Math 302 Lecture Notes Kenneth Kuttler October 6, 2006 2 Contents 1 Introduction 11 I Vectors, Vector Products, Lines 13 2 Vectors And Points In R n 5 Sept. 19 2.1 R n Ordered n− tuples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Vectors And Algebra In R n . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3 Geometric Meaning Of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4 Geometric Meaning Of Vector Addition . . . . . . . . . . . . . . . . . . . . . 22 2.5 Distance Between Points In R n . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.6 Geometric Meaning Of Scalar Multiplication . . . . . . . . . . . . . . . . . . 26 2.7 Unit Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.8 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.9 Vectors And Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3 Vector Products 39 3.1 The Dot Product 6 Sept. . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.1.1 Definition In terms Of Coordinates . . . . . . . . . . . . . . . . . . . . 39 3.1.2 The Geometric Meaning Of The Dot Product, The Included Angle . . 40 3.1.3 The Cauchy Schwarz Inequality . . . . . . . . . . . . . . . . . . . . . . 42 3.1.4 The Triangle Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.1.5 Direction Cosines Of A Line . . . . . . . . . . . . . . . . . . . . . . . . 44 3.1.6 Work And Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 The Cross Product 7 Sept. . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2.1 The Geometric Description Of The Cross Product In Terms Of The Included Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2.2 The Coordinate Description Of The Cross Product . . . . . . . . . . . 50 3.2.3 The Box Product, Triple Product . . . . . . . . . . . . . . . . . . . . 52 3.2.4 A Proof Of The Distributive Law For The Cross Product ∗ . . . . . . . 53 3.2.5 Torque, Moment Of A Force . . . . . . . . . . . . . . . . . . . . . . . 54 3.2.6 Angular Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.2.7 Center Of Mass ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.3 Further Explanations ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.3.1 The Distributive Law For The Cross Product ∗ . . . . . . . . . . . . . 57 3.3.2 Vector Identities And Notation ∗ . . . . . . . . . . . . . . . . . . . . . 59 3.3.3 Exercises With Answers . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3 4 CONTENTS II Planes And Systems Of Equations 69 4 Planes 11 Sept. 73 4.1 Finding Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.1.1 Planes From A Normal And A Point . . . . . . . . . . . . . . . . . . . 73 4.1.2 The Angle Between Two Planes . . . . . . . . . . . . . . . . . . . . . 74 4.1.3 The Plane Which Contains Three Points . . . . . . . . . . . . . . . . . 75 4.1.4 Intercepts Of A Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.1.5 Distance Between A Point And A Plane Or A Point And A Line ∗ . . 77 5 Systems Of Linear Equations 12,13 Sept. 79 5.1 Systems Of Equations, Geometric Interpretations . . . . . . . . . . . . . . . 79 5.2 Systems Of Equations, Algebraic Procedures . . . . . . . . . . . . . . . . . . 82 5.2.1 Elementary Operations . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.2.2 Gauss Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.3 The Rank Of A Matrix 14 Sept. . . . . . . . . . . . . . . . . . . . . . . . 94 5.4 Theory Of Row Reduced Echelon Form ∗ . . . . . . . . . . . . . . . . . . . . . 96 5.4.1 Exercises With Answers . . . . . . . . . . . . . . . . . . . . . . . . . . 99 III Linear Independence And Matrices 107 6 Spanning Sets And Linear Independence 18,19 Sept. 111 6.0.2 Spanning Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.0.3 Linear Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.0.4 Recognizing Linear Dependence . . . . . . . . . . . . . . . . . . . . . . 118 6.0.5 Discovering Dependence Relations . . . . . . . . . . . . . . . . . . . . 119 7 Matrices 121 7.1 Matrix Operations And Algebra 20,21 Sept. . . . . . . . . . . . . . . 