concave direction and inflection point of curve
Fourth, indefinite integral
Concept and properties of indefinite integral
The method of finding indefinite integral
Examples of several special function integrals
Five, definite integral and its application
The concept of definite integral
Integral formula of calculus
Partial substitution integral method for definite integral
Generalized integral
Six, spatial analytic geometry
Space rectangular coordinate system
Direction cosine and direction number
Plane and spatial straight line
Surfaces and space curves
Eight, multivariate function differential calculus
The concept of multivariate function
Limit and continuity of binary function
partial derivative
complete differential
Derivation method of multivariate composite function
Extreme value of multivariate function
Nine, multivariate function integral calculus
The Concept and Properties of Double Integral
Calculation method of double integral
The concept of triple integral and its calculation method
Ordinary differential equation
Basic concepts of differential equations
Differential equation and homogeneous equation of separable variables
linear differential equation
Degradable higher order equation
Structure of solutions of linear differential equations
Solution of second-order homogeneous linear equation with constant coefficients
Solution of second-order non-homogeneous linear equation with constant coefficients
XI。 infinite series
The concept of derivative
Before learning the concept of number, let's discuss the instantaneous speed of linear motion with variable speed in physics.
For example, suppose a particle moves along the X axis, and its position X is a function of time T, and y=f(x), and find the instantaneous velocity of the particle at t0?
We know that when the time increases from t0 to δt, the position of particles increases.
This is the displacement of the particle in the time period △ t, so the average velocity of the particle during this period is;
If the particle moves at a uniform speed, this is the instantaneous speed at t0; If the particle moves along a non-uniform straight line, this is not the instantaneous velocity at t0.
We think that when the time period △t is infinitely close to 0, the average velocity will be infinitely close to the instantaneous velocity of the particle t0.
Namely: instantaneous velocity of particles at t0 =
To this end, the definition of derivative is produced as follows:
Definition of derivative
Let the function y=f(x) be defined in the neighborhood of point x0. When the independent variable x has an increment △ x at x0 (x+△ x is also in this neighborhood), correspondingly,
Function has increment
If the ratio of △y to △x has a limit when △x→0, it is called the derivative of y=f(x) at x0.
Remember as follows:
It can also be recorded as:
The function f(x) has a derivative at point x0. The function f(x) is simply derivable at point x0, otherwise it is not derivable.
If the function f(x) is differentiable at every point in the interval (a, b), it is said that the function f(x) is differentiable in the interval (a, b). At this time, the function y=f(x) is different for this region.
Each definite value of x in (a, b) corresponds to a definite derivative, which constitutes a new function.
Let's call this function the derivative of the original function y=f(x).
Note: the derivative is also the limit of the difference quotient.
Left and right derivatives
We have the concept of left and right limit, and the derivative is the limit of difference quotient, so we can give the concept of left and right derivative.
Ruoji
Existence, we call it the left derivative of the function y=f(x) at x=x0.
Ruoji
Existence, we call it the right derivative of the function y=f(x) at x=x0.
Note: The existence and equality of the left and right derivatives of the function y=f(x) at x0 is a necessary and sufficient condition for the function y=f(x) to be derivable at x0.
Hope to adopt thank you.