mathematical analysis – Serendipities

Mathematical analysis -1

Calculus and mathematical analysis

Derivatives, chain rule; maxima and minima, Lagrange multipliers; line and surface integrals of scalar and vector functions; Gauss’, Green’s and Stokes’ theorems. Sequences and series, Cauchy sequences, uniform convergence and its relation to derivatives and integrals; power series, radius of convergence, convergence of improper integrals. Inverse and implicit function theorems and applications; the derivative as a linear map; existence and uniqueness theorems for solutions of ordinary differential equations, explicit solutions of simple equations.; elementary Fourier series.

傅立叶分析原理及题解

1 , Symbolic logic, collections, and a collection of elementary operations, collection of the Cartesian product, function and mapping, collection of potential, equivalent of the collection, and axiomatic set theory.

2 。 Countable sets, equivalence and its mounting is not a subset of the collection of Cantor theorem. Countability of the set of rational numbers. The incalculability of the continuum.

3 。 The natural numbers, axioms of natural numbers and operations, the principle of mathematical induction.

4 。 Collection can be divided into rational lemma, set partition lemmas.

11 , Irrational and rational numbers.

2 , Axioms of the real number system, the decimal representation of a real number, upper and lower bounds, the natural numbers, the set of rational numbers, irrational numbers set, mathematical induction,Archimedes principle, number lines, real number q in binary representation, Dedekindsegmentation. Existence of bounded sets.

3 , Close the nested intervals theorem, finite covering theorem, the theorem of limit points .

6 。 Series, and its properties of infinite and infinitesimal sequence,Bernoullis inequalities and Newton binomial.

7 。 Number sequence limit definition and properties, a convergent sequence with their arithmetical properties, limits of arithmetic,Stolz theorem. Squeeze theory,Cauchy columns,

8 。 Method for algebraic equations iteration.

9 。 Complex convergent sequence of Toeplitz transformToeplitz theorem, limit of sequence of Cauchy theorem, the sequence of Cauchy andCauchy Guidelines,Abel theorem.

10 。 Limit definition of monotone sequences,Weierstrass theorems.

4 , Column limit Natural base e 、 Bolzano-Weierstrass Theorem, upper and lower limit of the sequence.

5 , Heine due principle, limits of arithmetic and the filter limits, series, convergence Cauchy criterion.

5 。 Principle of the uniqueness of the real numbers, the completeness of the real numbers.

6 , The limit of function Cauchy Definition of function at a point The definition of continuity, Infinitesimal function Infinity and infinitesimals and their order. Limitation of function limit, intermittent, continuous function properties, limiting nature of monotone function, continuous function of arithmetic operations. Composite functions and monotone functions limits and continuity of the trigonometric and exponential functions. Monotonic function continuous, monotonic function of discontinuity. The continuity of the function and its inverse function. The inverse function of the continuity and continuity of elementary function,Kepler equation of continuity.

Function limit Cauchy Guidelines. Intermediate value theorem, maximum value theorem, uniform continuity and theCantor-Heinetheorem and the limit of function of Cauchy and Heine definition of equivalence.

22 。 Interval continuous function intermediate value on the Cauchy theorem.

23 。 Uniform continuity for continuous functions on the interval of the Cantor theorem.

24 。 Khinchin principle of induction, connected set on a number line, continuous function of connectivity.

7 , With filters based on Heine definition of function limit promotion, some important limits. The upper and lower limits of the function.

8 , Calculus physics background, differential and the definition of the derivative, a differentiable function, derivative and differential geometric meaning, notion of differentiability of the function and its derivative, derivative calculation,derivatives of the sum, difference, product, commercial,Leibnitz formula. Higher order derivatives.

25 。 Contact the continuity and differentiability of functions, inverse functions and the derivative of composite functions,Kepler equation differentiability.

9 , Increasing function Darboux Theorem Fermat Theorem, Rolle Theorem, Rollet Theorem Increment theorem, The mean value theorem for derivatives. L ‘ Hospital rule,l ‘ Hopital first formula of Peano remainder of Taylor formula, L ‘ Hopital second formula. Roth theorem,Schlomilch-Routh more than the Taylor formula, Lagrange remainder term Taylor formula of elementary functions by Taylor formula. Lagrange remainder term with Cauchy remainder. Cauchy theorem of Lagrange theorem.

10 , Constant, monotonous and strictly monotonic function of judgement criteria. The extremum of function monotonicity condition, function point, judgment and necessary condition for extreme value of function. Young inequality,Holder inequality,Minkowskiinequalities, convex function, function of convexity, singularities, asymptotic lines,Jensen inequality , A function mapping

11 , Some examples of using differential calculus to study natural sciences.

12 , Primitive function and indefinite integral, primitive function calculation method, elliptic integrals.

Mathematical analysis -2

1 , Integrating physical and geometric background,Riemann integral definition,Riemann integrals,Riemann integrable functions, bounded Riemann Points. Integrable function space,

5 。 Riemann integrable function (continuous, monotonous limit, limit point continuous on the interval).

Lebesgue Theorem, Riemann Additivity of integration the integration interval, and omega And, integrability of functions on the interval RiemannGuidelines.

3 。 Function Riemann integrability of the equivalence of the three criteria.

4 。 Segmentation of the range judging function is Riemann integrable.

Estimation of integral mean value theorem for integrals and some important integral inequality.

6 。 Definite integral properties (linear, integrability of modules, functions and product, norm inequalities for the integration of non-negative integral of a function, monotonicity of the integral).

7 。 Integrability theorems for complex functions, integral as a function of the upper (lower) limit and this theory of the continuity and differentiability of a function.

8 。 Indefinite integrals,Newton-Leibnitz equation,Euler and Abel summation,Stirling formula.

2 , Maximum points, andNewton-Leibniz , definite integral formula of integration by parts and variable substitution,variable replaces the formula with the definite integral, integration by parts.

10 。 The first and the second integral mean value formula for cases prove integral mean value theorem for smooth functions.

Integral form Taylor Formula. Score more than Talyor formula, area principle, progressive formula for calculating integrals, parabola interpolation. One application of integral calculus.

3 , Definition and basic properties of generalized integral of generalized integral and generalized integral variable substitution and integration by parts formula, convergence of generalized integral methods,

13 。 Convergence of improper integral Cauchy criterion and the second order condition.

14 。 Absolute and conditional convergence of improper integrals.

There is more than one singularity of principal value improper integral and generalized integral.

4 , Metric spaces and Euclidean space, As a metric space R^n 、 R^n The open sets and closed sets, R^n Compact set, in Space caused by tightening set of judgments. R^n norm, and Euclid space R^n.

20 。 Space series, convergence of the lemma,

21 。 Metric space compactness,n- compactness of the dimensions on a cube.