121 7.1.1 Addition And Scalar Multiplication Of Matrices . . . . . . . . . . . . 121 7.1.2 Multiplication Of Matrices . . . . . . . . . . . . . . . . . . . . . . . . 124 7.1.3 The ij th Entry Of A Product . . . . . . . . . . . . . . . . . . . . . . . 127 7.1.4 Properties Of Matrix Multiplication . . . . . . . . . . . . . . . . . . . 129 7.1.5 The Transpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 7.1.6 The Identity And Inverses . . . . . . . . . . . . . . . . . . . . . . . . . 131 7.2 Finding The Inverse Of A Matrix, Gauss Jordan Method 21,22 Sept.133 7.3 Elementary Matrices 22 Sept. . . . . . . . . . . . . . . . . . . . . . . . . 138 7.4 Block Multiplication Of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 145 7.4.1 Exercises With Answers . . . . . . . . . . . . . . . . . . . . . . . . . . 146 IV LU Decomposition, Subspaces, Linear Transformations 151 8 The LU Factorization 25 Sept. 155 8.0.2 Definition Of An LU Decomposition . . . . . . . . . . . . . . . . . . . 155 8.0.3 Finding An LU Decomposition By Inspection . . . . . . . . . . . . . . 155 8.0.4 Using Multipliers To Find An LU Decomp osition . . . . . . . . . . . . 156 8.0.5 Solving Systems Using The LU Decomposition . . . . . . . . . . . . . 157 CONTENTS 5 9 Rank Of A Matrix 26,27 Sept. 159 9.1 The Row Reduced Echelon Form Of A Matrix . . . . . . . . . . . . . . . . . . 159 9.2 The Rank Of A Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 9.2.1 The Definition Of Rank . . . . . . . . . . . . . . . . . . . . . . . . . . 163 9.2.2 Finding The Row And Column Space Of A Matrix . . . . . . . . . . . 164 9.3 Linear Independence And Bases . . . . . . . . . . . . . . . . . . . . . . . . . . 166 9.3.1 Linear Independence And Dependence . . . . . . . . . . . . . . . . . . 166 9.3.2 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 9.3.3 The Basis Of A Subspace . . . . . . . . . . . . . . . . . . . . . . . . . 170 9.3.4 Finding The Null Space Or Kernel Of A Matrix . . . . . . . . . . . . 172 9.3.5 Rank And Existence Of Solutions To Linear Systems ∗ . . . . . . . . . 174 9.3.6 Exercises With Answers . . . . . . . . . . . . . . . . . . . . . . . . . . 175 10 Linear Transformations 27 Sept. 181 10.1 Constructing The Matrix Of A Linear Transformation . . . . . . . . . . . . . 182 10.1.1 Rotations of R 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 10.1.2 Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 10.1.3 Matrices Which Are One To One Or Onto . . . . . . . . . . . . . . . . 186 10.1.4 The General Solution Of A Linear System . . . . . . . . . . . . . . . . 187 10.1.5 Exercises With Answers . . . . . . . . . . . . . . . . . . . . . . . . . . 190 V Eigenvalues, Eigenvectors, Determinants, Diagonalization 193 11 Determinants 2,3 Oct. 197 11.1 Basic Techniques And Prop erties . . . . . . . . . . . . . . . . . . . . . . . . . 197 11.1.1 Cofactors And 2 ×2 Determinants . . . . . . . . . . . . . . . . . . . . 197 11.1.2 The Determinant Of A Triangular Matrix . . . . . . . . . . . . . . . . 200 11.1.3 Properties Of Determinants . . . . . . . . . . . . . . . . . . . . . . . . 201 11.1.4 Finding Determinants Using Row Operations . . . . . . . . . . . . . . 203 11.1.5 A Formula For The Inverse . . . . . . . . . . . . . . . . . . . . . . . . 204 12 Eigenvalues And Eigenvectors Of A Matrix 4-6 Oct. 209 12.0.6 Definition Of Eigenvectors And Eigenvalues . . . . . . . . . . . . . . . 209 12.0.7 Finding Eigenvectors And Eigenvalues . . . . . . . . . . . . . . . . . . 211 12.0.8 A Warning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 12.0.9 Defective And Nondefective Matrices . . . . . . . . . . . . . . . . . . . 215 12.0.10 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 12.0.11 Migration Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 12.0.12 Complex Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 12.0.13 The Estimation Of Eigenvalues . . . . . . . . . . . . . . . . . . . . . . 228 12.1 The Mathematical Theory Of Determinants ∗ . . . . . . . . . . . . . . . 