【正文】 g independent learners, make mathematical models and skilled performance and practiced expression in signs and mathematical languages. At last, students have good writing of proofs in college mathematics and good clarity of thought and language when writing or municating mathematics. The students should have the following: be familiar with the basic concepts and results of Advanced Mathematics in a manner concurring with the mathematical way of reasoning and abstraction. achieve mand of the fundamental definitions and concepts of calculus theory. achieve proficiency in writing proofs, including those using basic proof techniques of Advanced Mathematics such as bijections, minimal counterexamples, and loaded induction.Contents. What are the abilities of students after learning Advanced Mathematics? We think of students should:Part One: Real Set, Functions, Limits and Continuity1. Understand deeply the concept and prosperities of functions。 3. Work with functions graphically, numerically, and analytically, operate basic elementary functions and four basic operations and pound operation to product an elementary function。5. Demonstrate understanding of properties of limits, such as, rational and pounded operations, the uniquely limit value, boundaries, preserving signs, two convergent principles (squeezing, monotonic and bounded), and use these properties to pute simple limits。7. Demonstrate understanding of infinitesimal and infinity, hold the cooperation between two infinitesimals (infinities), and apply infinitesimal in putation of limits。 recognize distinct types of discontinuous points。10. Be acquaint him/herself with limit as a tool for solving problems.Part Two: Differentiation of one real variable and Its Application1. Grasp the meaning of derivatives in different problems。 3. Take the basic formulas of derivatives in mind, control efficiently finding derivatives by the operation rules on derivatives。5. Apply rules of differentiation, and use derivatives to solve varied problems。7. Demonstrate understanding of mean theorems (core theorems in differentiation), especially, Lagrange’s mean theorem, and employ them to prove various problems。 working out a great number of exercises is needed to most of students。10. Distinguish well local extremum and absolute (global) extremum。11. Judge the convexity of a function, and find inflective points。 and draw sketching graph of a function。13. Master what is an approximate solution of an equation, and two important methods (dichotomy, tangent method).Part Three: Integration of one real variable1. Demonstrate understanding of the concept and geometric significance of definite integrals, and prehend some properties of definite integrals and the mean theorem of integration.2. Be acquainted with antiderivative (primary function, indefinite 1ntegral) of a function, and skilled putation in finding indefinite integrals (substitution, and integral by parts)。5. Model scientifically some geometric and physical problems by the element method of integration, and solve simple problem from the real world。 2. Be acquainted with the operations of vectors, such as dot product, cross multiplication, and judge vertical or horizontal vectors。4. Demonstrate understanding of a surface and its equation, learn particular surfaces and their graphs by heart，such as rotational surfaces, cylindrical surfaces。 6. Understand the projection of curves of intersect surfaces。2. Comprehend limit and continuity of functions of multiple real variables, and properties of functions defined on bounded and closed regions。4. Study a vector function of one real variable for the concept and derivative。6. Compute flexibly partial derivatives of first and second order about explicit functions and implicit functions of multiple real variables。 8. Apply partial derivative to find extremum of functions of two real variables and some application problems. Part Six: Integration of multiple variables1. Demonstrate understanding of double integrals and trip1e integrals and property of multiple integrals。3. Be acquainted with two types of line integrals and the properties and connection between them, and work out putation of these line integrals。 5. Know well two types of surface integrals, and recognize the Gauss’ formula and Stokes’ formula。 be able to estimate divergence and rotation。2. Control the method of differential equation by separable variable and solve oneorder differential equations。4. Demonstrate understanding of the structure of solutions of linear differential equations。 and be able to solve some particular linear differential equations of second order with constant coefficients.Part Eight: Series1. Demonstrate understanding of the convergence and divergence of infinite series。2. Work well on series of positive number terms, learn particular geometric series, harmonic series and Pseries by heart。3. Be acquaint with alternating series and the Leibniz’s Test, and know the difference between absolute convergent series and conditional convergent series。 5. Be familiar with the Taylor’s series and Maclaurin39。 work well on Maclaurin39。6. Understand the thought of approximate putation by means of power ser