Advanced mathematics.Ⅱ

Chapter 7 Differential equations
  7.1 Basic concepts of differential equations
    7.1.1 Examples of differential equations
    7.1.2 Basic concepts
    7.1.3 Geometric interpretation of the first-order differential equation
  Exercises 7.1
  7.2 First-order differential equations
    7.2.1 First-order separable differential equation
    7.2.2 Homogeneous first-order equations
    7.2.3 Linear first-order equations
    7.2.4 Bernoulli's equation
    7.2.5 Some other examples that can be reduced to linear first-order equations
  Exercises 7.2
  7.3 Reducible second-order differential equations
  Exercises 7.3
  7.4 Higher-order linear differential equations
    7.4.1 Some examples of linear differential equation of higher-order
    7.4.2 Structure of solutions of linear differential equations
  Exercises 7.4
  7.5 Higher-order linear equations with constant coefficients
    7.5.1 Higher-order homogeneous linear equations with constant coefficients
    7.5.2 Higher-order nonhomogeneous linear equations with constant coefficients
  Exercises 7.5
  7.6 Euler's differential equation
  Exercises 7.6
  7.7 Applications of differential equations
  Exercises 7.7
Chapter 8 Vectors and solid analytic geometry
  8.1 Vectors in plane and in space
    8.1.1 Vectors
    8.1.2 Operations on vectors
    8.1.3 Vectors in plane
    8.1.4 Rectangular coordinate system
    8.1.5 Vectors in space
  Exercises 8.1
    Part a
    Part b
  8.2 Products of vectors
    8.2.1 Scalar product of two vectors
    8.2.2 Vector product of two vectors
    8.2.3 Triple scalar product of three vectors
    8.2.4 Applications of products of vectors
  Exercises 8.2
    Part a
    Part b
  8.3 Planes and lines in space
    8.3.1 Equations of planes
    8.3.2 Equations of lines in space
  Exercises 8.3
    Part a
    Part b
  8.4 Surfaces and space curves
    8.4.1 Cylinders
    8.4.2 Cones
    8.4.3 Surfaces of revolution
    8.4.4 Quadric surfaces
    8.4.5 Space curves
    8.4.6 Cylindrical coordinate system
    8.4.7 Spherical coordinate system
  Exercises 8.4
    Part a
    Part b
Chapter 9 The differential calculus for multi-variable functions
  9.1 Definition of multi-variable functions and their basic properties
    9.1.1 Spacer2 andrn
    9.1.2 Multi-variable functions
    9.1.3 Visualization of multi-variable functions
    9.1.4 Limits and continuity of multi-variable functions
  Exercises 9.1
    Part a
    Part b
  9.2 Partial derivatives and total differentials of multi-variable functior
    9.2.1 Partial derivatives
    9.2.2 Total differentials
    9.2.3 Higher-order partial derivatives
    9.2.4 Directional derivatives and the gradient
  Exercises 9.2
    Part a
    Part b
  9.3 Differentiation of multi-variable composite and implicit functions
    9.3.1 Partial derivatives and total differentials of multi-variable composit functions
    9.3.2 Differentiation of implicit functions
    9.3.3 Differentiation of implicit functions determined by equation systems
  Exercises 9.3
    Part a
    Part b
Chapter 10 Applications of multi-variable functions
  10.1 Approximate function values by total differential
  10.2 Extreme values of multi-variable functions
    10.2.1 Unrestricted extreme values
    10.2.2 Global maxima and minima
    10.2.3 The method of least squares
    10.2.4 Constrained extreme values
    10.2.5 The method of lagrange multipliers
  Exercises 10.2
    Part a
    Part b
  10.3 Applications in geometry
    10.3.1 Arc length along a curve
    10.3.2 Tangent line and normal plane of a space curve
    10.3.3 Tangent planes and normal lines to a surface
    10.3.4 Curvature for plane curves
  Exercises 10.3
    Part a
    Part b
    Synthetic Exercises
Chapter 11 Multiple integrals
  11.1 Concept and properties of double integrals
    11.1.1 Concept of double integrals
    11.1.2 Properties of double integrals
  Exercises 11.1
  11.2 Evaluation of double integrals
    11.2.1 Geometric meaning of double integrals
    11.2.2 Double integrals in rectangular coordinates
    11.2.3 Double integrals in polar coordinates
    11.2.4 Integration by substitution for double integrals in general
  Exercises 11.2
    Part a
    Part b
  11.3 Triple integrals
    11.3.1 Concept and properties of triple integrals
    11.3.2 Triple integrals in rectangular coordinates
    11.3.3 Triple integrals in cylindrical and spherical coordinates
    11.3.4 Integration by substitution for triple integrals in general
  Exercises 11.3
    Part a
    Part b
  11. 4 Applications of multiple integrals
    11.4.1 Surface area
    11.4.2 The center of gravity
    11.4.3 The moment of inertia
  Exercises 11.4
    Part a
    Part b
Chapter 12 Line integrals and surface integrals
  12.1 Line integrals
    12.1.1 Line integrals with respect to arc length
    12.1.2 Line integrals with respect to coordinates
    12.1.3 Relations between two types of line integrals
  Exercises 12.1
    Part a
    Part b
  12.2 Green's formula and its applications
    12.2.1 Green's formula
    12.2.2 Conditions for path independence of line integrals
  Exercises 12.2
    Part a
    Part b
  12.3 Surface integrals
    12.3.1 Surface integrals with respect to surface area
    12.3.2 Surface integrals with respect to coordinates
  Exercises 12.3
    Part a
    Part b
  12.4 Gauss' formula
  Exercises 12.4
    Part a
    Part b
  12.5 Stokes' formula
    12.5.1 Stokes' formula
    12.5.2 Conditions for path independence of space line integrals
  Exercises 12.5
Bibliography