229 12.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 12.2 The Cayley Hamilton Theorem ∗ . . . . . . . . . . . . . . . . . . . . . . . 241 12.2.1 Exercises With Answers . . . . . . . . . . . . . . . . . . . . . . . . . . 242 VI Curves, Curvilinear Motion, Surfaces 253 13 Quadric Surfaces 9 Oct. 257 6 CONTENTS 14 Curves In Space 10,11 Oct. 261 14.1 Limits Of A Vector Valued Function Of One Variable . . . . . . . . . . . . . 261 14.2 The Derivative And Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 14.2.1 Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 14.2.2 Geometric And Physical Significance Of The Derivative . . . . . . . . 267 14.2.3 Differentiation Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 14.2.4 Leibniz’s Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 14.2.5 Exercises With Answers . . . . . . . . . . . . . . . . . . . . . . . . . . 271 15 Newton’s Laws Of Motion ∗ 273 15.0.6 Kinetic Energy ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 15.0.7 Impulse And Momentum ∗ . . . . . . . . . . . . . . . . . . . . . . . . . 278 15.0.8 Conservation Of Momentum ∗ . . . . . . . . . . . . . . . . . . . . . . . 278 15.0.9 Exercises With Answers . . . . . . . . . . . . . . . . . . . . . . . . . . 279 16 Physics Of Curvilinear Motion 12 Oct. 281 16.0.10 The Acceleration In Terms Of The Unit Tangent And Normal . . . . . 281 16.0.11 The Curvature Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 16.0.12 The Circle Of Curvature* . . . . . . . . . . . . . . . . . . . . . . . . . 286 16.1 Geometry Of Space Curves ∗ . . . . . . . . . . . . . . . . . . . . . . . . . 288 16.2 Independence Of Parameterization ∗ . . . . . . . . . . . . . . . . . . . . 291 16.2.1 Hard Calculus ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 16.2.2 Independence Of Parameterization ∗ . . . . . . . . . . . . . . . . . . . 295 16.3 Product Rule For Matrices ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 16.4 Moving Coordinate Systems ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 VII Functions Of Many Variables 301 17 Functions Of Many Variables 16 Oct. 305 17.1 The Graph Of A Function Of Two Variables . . . . . . . . . . . . . . . . . . . 305 17.2 The Domain Of A Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 17.3 Open And Closed Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 17.4 Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 17.5 Sufficient Conditions For Continuity . . . . . . . . . . . . . . . . . . . . . . . 312 17.6 Properties Of Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . 313 18 Limits Of A Function 17-23 Oct. 315 18.1 The Directional Derivative And Partial Derivatives . . . . . . . . . . . . . . . 318 18.1.1 The Directional Derivative . . . . . . . . . . . . . . . . . . . . . . . . 318 18.1.2 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 18.1.3 Mixed Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 323 18.2 Some Fundamentals ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 18.2.1 The Nested Interval Lemma ∗ . . . . . . . . . . . . . . . . . . . . . . . 328 18.2.2 The Extreme Value Theorem ∗ . . . . . . . . . . . . . . . . . . . . . . 329 18.2.3 Sequences And Completeness ∗ . . . . . . . . . . . . . . . . . . . . . . 330 18.2.4 Continuity And The Limit Of A Sequence ∗ . . . . . . . . . . . . . . . 333 CONTENTS 7 VIII Differentiability 335 19 Differentiability 24-26 Oct. 339 19.1 The Definition Of Differentiability . . . . . . . . . . . . . . . . . . . . . . . . 339 19.2 C 1 Functions And Differentiability . . . . . . . . . . . . . . . . . . . . . . . . 341 19.3 The Directional Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 19.3.1 Separable Differential Equations ∗ . . . . . . . . . . . . . . . . . . . . . 344 19.3.2 Exercises With Answers ∗ . . . . . . . . . . . . . . . . . . . . . . . . . 347 19.3.3 A Heat Seaking Particle . . . . . . . . . . . . . . . . . . . . . . . . . . 