Language: English
ISBN/ISSN: 7040177943
Published on: 2006-01
Paperback
This is the second volume of the textbook "Fundamentals of Advanced Math-ematics" written by the same authors. It includes vector algebra and analytic geometry in space, multivariable calculus, and linear ordinary differential e-quations. The intentions and features are as introduced in the preface to the first volume. We repeat here the important advice to students in the first vol-ume, as it is equally important for this second volume.In order to learn calculus, it is not enough to read the textbook as if it were a newspaper. Learning requires careful reading, working through exam-ples step by step, and solving problems. Solving problems requires more than imitation of examples. It is necessary to think about what the problem really asks and to develop a method for that particular problem.If something is still not clear after you have tried to understand it, you should ask a classmate, a more advanced student, or your teacher. If a classmate asks you a question, you may learn a great deal from explaining the answer.
The following two additional remarks might be helpful to readers in u-sing the second volume.
(1) The material on linear systems of ordinary differential equations (Section 9.2) is not included in the fundamental requirements. Before study-ing it, readers will need some basic knowledge of linear algebra.
(2) Some of the material in this volume has been stated in terms of ma-trices and determinants. For readers who are not yet familiar with the basic concepts and operations for matrices and determinants we have included a brief outline in Appendix A.
Chapter 5 Vector Algebra and Analytic Geometry in Space
5.1 Vectors and Their Linear Operations
5.1.1 The concept of vector
5.1.2 Linear operationS on vectorS
5.1.3 Projection of vectors
5.1.4 Rectangular coordinate systems in space and components of Vectors
Exercises 5.1
5.2 Multiplicative Operations on Vectors
5.2.1 The scalar product(dot product，inner product)of two vectorS
5.2.2 The vector product(cross product，outer product)of two vectors
5.2.3 The mixed product of three vectors
Exercises 5.2
5.3 Planes and Lines in Space
5.3.1 Equations of planes
5.3.2 Position relationships between two planes
5.3.3 Equations of straight lines in space
5.3.4 Position relationships between two lines
5.3.5 Position relationships between a line and a plane
5.3.6 Distance from a point to a plane(1ine)
Exercises 5.3
5.4 Surfaces and Space Curves
5.4.1 Equations of surfaces
5.4.2 Quadric surfaces
5.4.3 Equations of space curves
Chapter 6 The Multivariable Differential Calculus and its Applications
6.1 Limits and Continuity of Multivariable Functions
6.1.1 Primary knowledge of point sets in the space Rn
6.1.2 The concept of a multivariable function
6.1.3 Limit and continuity of multivariable functions
Exercises 6.1
6.2 Partial Derivatives and Total Differentials of Multivariable Functions
6.2.1 Partial derivatives
6.2.2 Total differentials
6.2.3 Higher-order partial derivatives
6.2.4 Directional derivatives and the gradient
Exercises 6.2
6.3 Differentiation of Multivariable Composite Functions and Implicit Functions
6.3.1 Partial derivatives and total differentials of multivariable composite functions
6.3.2 Differentiation of implicit functions defined by one equation
6.3.3 Differentiation of implicit functions determined by more than one equation
Exercises 6.3
6.4 Extreme Value Problems for Multivariable Functions
6.4.1 Unrestricted extreme values
6.4.2 Global maxima and minima
6.4.3 Extreme values with constraints; The method of Lagrange multipliers
Exercises 6.4
* 6.5 Taylor's Formula for Functions of Two Variables
6.5.1 Taylor's formula for functions of two variables
6.5.2 Proof of the sufficient condition for extreme values of function of two variables
Exercises 6.5
6.6 Derivatives and Differentials of Vector-valued Functions
6.6.1 Derivatives and differentials of vector-valued functions of one variable
6.6.2 Derivatives and differentials of vector-valued functions of two variables
6.6.3 Rules for differential operations
Exercises 6.6
6.7 Applications of Differential Calculus of Multivariable Functions in Geometry
6.7.1 Tangent line and normal plane to a space curve
6.7.2 Arc length
6.7.3 Tangent planes and normal lines of surfaces
6.7.4 Curvature
* 6.7.5 The Frenet frame
* 6.7.6 Torsion
Exercises 6.7
Synthetic exercises
Chapter 7 The Integral Calculus of Multivariable Scalar Functions and Its Applications
7.1 The Concept and Properties of the Integral of a Multivariable Scalar Function
7.1.1 Computation of mass of an object
7.1.2 The concept of the integral of a multivariable scalar function
7.1.3 Properties of integrals of multivariable scalar functions
Exercises 7.1
7.2 Computation of Double Integrals
7.2.1 Geometric meaning of the double integral
7.2.2 Computation methods for double integrals in rectangular coordinates
7.2.3 Computation of double integrals in polar coordinates
7.2.4 Integration by substitution for double integrals in general
Exercises 7.2
7.3 Computation of Triple Integrals
7.3.1 Reduction of a triple integral to an iterated integral consisting of a single integral and a double integral
7.3.2 Computation of triple integrals in cylindrical and spherical coordinates
7.3.3 Computation of triple integrals by general substitutions
Exercises 7.3
7.4 Applications of Multiple Integrals
7.4.1 The method of elements for multiple integrals
7.4.2 Examples of applications
Exercises 7.4
7.5 Line and Surface Integrals of the First Type
7.5.1 Line integrals of the first type
7.5.2 Surface integrals of the first type
Exercises 7.5
Synthetic exercises
Chapter 8 The Integral Calculus of Multivariable Vectorvalued Functions and its Applications in the Theory of Fields
8.1 Line and Surface Integrals of the Second Type
8.1.1 The concept of field
8.1.2 Line integrals of the second type
8.1.3 Surface integrals of the second type
Exercises 8.1
8.2 The Relations Between Different Kinds of Integrals and their
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Chapter 9 Linear Ordinary Differential Equations
Appendix A Basic Properties of Matrices and Determinants
Appendix B Answers and Hints for Exercises