22 。 On the function of the (lower) defines and limits the existence, continuity of functions on Compact sets, continuous function on a compact set of mean value theorem.

23 。 Uniform convergence theorem for continuous functions on Compact sets.

5 , Euler theorem, topological equivalence andEuclid space mapped in continuity, with the embryo, the classification theorem of closed surfaces, topological invariants.

15 。 Simply issue the arc length of the curve, differentiability of the arc length of the curve.

16 。 Boundary is rectifiable curve graphics of the separability principle.

17 。 Jordan curved trapezoidal judge integrability.

6 , Topological space And the definition of metric space, a topological space The concept of topological space, Open set andbasic properties of open sets, closed sets, borders, examples of topological radicals, the convergence of the series,Hausdorff spaces, topology, A subspace of a topological space. Metric spaces and topological space direct product, second-countable space.

24 。 The concept of limit mapping in topological space, limit the properties of topological spaces,

7 , Continuous mapping, complex maps. Continuous mapping with the same germ, limit of compound mapping,Hausdorff spaces and normed space maps to the limit. Peano curve, theTietze extension theorem and the compactness of topological space,Heine-Boreltheorem, the compact nature of the space,Bolzano-Weierstrass Properties, Lebesgue Lemma, locally compact spaces, LindelofTheorem.

25 。 Continuous mapping in topological spaces, for example, the upper limit and lower limit of continuous functions, continuity of the function.

26 。 Connected sets of a topological space and connectivity maps. Topological spaces caused by tightening of criteria of properties of compact sets, semi-continuous function,Hausdorff space compactness and separation theorems.

8 , Product space, the product topology,Tychonoff product theorem, connected topological spaces and quotient topological,Alexandroff theorem, bond topology, complete metric spaces and metric space completion, close the ball set lemmas, first class and second class sets,Baire Class mapping theorem, a topological space limit, topological maps on the space of continuous and continuous, repeated limit and double limit, contraction mapping principle.

9 , Normed linear spaces,Banach space andEuclid space,Hilbert spaces, linear operators, operator norm, the continuous operator on spaces, normed spaces of differentiable maps, maps of differential and derivative,

27 。 Map of differential, partial derivatives and differentiability of functions necessary, second-order conditions for differentiability. 28。 Differentiability theorem for complex functions, first-order differential invariantJacobian matrix.

Map of differential Jacobi Matrix, functions, continuity and differentiability, differential arithmetic, complex mapping of differentiation, inverse differential, mapping the partial derivative and gradient, directional derivative.

10 , Increment theorem, in a continuously differentiable map, higher order differential mean value theorem, mappings and partial derivatives, differential operation and

29 。 Mixed derivatives of Schwarz and Young theorem.

Map Taylor Formulas, Peano Lagrange Remainder of function of several variables Taylor Theorem. Mapping local maxima,, tangent plane, normal vectors, tangent vectors.

11 , Implicit mapping theorem, the implicit function theorem, mapping of the implicit function theorem. A diffeomorphism, the inverse mapping theorem, rank theorem, function, relevance,Morse lemmas.

12 , R^n k Koreko manifolds, tangent space definition, necessary conditions of extreme value of multivariate function, conditions are extremum values, extremum of second order conditions. Lagrange multiplier method.

Mathematical analysis -3

1 , Series and more, necessary conditions for convergence, Convergence and divergence, the absolute convergence of numerical series, non-negative number a series converges if and only if, the comparison method, Weierstrass The comparison method, The series converges Cauchy Sentenced to According to series of judgments. D ‘ Alembert discriminant method andGaussdiscriminance, andRabbe discriminant method,Kummer discriminant method,Bertrand Judging method and generalized integralCauchy-Maclaurin integral criterion.

2 , Several series of Leibniz series test , andAbel discriminant method andDirichlet discriminant method, the rearrangement of the series, Absolute convergence theorem of the rearrangement of the series, conditionally convergent series of Riemann theorem,Mertenstheorem,

Double series, double series and tired relationships between secondary number, Two absolute convergence theorem of series of products, Dual rearrangements of absolutely convergent series, Two product series Mertens Theorem. Absolute convergence theorem of multiple series. Infinite product, and the nature of the infinite product, necessary condition of infinite product converges, infinite product of absolute convergence,gamma function and Euler function infinite product definitions,Euler Formula. Gammafunction equation.

3 , Sets, variables with parameters of convergence of function sequence function and the continuity theorem for function series and, convergence and uniform convergence, uniform convergence of function series of Cauchy convergence criterion, plural domains with complex series, power series, Cauchy-Hadamard formula,Abel theorem and function analysis of the power series representation, power series, uniform convergence of function series Weierstrass order high criteria,Abel And theDirichletand theAbel-Dirichlet method.

4 , Double limit Exchange conditions, function limit the continuity of function, power series and function continuity and uniform convergence of the series of non-negative continuous functions on the interval criterion of uniform convergence of Dini theorem, limit function is integrable functions, and integral theorem of function series, Functions of the differentiability of limit function, and micro levels of differentiability theorem for continuous functions.

11 。 Based on the dual and multiple limits.

12 。 The radius of convergence of power series Cauchy-Hadamard theorem on an open interval of convergence of power series and continuity theorem, continuity on an open interval of convergence of power series of Abel theorem, product of the power series.

13 。 The differentiability and integrability theorem of power series, derivation and itemized the quadrature power series case-by-case, the function expanded into a Taylor series, elementary functions expanded into a Taylor series.

Power series and the differentiability of the function, Cesaro And, Tauber Theorem.

16 。 Characteristics of uniformly convergent power series.

5 , Totally bounded and equicontinuous, andArzela-Ascoli theorem, theWeierstrass approximation theorem, theStone-Weierstrass theorem, the application of power series in Combinatorial mathematics.

6 , Step function of the integral, the integral of a function on, the General range of the Lebesgue integrable functions and theLebesgue integral of the basic properties,Levi monotone convergence theorem andLebesgue Control convergence theorem andLebesgue integrals.

7 , Parametric definition of integral, variable points of continuity related to differentiability, integrability theorems. Parametric integral integral, variable generalized integral of uniform convergence, variable generalized uniform convergence of integral of Weierstrass, andAbel, andDirichlet discriminant method. Improper integrals, limit, variable generalized integral under the continuity and differentiability, variable generalized integral points.

20 。 The theory of infinite integral,Dirichlet integrals.

8 , Lebesgue measurable function, the relationship between measurability and integrability,Lebesgue integrals, limits, under Exchange integral order,Lebesgue measure,Lebesgue measurable sets, Set of square-integrable functions,Riesz-Fischer theorem.

