1、微积分课程教学基本要求(讨论稿) 2003,8（Basic requirements of Calculus Teaching (discussion draft) 2003,8）Reading is like eating; eating well is the soul; eating ill is worse than eating. - Zhang XuechengBasic requirements of Calculus Teaching (discussion draft) 2003,8(1) basic requirements of calculus (I) teaching
2、 (3 hours / week, 48 hours);(a) explanation“Calculus“ (I) called “intuitive calculus“, its characteristic is to limit to intuitive definition to understand, difficult across the limit theory of proof, the most basic calculus as soon as possible into the main content: a function of differential and i
3、ntegral and simple differential equations. This allows students to easily grasp the actual entry first the application of the basic calculus with extensive content, outstanding processing method of mathematics and physics concepts, Newton type of geometric combination, not rigidly adhere to the stri
4、ct mathematical proof, the emphasis on the analysis of basic calculation skills and application of calculus method and cultivate the ability to solve practical problems.(1) the limit concept of this part is mainly defined as “infinite tendency“. The exact definition of limit is introduced, and the l
5、imit proof is not required, but the use of limit number is required.(2) the boundedness of the continuous function on the closed interval, and the conclusion of the minimum value and the betweenness value will be applied(3) this section calls for outstanding calculations and applications.The student
6、s from middle school to university has great changes in the learning method, to adapt to this process, suggest that learning methods and guidance of reading textbooks and reference books for the students in the class teaching, there should be appropriate examples to explain.(two) content1. function:
7、Function definitions, elementary elementary functions, implicit functions, functions expressed by parametric equations, composite functions.Several main properties of a function are boundedness, parity, monotonicity, periodicity, convexity and concavity.2 limit:Only discuss the limit of function, em
8、phasize “infinity approach“, do not require the proof problem defined by “, just use “thought“ to explain the limit of number and boundedness of the limitThe operational properties of limit, two important limits, infinite small quantity, infinite large quantity, use limit property, equivalent infini
9、tesimal, high order infinitesimal computing limit.3. continuous:The concepts of continuity and discontinuity (not consistent convergence), the properties of closed interval continuous functions4. derivatives and differentialsThe concept of derivative and differential, geometric meaningDerivative and
10、 differential calculus: basic derivatives, differential formulas, four operations, compound functions, chain laws, parametric equations, derivatives, implicit functions, derivatives, derivatives of higher order, Leibniz formula5. differential mean value theorem and derivative applicationsThe proof a
11、nd application of three differential mean value theoremsLHospital law, Taylor formula, function at the Taylor formula, using Taylor formula for the limit of the functionStudy on the function of the state: the extremum, convexity, inflection point, curve; function image is discussed and omitted paint
12、ing.The extreme value and the most value problem of a unary function.6. integralThe concepts and properties of primitive functions and indefinite integrals; calculations of indefinite integrals: together with differential equations, variable substitution, partial integrals, and ideas and conclusions
13、 on the integration of rational functions7. definite integral concept and its basic properties, variable limit integral and calculus, basic theorem, Newton-Leibniz formulaThe calculation of definite integral: take together differential, variable substitution, partial integral, and understand the int
14、egral that can not be accumulated into elementary functions.Application of definite integralGeometric applications: area, mean, rotation volume, curve arc length, rotation of the side of the bodyPhysical applications: mass centers, moments of inertia, gravity, doing work8. simple differential equati
15、onsThe practical background and basic concepts of differential equationsElementary solutions of differential equations: separation of variables, homogeneous equations, first order linear equations, constant variation method, Bernoulli equation, reduced order two order equations:(2) basic requirement
16、s of calculus (I) teaching (4 hours / week, 64 hours);(a) explanationThis is a course for information and science.(two) contentThe basic content is the same as that of calculus (I) (3 hours / week, 48 hours), and the following contents are added:Introduction to the existence and uniqueness of soluti
17、ons of 1. differential equations.The structure of high order linear equations, constant variation Faqiute solution.The solution of homogeneous higher order linear equation with constant coefficients and the method of comparison coefficients for non-homogeneous higher order linear equation with const
18、ant coefficientsApplications of differential equations2. in addition, the content and application in front can be a little deeper and a little more appropriate.(3) basic requirements of calculus (II) teaching (3 hours / week, 48 hours);(a) explanation“Calculus“ (II) called “rational calculus“, its c
19、haracteristic is discussed on the strict mathematics theory is based on the limit function, integrability and series etc., mathematical rational thinking and logical reasoning is a rigorous training to the students, to strengthen students mathematics accomplishment.This course serves as a basic cour
20、se in mathematical thinking and methodology, requiring simultaneous mastery of basic content,Let the students understand as much as possible about some basic ideas for dealing with continuous models.(two) contentThe extensions of 1. number systems, the bounds and the supremum of the set of numbers,
21、and the existence theorem of the supremum.Note: the main concept of clarifying the set of real numbers is bounded and unbounded, given the supremum definition, admit supremum theorem.2. limit and continuity of functionSequence limit: concepts, properties, monotone bounded, bounded theorem, forced th
