Multivariable Calculus

“이 책의 가장 큰 특징은 다변수 미적분학을 계산 기법의 나열로 가르치지 않는다는 점이다. 대신 선형대수학, 기하학적 의미, 변수변환, 벡터해석, 그리고 물리 법칙을 하나의 연속적인 흐름 속에서 연결하여 설명한다. 미적분학 교재는 편미분, 중적분, 좌표변환, 선적분, 면적분, 그린 정리, 스토크스 정리, 그리고 발산 정리를 서로 독립된 계산 주제로 다루지만 이 책에서는 행렬, 선형사상, 도함수, 야코비안, 스케일링 인자, 좌표계, 선적분, 면적분, 플럭스, 컬, 발산, 그리고 보존 법칙을 각각 고립된 공식으로 제시하지 않고, 하나로 연결된 개념 체계의 구성 요소로 다룬다. 따라서 학생들은 단순히 ‘어떻게 계산하는가’뿐만 아니라 ‘왜 이러한 공식이 등장하는가’, ‘어떠한 구조 아래에서 이러한 계산이 가능한가’, 그리고 ‘이러한 개념들이 물리적 모형과 어떻게 연결되는가’를 반복적으로 접하며 이해할 수 있도록 돕는다.” - 저자의 말 중에서

Part I Vectors and Linear Functions 1 Multi-variable Vector-valued Functions (review) 3 1.1 Vectors and inner product 3 1.2 Domain, range, and codomain 7 1.3 Graphs, images, level sets, and contours 10 2 Matrix Multiplication and Determinant 15 2.1 Notation 15 2.2 Matrix multiplication 17 3 Matrix Multiplication and Linear Functions 23 3.1 Matrix multiplication (continued) 23 4 Eigenvalues and Composition of Linear Functions 31 4.1 Composition of linear functions 31 4.2 Eigenvalues and eigenvectors 35 5 Parallelotopes and Volume Scaling Factor 41 5.1 Cross product 41 5.2 Cubes and parallelotopes 44 5.3 Volume scaling factor and parallelotopes 46 Part II Derivative of Multi-variable Functions 6 Continuity of Multi-variable Functions 55 6.1 Limit and continuity in Rn 55 6.2 Discontinuous multi-variable functions 62 6.3 Composition of two functions 66 7 Directional and Partial Derivatives 69 7.1 Directional derivative 69 7.2 Partial derivative 72 7.3 Partial derivatives with constrained variables 75 7.4 Change of variables 78 8 Differentiability 83 8.1 Derivative matrix and gradient vector 83 8.2 Differentiability and derivatives 86 9 The Chain Rule 95 9.1 The chain rule 95 9.2 Zero-level set and graph of derivatives 100 10 Line Integral 109 10.1 Parametrized curves and chain rule 109 10.2 Directional derivative and chain rule 111 10.3 Line integral 114 11 Extreme Values 121 11.1 Extreme values and the Hessian matrix 121 11.2 Criterion for maximum, minimum, and saddle 124 11.3 Lagrange multiplier 129 12 Taylor’s Formula for Multi-Variable Functions 135 12.1 Taylor’s formula for one-variable functions 135 12.2 Higher order directional derivatives 137 12.3 Taylor’s formula for n-variable functions 139 Part III Integration of Multi-variable Functions 13 Double and Triple Integration on Rectangular Domains 147 13.1 Riemann integration in (a review) 147 13.2 Riemann integration in 149 13.3 Riemann integration in 152 13.4 Iterated integrals 153 14 Integration over General Domains 159 14.1 Two types of domains 160 14.2 Double integration 161 14.3 Triple integration 165 15 Integration with Variable Changes 171 15.1 Volume scaling factor 171 15.2 Linear approximation 174 16 Coordinate Systems 181 16.1 Variable change for multiple integrals 181 16.2 Polar coordinates 184 17 Cylindrical and Spherical Coordinates 191 17.1 Cylindrical coordinates 191 17.2 Spherical coordinates 195 18 Surface Integral 199 18.1 Functions on parameterized surfaces 200 18.2 Area scaling factor when 202 18.3 Surface integral 203 Part IV Integration of Vector Fields 19 Line Integral for Tangential Component 211 19.1 Line integral for a scalar function 211 19.2 Line integral for a force field 213 19.3 Path independence, potential, and conservative fields 217 20 Potential Field versus Fluid Flow 221 20.1 Potential field is conservative 221 20.2 Line integral and closed curves 223 20.3 Flow and circulation 226 21 Surface Integral for Normal Components 231 21.1 Surface integral for a scalar function 231 21.2 Surface, normal vector, and tangent plane 233 21.3 Surface integral for a vector field 235 22 Divergence Theorem 241 22.1 Boundary of curves and surfaces 241 22.2 Divergence and the divergence theorem 243 22.3 Divergence theorem in 249 23 Divergence Theorem and Conservation Laws 253 23.1 Flux and conservation laws 253 23.2 Mass conservation 256 23.3 Gauss law with concentrated charge density 259 24 Stokes’ Theorem 263 24.1 Curl of a vector field 263 24.2 Stokes’ theorem 265 25 Stokes’ Theorem and Applications 275 25.1 Simply connected domain 275 25.2 Examples 276 25.3 Faraday’s Law of Electromagnetic Induction 279 25.4 Maxwell Equations 280 Part V Appendix 26 Miscellaneous 287 26.1 Moments and Center of Mass 287 26.2 Green’s Theorem 289 26.3 The Speed of Light and the Problem of Coordinate Systems 290 27 Heat Equation and Diffusion Equation 293 27.1 Homogeneous Heat Equation 293 27.2 Nonlinear Heat Equation 296 27.3 Heat Equation in a Heterogeneous Environment 297 27.4 Random Walk and Diffusion Equation 299 Index 303