9 , Beta function and Gamma function,Gauss-Euler equation,gamma and Euler The integral of a function definition, formulae,Stirlingformula and the Wallis formula, convolution and convolution differential andDelta functions, using Delta Function approximation of functions, generalized functions, space of generalized functions, basic solutions.

10 , Orthogonal functions system, strictly-continuous functions of Fourier coefficient of Bessel inequality,Lyapunov-Parseval equation, the completeness of orthogonal function. Pythagoras theorem,Fourier series and Fourier coefficients,Fourier series of limit properties, complete orthogonal system, v-series, Trigonometric average convergence and pointwise convergence, strictly point-like smooth functions of Fourier series on the uniform convergence theorem.

27 。 Fourier series of Jordan and Dirichlet discriminant method.

28 。 Fourier series converges Dini and Lipschitz criterion.

Dirichlet Nuclear, Riemann Lemma, promotion Fourier Lemma, localization principle, Principle of locality. Fejer theorem, theWeierstrass second approximation theorems, trigonometric functions completeness, closed theorems of trigonometric functions.

29 。 The infinite product of the sine function, series and definition of the cotangent function.

30 。 Fejer , polynomial approximation theorems of trigonometric polynomial form of sine and cosine function definition.

Parseval Equations and the isoperimetric inequality.

31 。 Summation of divergent series, exponential sums,Tauber theory, arithmetic average summation method,Poisson-Abel and Cesaro ,Hardy-Landautheorem Application of generalized and.

32 。 Voronogo method,Cesaro generalized method,Borel , series Euler inversion.

33 。 Bessel functions, generating functions,Bessel function of integral form ofBessel functions of nature.

34 。 Heat conduction on the disc.

11 , Fourier transform,Fourier integrals,Fourier series and Fourier integral, orthogonal,Fourier Point of convergence theorem for integrals, downhill function space, Fourier Operational properties of the transform, inverse formula, ParsevalEquation, Fourier Transform, convolution, Fourier Transformation applications in mathematical physics equations,Trigonometric polynomial approximation theorem, Possion A summation.

12 , Gradual, progressive power series,Laplace integral,Laplace integral principles of localization,Watsonlemmas,Laplace The asymptotic expansions of integrals, stable phase.

Mathematical analysis -4

1 , R^n Jordan measure, multiple Riemann integral, as the base of the limit of the Riemann integral, Riemann integrability, andmultiple function integrability of Riemann guidelines. Multiple function integrability criterion of equivalence. Special methods of multiple function integrability and its links with consistent split.

4 。 Jordan curve graphics judging integrability.

5 。 Two multiple integral definition of equivalence,Jordan measure

Lebesgue Under the integral and integral theorem, and the Darboux Integrability theorems, on the set of admissible set, Riemann Integrals, multiple Riemann Additivity of integration, multiple Riemann Integral estimate.

6 。 The basic properties of multiple integrals (linear, mean value theorem, additivity, integral inequality), the multiple integral as repeated integral.

7 。 Riemann multiple integral of Lebesgue integrability criterion.

8 。 Use of compact convex sets of smooth maps differential estimation error of variable substitution, as convexity theorem for smooth maps,

2 , Fubini theorem, the integral variable substitution, variable substitution formulas,Sard ‘s lemma.

9 。 Improper Integrals of the first and second forms, non-negative function first improper integrals in the form of comparison discriminant method.

3 , Generalized multiple Riemann integrals, convergence of generalized integral control method, generalized integrals of variables formula.

4 , Variable substitution in differential form, Definition of manifold, edge and boundless manifolds, smooth manifolds and smooth maps, Manifold orientation, Orientable and nonorientable manifolds, surface boundary-oriented coordination unit, the second axiom of countability, decomposition.

5 , Tangent vector, the tangent space and the cotangent space, the tangent bundle and the cotangent bundle, sub-manifold and the manifold of smooth maps. Immersion and embedding, a wide range of the implicit function theorem.

6 , Rn in the surface area, represented by a double integral of surface area. Vector fields, the lie bracket and theFrobeniustheorem and tensor fields, on manifolds of differential forms and exterior differential forms and Lie derivative.

7 , Integral of a differential form of physical origin, differential forms, a differential form of variable substitution. Differential forms on a manifold of integral quality, volume, distribution on the surface of the form.

8 , First type curved surface and curve integral, the second type curved surface and curve points, first and second surface integrals, integral and differential forms.

11 。 Line integrals of the first and second-nature of the curve integral for definite integrals.

On a closed curve line integrals of the second form, Green Formulas, Gauss-Ostrogradsky Formula, General Stokes Formulas, RiemannManifold, and The integration of differential forms on a manifold. Stokes formulas. Riemann manifold of Stokes formula, lie on the score.

9 , The divergence of a vector field, and Gradient, Curl and theHamilton operator,Laplace operator, orthogonal curvilinear coordinates on the curvilinear coordinates (gradient, divergence and curl, and vector analysis of the basic formulas. ）

10 , There are potential fields, a gradient vector field, conservative, and condition of integral is independent of the path of the curve. Homotopy, tubes, proper form,Poincare ‘s lemma, irrotational field, potential functions.

11 , Poincare theorem and thede Rham cohomology, andde Rham theorem.

12 , Derivation of the heat equation, the continuity equation derivation and derivation of basic equations, Continuum Mechanics

Derivation of the wave equation.