22、eorem, bounded sequence, convergent subsequence, Cauchy criterion.A typical example of a concept, property, or limit.The proof of the concept of uniformly continuous concept and continuous function on bounded closed interval.Note: strict limit proof should be emphasized, and methods of dealing with
23、problems with limit thought should be strengthened.From continuous to consistent continuous, should be a relatively big jump, if can handle, meaning is not just understand a concept. It is at least able to make students understand the nature of the point and the nature of the whole, which are two co
24、ncepts.3. definite integralThe concept of definite integral: definition and necessary condition.Necessary and sufficient condition of integrability: sufficient and necessary condition and common integrable function class.The proof of the property of definite integral:The concept and property of gene
25、ralized integral and the convergence rule of two kinds of generalized integral.4. number seriesThe basic concepts and properties of series of numbers;The positive series and its comparison, the convergence method, the laws of Da, the two law of law, etc.;Any series of convergence methods and propert
26、ies of the alternating series Leibniz law, absolute convergence and conditional convergence.5. series of function termsThe series of functional terms: the concept of uniform convergence and the analytic properties of the series of function terms;Power series: the concept of convergence radius, and t
27、he analytic property of power series;The series of functions at Taylor, the direct and indirect methods of generating power seriesFourier series: the orthogonality of functions, the concept of orthogonal functions, the orthogonality of trigonometric functions, the extension of functions into Fourier
28、 series, the convergence theorem. The mean convergence of Fourier seriesNote: add the average convergence, at least in the following points to make students gain:1., we can contact the concepts of norms in linear space, and associate the intuition with the thought of pumping.2., exposure to the same
29、 thing can be judged by different standards.3., realize how to prove (solve) a problem with what you have learned.(4) basic requirements of calculus (II) teaching (2 hours / week, 32 hours);(a) explanationThis is a course for information and science.(two) contentThe basic content and the calculus (I
30、I) (3 hours / week, 48 hours) content, the content of the appropriate adjustment. With the preceding calculus (I) (4 hours / week, 64 hours) comprehensive consideration.(5) basic requirements of calculus (III) teaching (4 hours / week, 64 hours);(a) explanation“Calculus“ (III) including multivariabl
31、e calculus and calculus of further application. The characteristics of paying more attention to broaden the knowledge and knowledge of modern mathematics, introducing the interface, while strengthening the cultivation of mathematics application consciousness and ability.(two) content(1) differential
32、 calculus of multivariate functions(1.) distance and convergence, and the open neighborhood, connected sets and area. The limit of function of many variables, the definition of continuous functionAnd property;2. partial derivatives and (all) differentials, higher order partial derivatives;3. compoun
33、d function differential method, directional derivative and gradient;* 4. differential mapping, Jacobian matrix *;5. implicit function differential method, implicit number differential method determined by equation;6*. consists of equationsImplicit number differential method;7. differential calculus
34、applications (1):The tangent vector of a space curve, the normal vector and tangent plane of a space surfaceReading is like eating; eating well is the soul; eating ill is worse than eating. - Zhang Xuecheng* moving markers: curvature and torsion of curves;8. applications of differential calculus (2)
35、: extreme value and conditional extremum.(2) re integration1. definition and properties of double integral;2. calculation of double integral:As the repeated integral double integrals in Cartesian coordinates and polar coordinates;3. double integral variable substitution;4. calculate three integrals
36、in Cartesian coordinate system, cylindrical coordinate system and spherical cylindrical coordinate system;5. area of the surface, the area Infinitesimal in the Cartesian coordinate system and the area infinitesimal under the parameter equation.(3) line, area and vector functionsThe concept of 1. vec
37、tor field, the integration of the first and second type curves, the Green formula, the integral of the plane curve and the path independent condition;Concept and calculation of surface integral of type 2. and second;3. Gauss formula in Stokes formula;4. vector field preliminary:The number of gradien
38、t field, vector field curl and divergence, conservative field, irrotational field.(4) reference integral1., the concept of integral with reference, the basic nature;ThreeAnalytic properties of functions represented by integrals: continuity, differentiability and integrability;3. - function and funct
39、ion.(5) differential equations1. basic concepts of differential equations, the existence and uniqueness theorem (not);2. structure of solutions of higher order linear equations, homogeneous and non-homogeneous constant coefficients, higher order linear equations, applications;3. linear differential
40、equations with constant coefficients are solved by eigenvalues and eigenvectorsNote: for the class of calculus (I) (4 hours / week, 64 hours), the differential equation is changed into:(5) differential equations1. linear differential equations with constant coefficients are solved by eigenvalues and
41、 eigenvectors:Structure of solution * constant variation method;2. concept and meaning of stability, stability of solutions of linear differential equations.You can also increase the number of selected candidates.The basic requirements of College Mathematics Series TeachingOneOneBasic requirements of calculusReading is like eating; eating well is the soul; eating ill is worse than eating. - Zhang Xuecheng
