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Chapter 7 Analytic Geometry in Space and Vector Algebra 1

7.1 Vector and their linear operations 1

7.2 Rectangular coordinate systems in space and components of vectors 6

7.3 The scalar product vector product mixed product 15

7.4 Planes and their equations 23

7.5 Straight lines in space and their equations 28

7.6 Surfaces and their equations 34

7.7 Space curves and their equations 39

7.8 Quadric surfaces 41

Chapter 8 The Multivariable Differential Calculus and its Applications 47

8.1 Basic concepts of multivariable functions 47

8.2 Limit and continuity for function of several variables 57

8.3 Partial derivatives and higher-order partial derivatives 63

8.4 Total differentials 71

8.5 Directional derivatives and the gradient 77

8.6 Differentiation of multivariable composite functions 84

8.7 Differentiation of implict functions 89

8.8 Applications of differential calculus of multivariable functions in geometry 100

8.9 Extreme value problems for multivariable functions 107

Chapter 9 Multiple Integrals 119

9.1 Double integral 119

9.2 Evaluation of a double integral by iterated integration 123

9.3 Change of variables in a double integral 132

9.4 Improper double integrals 140

9.5 Applications of double integrals 142

9.6 Extensions to higher dimensions 147

9.7 Change of variables in a triple integral 150

Chapter 10 Line Integrals and Surface Integrals 163

10.1 Line integrals with respect to arc lengths 163

10.2 Line integrals with respect to coordinates 169

10.3 Green's theorem,Path independence 174

10.4 Surface integrals with respect to surface areas 179

10.5 Surface integrals with respect to coordinates 183

10.6 The divergence theorem 186

10.7 Stokes theorem 192

Chapter 11 Differential Equations 199

11.1 Differential equations and their solutions 199

11.2 Separable equations 204

11.3 Linear first-order equations 209

11.4 Homogeneous equations 214

11.5 Exact equations 220

11.6 Reducible second-order equations 224

11.7 second-order linear equations 228

Answers 242