数学分析-1

Calculus and mathematical analysis

1，逻辑符号、集合与集合的初等运算、集合的Cartesian积，函数与映射、集合的势、集合的等价，公理集合论。
2。可数集、集合及其子集蔟的不等价性的Cantor定理。有理数集的可数性。连续统的不可数性。
3。自然数，自然数的公理与运算，数学归纳原理。
4。集合的可分理性引理，数集的分割引理。
11，无理数与有理数。
2，实数的公理系统、实数的小数表示，上下确界、自然数集、有理数集、无理数集、数学归纳法、Archimedes原理、数直线、实数的q进制表示、Dedekind分割。有界数集的确界存在性。
3，闭区间套定理、有限覆盖定理、极限点定理。
10.6。数列，无穷大与无穷小数列及其性质，Bernoullis不等式与Newton二项式。
11.7。数列极限的定义及其性质、收敛数列与他们的算术性质，极限的算术运算、Stolz定理。夹逼原理、Cauchy列、
12.8。求代数方程根的迭代方法。
13.9。复合收敛数列的Toeplitz变换，Toeplitz定理，数列极限的Cauchy定理，数列的Cauchy和，Cauchy准则、Abel定理。
14.10。极限的定义，单调数列，Weierstrass定理。
15.
16.4，子列的极限，自然对数底e、Bolzano-Weierstrass定理，数列的上下极限。
17.5， Heine归结原理、极限的算术运算、滤子极限、数列收敛的Cauchy准则。
18.5。实数域的唯一性原理，实数域的完备性。
19.
20.6，函数极限的Cauchy定义，函数在某一点上连续性的定义、无穷小函数，无穷大与无穷小量及其阶。函数极限的有限性，间断点、连续函数的性质、单调函数极限的性质，连续函数的算术运算。复合函数与单调函数的极限、三角函数与指数函数的连续性。单调函数的间断，单调函数的不连续性。单调函数与反函数的连续性。反函数的连续性，初等函数的连续性，Kepler方程解的连续性。
21.函数极限存在的Cauchy准则。中间值定理、最大值定理、一致连续、Cantor-Heine定理、函数极限的Cauchy与Heine定义的等价性。
22.22。区间上连续函数介值的Cauchy定理。
23.23。区间上连续函数一致连续性的Cantor定理。
24.24。Khinchin归纳原理，数直线上的连通集，连续函数的连通性，。
25.7，用滤子基对Heine定义的函数极限进行推广、一些重要的极限。函数的上下极限。
26.8，微分学的物理背景、微分与导数的定义、可微函数、微分与导数的几何意义、函数可微性的概念及其导数，导数的计算、导数的和、差、积、商，Leibnitz公式。高阶导数。
27.25。函数连续性与可微性的联系，反函数与复合函数的导数，Kepler方程解的可微性。
28.9，增函数的Darboux定理，Fermat定理、Rolle定理、Rollet定理，有限增量定理、导数的中值定理。l‘Hospital法则、l’Hopital第一公式，带Peano余项的Taylor公式、l’Hopital第二公式。Roth定理、带Schlomilch-Routh余项的Taylor公式、具有Lagrange余项的Taylor公式，初等函数按Taylor公式展开。Lagrange余项与Cauchy余项。Cauchy定理与Lagrange定理。
29.10，常数、单调与严格单调函数的判断准则。函数单调性的条件、函数的内极值点、函数极值的判断与必要条件。Young不等式、Holder不等式、Minkowski不等式、凸函数、函数的的凸性，奇点，渐进线，Jensen不等式、函数作图
30.11，用微分学研究自然科学的一些例子。
31.12，原函数与不定积分、原函数的计算方法、椭圆积分。

数学分析-2

1，积分的物理与几何背景、Riemann积分的定义、Riemann积分，Riemann可积函数、有界函数的Riemann积分。可积函数空间、

5。Riemann可积函数类（连续性、单调极限，区间上极限的点状连续）。

Lebesgue定理、Riemann积分积分区间的可加性、omega和，区间上函数可积性的Riemann准则。

3。函数的Riemann可积性的三个准则的等价性。

4。利用区间的一致分割判断函数的Riemann可积性。

积分的估计、积分中值定理、一些重要的积分不等式。

6。定积分的性质（线性，模的可积性，函数的商与积，积分的模不等式，非负函数的积分，积分的单调性）。

7。复合函数的可积性定理，积分作为函数的上（下）极限和，此函数的连续与可微性理论。

8。不定积分，Newton-Leibnitz公式，Euler与Abel求和公式，Stirling公式。

2，变上限的积分、Newton-Leibniz公式、定积分的分部积分与变量替换、变量替换公式与定积分的分部积分法。

10。第一与第二积分中值公式，在光滑函数情况下证明积分中值定理。

积分形式的Taylor公式。积分余项的Talyor公式、面积原理、积分的渐进计算公式，抛物线内插。一元积分学的应用。

3，广义积分的定义、广义积分的基本性质、广义积分的变量替换与分部积分公式、广义积分收敛性的判别法、

13。反常积分收敛性的Cauchy准则与二阶条件。

14。反常积分的绝对与条件收敛。

有多个奇异点的广义积分、广义积分的主值。

4，度量空间与欧氏空间，作为度量空间的R^n、R^n中的开集和闭集、R^n中的紧致集、空间上紧致集的判断。R^n中的范数、作为Euclid空间的R^n。

20。空间上数列的收敛性引理，

21。度量空间上的紧致性，n维度空间上立方体的紧致性。

22。函数的上（下）确界定理与极限存在性，紧致集上函数的连续性，紧致集上连续函数的中值定理。

23。紧致集上连续函数的一致收敛定理。

5， Euler定理、拓扑等价、Euclid空间中映射的连续性、同胚、闭曲面的分类定理、拓扑不变量。

15。简单光华曲线的弧长，曲线弧长的可微性。

16。边界为可求长曲线的图形的可分离性原理。

17。利用Jordan曲边梯形判断可积性。

6，拓扑空间，拓扑空间与度量空间的定义、拓扑空间的概念，开集、开集的基本性质，闭集、边界、例子，拓扑基、数列的收敛性，Hausdorff空间、子拓扑、拓扑空间的子空间。度量空间与拓扑空间的直积、第二可数空间。

24。拓扑空间上映射的极限的概念，拓扑空间上的极限的性质，

7，连续映射、复合映射。连续映射与同胚、复合映射的极限，Hausdorff空间与赋范空间上映射的极限。Peano曲线、Tietze扩张定理、拓扑空间的紧致性、Heine-Borel定理、紧致空间的性质、Bolzano-Weierstrass性质、Lebesgue引理、局部紧空间、Lindelof定理。

25。拓扑空间上的连续映射，例子，连续函数的上极限与下极限，复合函数的连续性。

26。拓扑空间的连通集与连通映射。拓扑空间上紧致性的判据，紧致集的性质，半连续函数，Hausdorff空间上的紧致性，分离性定理。

8，乘积拓扑、乘积空间、Tychonoff乘积定理、连通的拓扑空间、商拓扑、Alexandroff定理、粘合拓扑、完备的度量空间、度量空间的完备化、闭球套引理、第一纲集与第二纲集、Baire纲定理、拓扑空间上的映射的极限、拓扑空间上的映射的连续与一致连续、二重极限与累次极限、压缩映像原理。

9，线性赋范空间、Banach空间、Euclid空间、Hilbert空间、线性算子、算子的范数、连续算子空间、赋范空间上的可微映射、映射的微分与导数、

27。映射的微分，偏导数与函数可微性的必要条件，可微性的二阶条件。28。复合函数的可微性定理，一阶微分的不变性，Jacobian矩阵。

映射的微分的Jacobi矩阵、函数的连续性与可微性、微分的算术运算、复合映射的微分、逆映射的微分、映射的偏导数与微分、方向导数与梯度。

10，有限增量定理、连续可微映射、中值定理、映射的高阶微分与偏导数、高阶微分的运算、

29。混合导数的Schwarz和Young定理。

映射的Taylor公式、具有Peano和Lagrange余项的多元函数的Taylor定理。映射的局部极值、、切平面、法向量、切向量。

11，隐映射定理、隐函数定理，映射的隐函数定理。微分同胚、逆映射定理、秩定理、函数相关性、Morse引理。

12， R^n中的k维子流形、切空间的定义、多元函数极值存在的必要性条件，条件极值、极值的二阶条件。Lagrange乘子法。

数学分析-3

1，数列和的余项，级数收敛的必要条件，数项级数的收敛与发散、绝对收敛、非负数项级数收敛的充要条件、比较判别法、Weierstrass比较判别法、级数收敛的Cauchy判据，级数收敛的判断。D‘Alembert判别法、Gauss判别法、Rabbe判别法、Kummer判别法、Bertrand判别法、广义积分的Cauchy-Maclaurin积分判别法。

2，数项级数的Leibniz级数判别法、Abel判别法、Dirichlet判别法、级数的重排、级数重排的绝对收敛定理，条件收敛级数的Riemann定理、Mertens定理、

二重级数、二重级数与累次级数之间的关系、二重乘积级数的绝对收敛定理，二重绝对收敛级数的重排、二重乘积级数的Mertens定理。多重级数的绝对收敛定理。无穷乘积、无穷乘积的性质，无穷乘积收敛的必要条件、无穷乘积的绝对收敛、gamma函数与Euler函数的无穷乘积定义，Euler公式。gamma函数的函数方程。

3，函数列的收敛集、含参变量的函数族、函数项级数和的连续性定理，收敛与一致收敛、函数项级数的一致收敛的Cauchy准则、复数域的收敛与复数项级数、幂级数、Cauchy-Hadamard公式、Abel定理、函数的幂级数表示、幂级数的解析性、函数项级数一致收敛的Weierstrass优级数判别法、Abel、Dirichlet、Abel-Dirichlet判别法。

4，二重极限可交换的条件、函数族的极限函数的连续性、幂级数的和函数的连续性、区间上非负连续函数级数的一致收敛的判据与一致收敛的Dini定理、函数族极限函数的可积性、函数项级数的可积性定理，函数族的极限函数的可微性、可微连续函数项级别数的可微性定理。

11。基上的二重与多重极限。

12。幂级数收敛半径的Cauchy-Hadamard定理，开收敛区间上幂级数和的连续性定理，开收敛区间上幂级数连续性的Abel定理，幂级数的乘积。

13。幂级数的可微性与可积性定理，幂级数的逐项求导和逐项求积分，函数展开成Taylor级数，初等函数展开成Taylor级数。

幂级数的和函数的可微性、Cesaro和、Tauber定理。

16。幂级数一致收敛的几个特征。

5，完全有界与等度连续、Arzela-Ascoli定理、Weierstrass逼近定理、Stone-Weierstrass定理、幂级数在组合数学中的应用。

6，阶梯函数的积分、上函数的积分、一般区间上的Lebesgue可积函数类、Lebesgue积分的基本性质、Levi单调收敛定理、Lebesgue控制收敛定理、Lebesgue 广义积分。

7，含参变量积分的定义、含参变量积分的连续性与可微性、可积性定理。含参变量积分的积分、含参变量广义积分的一致收敛性、含参变量广义积分的一致收敛的Weierstrass、Abel、Dirichlet判别法。反常积分号下取极限、含参变量广义积分的连续性与可微性、含参变量广义积分的积分。

20。无穷限积分的理论，Dirichlet积分。

8， Lebesgue可测函数、可测性与可积性之间的关系、Lebesgue积分号下取极限、交换积分顺序、Lebesgue测度、Lebesgue可测集、平方可积函数集、Riesz-Fischer定理。

9， Beta函数与Gamma函数、Gauss-Euler公式、gamma与Euler函数的积分定义，余元公式、Stirling公式与Wallis公式、卷积、卷积的微分、Delta函数族、用Delta函数族逼近函数、广义函数、广义函数空间、基本解。

10，正交函数系、严格点状连续函数的Fourier系数的Bessel不等式，Lyapunov-Parseval等式，正交函数系的完备性。Pythagoras定理、Fourier级数与Fourier系数、Fourier级数的极限性质、完备正交系、三角级数、三角级数的平均收敛性与逐点收敛、严格点状光滑函数的Fourier级数的一致收敛定理。

27。Fourier级数收敛的Jordan与Dirichlet判别法。

28。Fourier级数收敛的Dini和Lipschitz判别法。

Dirichlet核，Riemann引理、推广的Fourier引理、局部化原理、局部性原理。Fejer定理、Weierstrass第二逼近定理、三角函数系的完备性、三角函数系的封闭性定理。

29。正弦函数的无穷乘积，余切函数的级数和定义。

30。Fejer和，多项式逼近定理的三角函数多项式形式，正弦与余弦函数解析定义。

Parseval等式、等周不等式。

31。发散级数求和，指数级数的求和方法，Tauber理论，算术平均求和方法，Poisson-Abel和Cesaro方法的比较，Hardy-Landau定理，广义和的应用。

32。Voronogo方法、Cesaro一般化方法，Borel方法、级数的Euler反演。

33。Bessel函数，母函数，Bessel函数的积分形式，Bessel函数的性质。

34。圆盘上的热传导。

11， Fourier变换、Fourier积分、Fourier级数与Fourier积分，正交系，Fourier积分的点状收敛定理、速降函数空间、Fourier变换的运算性质、反演公式、Parseval等式、Fourier变换与卷积、Fourier变换在数学物理方程中的应用、三角多项式逼近定理，Possion求和公式。

12，渐进展开、渐进幂级数、Laplace积分、Laplace积分的局部化原理、Watson引理、Laplace积分的渐进展开、稳定相位法。

数学分析-4

1， R^n中的Jordan测度、多重Riemann积分、作为基的极限的Riemann重积分，Riemann可积性、多重函数可积性的Riemann准则。多重函数可积性判据的等价性。多重函数可积性的特殊判别法及其与一致分割的联系。

4。Jordan曲边图形判断可积性。

5。两种多重积分定义的等价性，Jordan测度，

Lebesgue定理、上积分与下积分、Darboux可积性定理、容许集、集合上的Riemann积分、多重Riemann积分的可加性、多重Riemann积分的估计。

6。多重积分的基本性质（线性、中值定理、可加性、积分不等式），化多重积分为累次积分。

7。Riemann多重积分的Lebesgue可积性判据。

8。利用紧致凸集上光滑映射的微分估计变量替换的误差，光滑映射的像的凸性定理，

2， Fubini定理、多重积分的变量替换、变量替换公式、Sard引理。

9。第一与第二形式的反常积分，非负函数第一形式反常积分的比较判别法。

3，广义多重Riemann积分、广义重积分收敛性的控制判别法、广义重积分的变量替换公式。

4，微分形式的变量替换，流形的定义、带边与无边流形、光滑流形、光滑映射、流形的定向，可定向与不可定向流形、曲面边界定向的协调性、第二可数公理、单位分解。

5，切向量、切空间、余切空间、切丛与余切丛、子流形、子流形上的光滑映射。浸入与嵌入、大范围的隐函数定理。

6， Rn中曲面的面积、用二重积分表示曲面的面积。向量场、李括号、Frobenius定理、张量场、流形上的微分形式与外微分形式、李导数。

7，微分形式的积分的物理起源、微分形式，微分形式的变量替换。流形上的微分形式的积分、分布在曲面上的质量、体积形式。

8，第一型曲面与曲线积分、第二型曲面与曲线积分、第一、第二形曲面积分，与微分形式的积分的关系。

11。第一与第二型曲线积分的性质，化曲线积分为定积分。

闭曲线上第二形式的曲线积分，Green公式、Gauss-Ostrogradsky公式、一般的Stokes公式、Riemann流形、流形上微分形式的积分。一般的Stokes公式。Riemann流形上的Stokes公式、李群上的积分。

9，向量场的散度、梯度、旋度、Hamilton算子、Laplace算子、正交曲线坐标下与曲线坐标上的（梯度和散度及旋度、及向量分析的基本公式。）

10，有势场、有势向量场，保守场、曲线积分的道路无关性的条件。同伦、管量场、恰当形式、Poincare引理、无旋场、势函数。

11， Poincare定理、de Rham上同调、de Rham定理。

12，热传导方程的推导、连续性方程的推导、连续介质力学基本方程的推导、

波动方程的推导。

Calculus & Mathematical Analysis

Rudin, Principles of mathematical analysis, McGraw-Hill.
Courant, Richard; John, Fritz Introduction to calculus and analysis. Vol. I. Reprint of the 1989 edition.Classics in Mathematics. Springer-Verlag, Berlin, 1999.

Courant, Richard; John, Fritz Introduction to calculus and analysis. Vol. II. With the assistance of Albert A. Blank and Alan Solomon. Reprint of the 1974 edition. Springer-Verlag, New York, 1989.

Apostol, “Mathematical analysis”

Fikhtengolts, “微积分学教程”, “数学分析原理”

Demidovich, “Problems in mathematical analysis “

И.И.利亚什科，A.K.博亚尔丘克, “高等数学例题与习题集”

Zorich, “Mathematical Analysis “

Smirnov, “A Course of Higher Mathematics “

Arkhipov, Sadovnichy, “数学分析讲义”

Markarov, “Selected Problems in Real Analysis”

Khinchin, “A Course of Mathematical Analysis”, “Eight Lectures on Mathematical Analysis “

Arkhipov, “Trigonometric sums in number theory and analysis”

3，Valle Possin，Cours de Analyse Infinitesimale，Gauthier-Villars，1903。

5。L.D.Kudryavsev，数学分析教程，物理数学书籍出版社，1989。

6，A.N.Kolmogorov、P.S.Aleksandrov，实变函数论引论，科学出版社，1938。

7，L.Schwartz，Cours de Analyse，Hermann，1981。

9。L.D.Kudryavtsev，数学分析习题集，科学出版社，1984。

Thomas, “Thomas’ Calculus Early Transcendentals”

Varberg, “Calculus with Differential Equations”

小平邦彦, “An introduction to Calculus”

Hardy, “A Course of Pure Mathematics “

Spivak, “Calculus on manifolds”

Munkres, “Analysis on manifolds”

Courant, “Differential and Integral Calculus”

Apostol, “Calculus”

Spivak, “The Hitchhiker’s Guide to Calculus”

Solow, “How to Read and Do Proofs”

Velleman, “How to Prove It”

Stewart, “Calculus”

Larson, “Calculus”

Anton, “Calculus”

Edwards, “Calculus, Early Transcendentals

Rogawski, “Calculus: Early Transcendentals”

Briggs, “Calculus: Early Transcendentals”

Tan, “Calculus: Early Transcendentals”

Banner, “The Calculus Lifesaver”

Swokowski, “Calculus”

McCallum, “Calculus: Single and Multivariable”

Strang, “Calculus”

Wilson, “Advanced Calculus”

Murray, “Differential and Integral Calculus”

Comenetz, “Calculus: The Elements”

Gootman, “Calculus”

Bleau, “Forgotten Calculus”

Kline, “Calculus: An Intuitive and Physical Approach”

Silverman, “Essential Calculus with applications”

Patrick, “Calculus: Art of Problem Solving”

Lang, “A First Course in Calculus”

Ross, “Elementary Analysis: The Theory of Calculus”

Berman, “A Problem book in mathematical analysis”

Radulescu, “Problems in Real Analysis: Advanced Calculus on the Real Axis”

Erdman, “A ProblemText in Advanced Calculus”

Olympiad: The Harvard-MIT Mathematics Tournament

Euler, “Elements of Algebra”

Euler, “Foundations of Differential Calculus”

Euler, “Introduction to the Analysis of the Infinite”

Sloughter, “The Calculus of Functions of Several Variables”

Corral, “Vector Calculus”

Widder, “Advanced calculus”

Lang, “Calculus of Several Variables”

Bressoud, “Second Year Calculus”

Marsden, “Vector Calculus”

Schey, “Div, Grad, Curl and All That”

Shifrin, “Multivariable Mathematics”

Hubbard, “Vector Calculus, Linear Algebra, and Differential Forms”

Duistermaat, “Multidimensional Real Analysis”

Smith, “A Primer of Modern Analysis”

Fleming, “Functions of Several Variables”

Amann, “Analysis”

Edwards, “Advanced Calculus: A Differential Forms Approach”)

Buck, “Advanced Calculus”

Arfken, “Mathematical Methods for Physicists”

Kaplan, “Advanced Calculus”

Dieudonne, “Foundations of modern analysis”

Lay, “Analysis with an Introduction to Proof”

Finney, “Calculus: Graphical, Numerical, Algebraic”

Ostebee, “Calculus from Graphical, Numerical, and Symbolic Points of View”

Smirnov, “A Course of Higher Mathematics”

Polya, Szego, “Problems and Theorems in Analysis”

Loomis, Sternberg, “Advanced Calculus”

Klambauer, “Mathematical Analysis”

Nikolsky, “A course of mathematical analysis”

Dieudonne, “Treatise on Analysis.”

J.Dixmier, “高等数学”

Lusin, “实变函数论”

Goursat, “A Course in Mathematical Analysis”

Tao, “Analysis”

Pugh, “Real Mathematical Analysis “

龚昇, “简明微积分”, “话说微积分”

齐民友, “重温微积分”

方企勤,沈燮昌, “”数学分析”, “数学分析习题集”,”数学分析习题课教材”.”

张筑生, “数学分析新讲 (共三册)”

华罗庚, “高等数学引论第一卷”

何琛,史济怀,徐森林, “数学分析”

欧阳光中,姚允龙, “数学分析”

常庚哲, “数学分析教程”

许绍浦, “数学分析教程”

南大, “数学分析教程”

北大, “数学分析习题集”

林源渠, “数学分析”

陈传璋，金福临，朱学炎，欧阳光中, “数学分析”

欧阳光中，朱学炎，金福临，陈传璋, “数学分析”

陈纪修, “数学分析”

華東師範, “数学分析”

吴良森，毛羽辉, “数学分析习题精解”

刘玉琏, “数学分析讲义”

王昆扬, “简明数学分析（第一版）”

郇中丹，刘永平，王昆扬, “简明数学分析（第二版）”

邝荣雨, “微积分学讲义（第二版）”

林源渠，方企勤, “数学分析解题指南（第二版）”

邓东翱, “数学分析简明教程”

李成章，黄玉民, “数学分析”

陈省身, “在南开大学的演讲”

常庚哲，史济怀, “数学分析教程”

徐森林, “数学分析”

曹之江, “微积分学简明教程”

裴礼文, “数学分析中的典型问题与方法”

谢惠民, “数学分析习题课讲义”

徐利治, “数学分析的方法及例题选讲”

汪林的，王俊青的，还有B.R.盖尔鲍姆, “数学分析中的反例”

21《数学分析：定理•问题•方法》胡适耕，姚云飞著

22《数学分析原理与方法》胡适耕，张显文著

23《数学分析的理论、方法与技巧》邓乐斌编

25《数学分析内容、方法与技巧》孙清华, 孙昊著

【提高】

27《数学分析的方法及例题选讲:分析学的思想、方法与技巧》徐利治著

顺便提一下，徐教授的书，大多比较好，像《组合学讲义》就不错

29《数学分析问题研究与评注》汪林等编著

还有一本《数学分析拾遗》赵显曾著。

31《高等微积分》丘成桐主编

其他的还有《基础偏微分方程》、《分析学》、《有限群的线性表示》、《Markov过程导论》等。

32《高等数学》同济大学应用数学系

33 鉴于很多高等数学吧吧友询问一些书籍，这里特别说一下：

上海交大的《微积分》、《微积分之倚天剑》和《微积分之屠龙刀》，进一步可以看看《托马斯微积分》(很厚啊)。

龚升的《微积分五讲》和齐民友的《重温微积分》。

[56]。

【习题集】

【辅导书】、【习题集】中能做的部分。

34《高等数学例题与习题集.一,一元微积分》、《高等数学例题与习题集.二,多元微积分》

И.И.利亚什科等编著

【辅导书】

35《考研数学精编综合复习指南.理工类》余长安编著或《数学分析的理论、方法与技巧》邓乐斌著

36《高等数学中的若干问题解析》舒阳春编著

37《高等数学学习与提高指南:考研必读》陈鼎兴, 姚奎编著

38《高等数学内容、方法与技巧》(上下册) 孙清华, 郑小姣著

39 《微积分五讲》龚升著

作者另有《线性代数五讲》一书，与上书均为“中国科学技术大学数学教学丛书”之一。

【提高】

40《大学生数学竞赛试题研究生入学考试难题解析选编》李心灿等。

41《无穷级数与连分数》高建福著

42 《项武义基础数学讲义•单元微积分学》《项武义基础数学讲义•多元微积分学》

Calculus & Mathematical Analysis

1. Rudin, Principles of mathematical analysis, McGraw-Hill.

2. Courant, Richard ; John, Fritz Introduction to calculus and analysis. Vol. I. Reprint of the 1989 edition. Classics in Mathematics.Springer-Verlag, Berlin, 1999.

3. Courant, Richard ; John, Fritz Introduction to calculus and analysis. Vol. II. With the assistance of Albert A. Blank and Alan Solomon. Reprint of the 1974 edition. Springer-Verlag, New York, 1989.

Apostol, “Mathematical analysis”

Fikhtengolts, ” Calculus tutorials “, ” Principles of mathematical analysis “

Demidovich, “Problems in mathematical analysis “

И.И. Liashko, A.K. Boyarchuk , ” Examples of higher mathematics and problem sets”

Zorich, “Mathematical Analysis “

Smirnov, “A Course of Higher Mathematics “

Arkhipov, Sadovnichy, ” Lectures on mathematical analysis”

Markarov, “Selected Problems in Real Analysis”

Khinchin, “A Course of Mathematical Analysis”, “Eight Lectures on Mathematical Analysis “

Arkhipov, “Trigonometric sums in number theory and analysis”

3 , Valle Possin , Cours de Analyse Infinitesimale , Gauthier-Villars , 1903 。

5 。 L. D. Kudryavsev, mathematical analysis course, physics and mathematics Books Publishing House,1989.

6 , A.N.Kolmogorov 、 P.S.Aleksandrov , An introduction to theory of functions of a real variable, science press, 1938 。

7 , L.Schwartz , Cours de Analyse , Hermann , 1981 。

9 。 L. D. Kudryavtsev, the mathematical analysis problem set, science press,1984.

Thomas, “Thomas’ Calculus Early Transcendentals”

Varberg, “Calculus with Differential Equations”

Xiaopingbangyan , “An introduction to Calculus”

Hardy, “A Course of Pure Mathematics “

Spivak, “Calculus on manifolds”

Munkres, “Analysis on manifolds”

Courant, “Differential and Integral Calculus”

Apostol, “Calculus”

Spivak, “The Hitchhiker’s Guide to Calculus”

Solow, “How to Read and Do Proofs”

Velleman, “How to Prove It”

Stewart, “Calculus”

Larson, “Calculus”

Anton, “Calculus”

Edwards, “Calculus, Early Transcendentals

Rogawski, “Calculus: Early Transcendentals”

Briggs, “Calculus: Early Transcendentals”

Tan, “Calculus: Early Transcendentals”

Banner, “The Calculus Lifesaver”

Swokowski, “Calculus”

McCallum, “Calculus: Single and Multivariable”

Strang, “Calculus”

Wilson, “Advanced Calculus”

Murray, “Differential and Integral Calculus”

Comenetz, “Calculus: The Elements”

Gootman, “Calculus”

Bleau, “Forgotten Calculus”

Kline, “Calculus: An Intuitive and Physical Approach”

Silverman, “Essential Calculus with applications”

Patrick, “Calculus: Art of Problem Solving”

Lang, “A First Course in Calculus”

Ross, “Elementary Analysis: The Theory of Calculus”

Berman, “A Problem book in mathematical analysis”

Radulescu, “Problems in Real Analysis: Advanced Calculus on the Real Axis”

Erdman, “A ProblemText in Advanced Calculus”

Olympiad: The Harvard-MIT Mathematics Tournament

Euler, “Elements of Algebra”

Euler, “Foundations of Differential Calculus”

Euler, “Introduction to the Analysis of the Infinite”

Sloughter, “The Calculus of Functions of Several Variables”

Corral, “Vector Calculus”

Widder, “Advanced calculus”

Lang, “Calculus of Several Variables”

Bressoud, “Second Year Calculus”

Marsden, “Vector Calculus”

Schey, “Div, Grad, Curl and All That”

Shifrin, “Multivariable Mathematics”

Hubbard, “Vector Calculus, Linear Algebra, and Differential Forms”

Duistermaat, “Multidimensional Real Analysis”

Smith, “A Primer of Modern Analysis”

Fleming, “Functions of Several Variables”

Amann, “Analysis”

Edwards, “Advanced Calculus: A Differential Forms Approach”)

Buck, “Advanced Calculus”

Arfken, “Mathematical Methods for Physicists”

Kaplan, “Advanced Calculus”

Dieudonne, “Foundations of modern analysis”

Lay, “Analysis with an Introduction to Proof”

Finney, “Calculus: Graphical, Numerical, Algebraic”

Ostebee, “Calculus from Graphical, Numerical, and Symbolic Points of View”

Smirnov, “A Course of Higher Mathematics”

Polya, Szego, “Problems and Theorems in Analysis”

Loomis, Sternberg, “Advanced Calculus”

Klambauer, “Mathematical Analysis”

Nikolsky, “A course of mathematical analysis”

Dieudonne, “Treatise on Analysis.”

J.Dixmier, ” Advanced mathematics”

Lusin, ” Theory of functions of real variables”

Goursat, “A Course in Mathematical Analysis”

Tao, “Analysis”

Pugh, “Real Mathematical Analysis “

Gong Sheng , ” Brief calculus “, ” Words of calculus”

Qi , ” Review of calculus”

Enterprise Service , Shen Xiechang , “” Mathematical analysis “, ” Mathematical analysis problem set “,” Exercises in mathematical analysis teaching material”.”

Zhang Zhu , ” New speak of mathematical analysis ( Consists of three volumes)”

Hua luogeng , ” Introduction to the first volume of higher mathematics”

He Chen , Shi Jihuai , Xu Senlin , ” Mathematical analysis”

Ouyangguangzhong , Yao yunlong , ” Mathematical analysis”

Chang Gengzhe , ” Mathematical analysis tutorial”

Xu Shaopu , ” Mathematical analysis tutorial”

Nanda , ” Mathematical analysis tutorial”

North , ” Mathematical analysis problem set”

Linyuan Qu , ” Mathematical analysis”

Chen Chuanzhang, Jin Fulin, Zhu Xueyan, ouyangguangzhong , ” Mathematical analysis”

Europe Sun, Zhu Xueyan, Jin Fulin, Chen Chuanzhang , ” Mathematical analysis”

Chen jixiu , ” Mathematical analysis”

South China East Normal University area , ” Mathematical analysis”

Wu, liangshen, fuzzy glow , ” Exercises in mathematical analysis explained”

Liu Yulian , ” Lectures on mathematical analysis”

Wang Kunyang , ” Simple mathematical analysis (First edition)”

Xun Dan, Liu yongping, Wang Kunyang , ” Simple mathematical analysis (Second Edition)”

Kuangrong rain , ” Calculus lecture (Second Edition)”

Linyuan drains, Enterprise Service , ” Guide to solving problems of mathematical analysis (Second Edition)”

Deng Dongao , ” Brief course in mathematical analysis”

Li Chengzhang, Huang Yumin , ” Mathematical analysis”

Shiing-Shen Chern , ” Lecture at Nankai University”

Chang Gengzhe, Shi Jihuai , ” Mathematical analysis tutorial”

Xu Senlin , ” Mathematical analysis”

Cao Zhijiang , ” Simple tutorial on calculus”

Pei Liwen , ” Typical problems and methods in mathematical analysis”

Huimin Xie , ” Exercises in mathematical analysis class handouts”

Xu Lizhi , ” Selected topics in mathematical analysis method and examples”

Wang Lin, Wang Junqing, there B.R. B.r. gelbaum , ” Counterexamples in mathematics analysis”

21 Mathematical analysis: tillage, theorems, problems and methods of Hu Shi, Yao Yunfei a

22 The principles and methods of mathematical analysis of Hu Shi Geng, Zhang Xianwen with

23 The theory, methods and techniques of mathematical analysis Deng Lebin series

25 The contents, methods and techniques of mathematical analysis of Qinghua Sun , Sun h a

“Increase”

27 Selected topics in mathematical analysis method and examples : Analysis of Xu Lizhi on the ideas, methods and techniques

By the way, Professor Xu’s book, most good, like the combination of lectures on good

29 The research on mathematical analysis and commentary written by Wang Lin

There is a mathematical analysis notes Zhao Xianceng of the.

31 Shing-Tung Yau editor of the advanced calculus

Others are the basis of partial differential equations, the analysis and the linear representations of finite groups, and the Markov An introduction to the process.

32 Of the higher mathematics Tongji University, Department of applied mathematics

33 Since many friends ask higher mathematics books, special mention here:

Shanghai Jiao Tong University the calculus, the calculus of heaven sword and the Dragon Sabre of calculus, we can look at the Thomas calculus ( Very thick ) 。

Calculus of Gong Sheng five talk friends and align the reliving of calculus.

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“Onward”

“Books” and “sets” can be done in part.

34 Examples of higher mathematics and the problem set . , Single variable calculus, andthe examples in higher mathematics problem set . Two ,Multivariable calculus

И.И. Liashko, editor

“Books”

35 Mathematical knitting Guide for a comprehensive review of the grind . Of science written by Yu Changan Or the theory, methods and techniques of mathematical analysis of Deng Lebin with

36 The analysis of several problems in higher mathematics Shu written in early spring

37 Guide to advanced mathematics learning and improving : Grind required Chen Dingxing , Written by Yao Kui

38 Of the contents, methods and techniques of advanced mathematics ( Up and down ) Sun Qinghua , Zheng Xiaojiao with

39 Gong Sheng of the calculus five talk with

Author another linear algebra book, five talk, and the book is “China Science and Technology University math teaching books” one.

“Increase”

40 The math contest of college students graduate entrance exams selected Li Xincan, problem resolved.

41 Gao Jianfu with the infinite series and continued fractions

42 Lecture notes in mathematics • the basis of the Wuyi Wuyi foundations of calculus mathematics of handouts • Multivariable calculus