348 19.4 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 19.4.1 Related Rates Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 351 19.5 Normal Vectors And Tangent Planes 26 Oct. . . . . . . . . . . . . . . . 353 20 Extrema Of Functions Of Several Variables 30 Oct. 355 20.1 Local Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 20.2 The Second Derivative Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 20.2.1 Functions Of Two Variables . . . . . . . . . . . . . . . . . . . . . . . . 358 20.2.2 Functions Of Many Variables ∗ . . . . . . . . . . . . . . . . . . . . . . 359 20.3 Lagrange Multipliers, Constrained Extrema 31 Oct. . . . . . . . . . . 362 20.3.1 Exercises With Answers . . . . . . . . . . . . . . . . . . . . . . . . . . 367 21 The Derivative Of Vector Valued Functions, What Is The Derivative? ∗ 371 21.1 C 1 Functions ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 21.2 The Chain Rule ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 21.2.1 The Chain Rule For Functions Of One Variable ∗ . . . . . . . . . . . . 377 21.2.2 The Chain Rule For Functions Of Many Variables ∗ . . . . . . . . . . . 377 21.2.3 The Derivative Of The Inverse Function ∗ . . . . . . . . . . . . . . . . 381 21.2.4 Acceleration In Spherical Coordinates ∗ . . . . . . . . . . . . . . . . . . 381 21.3 Proof Of The Chain Rule ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 21.4 Proof Of The Second Derivative Test ∗ . . . . . . . . . . . . . . . . . . . 386 22 Implicit Function Theorem ∗ 389 22.1 The Method Of Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . 393 22.2 The Local Structure Of C 1 Mappings . . . . . . . . . . . . . . . . . . . . . . 394 IX Multiple Integrals 397 23 The Riemann Integral On R n 403 23.1 Methods For Double Integrals 1 Nov. . . . . . . . . . . . . . . . . . . . 403 23.1.1 Density Mass And Center Of Mass . . . . . . . . . . . . . . . . . . . . 410 23.2 Double Integrals In Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . 411 23.3 Methods For Triple Integrals 2-7 Nov. . . . . . . . . . . . . . . . . . . . 416 23.3.1 Definition Of The Integral . . . . . . . . . . . . . . . . . . . . . . . . . 416 23.3.2 Iterated Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 23.3.3 Mass And Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 23.3.4 Exercises With Answers . . . . . . . . . . . . . . . . . . . . . . . . . . 423 8 CONTENTS 24 The Integral In Other Coordinates 8-10 Nov. 427 24.1 Different Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 24.1.1 Review Of Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . 428 24.1.2 General Two Dimensional Coordinates . . . . . . . . . . . . . . . . . . 429 24.1.3 Three Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 24.1.4 Exercises With Answers . . . . . . . . . . . . . . . . . . . . . . . . . . 436 24.2 The Moment Of Inertia ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 24.2.1 The Spinning Top ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 24.2.2 Kinetic Energy ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 24.3 Finding The Moment Of Inertia And Center Of Mass 13 Nov. . . . 447 24.4 Exercises With Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 X Line Integrals 455 25 Line Integrals 14 Nov. 459 25.0.1 Orientations And Smooth Curves . . . . . . . . . . . . . . . . . . . . 459 25.0.2 The Integral Of A Function Defined On A Smooth Curve . . . . . . . 461 25.0.3 Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 25.0.4 Line Integrals And Work . . . . . . . . . . . . . . . . . . . . . . . . . 464 25.0.5 Another Notation For Line Integrals . . . . . . . . . . . . . . . . . . . 466 25.0.6 Exercises With Answers . . . . . . . . . . . . . . . . . . . . . . . . . . 467 25.1 Path Indep endent Line Integrals 15 Nov. . . . . . . . . . . . . . . . . . 468 25.1.1 Finding The Scalar Potential, (Recover The Function From Its Gradient)469 25.1.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 XI Green’s Theorem, Integrals On Surfaces 473 26 Green’s Theorem 20 Nov. 477 26.1 An Alternative Explanation Of Green’s Theorem . . . . . . . . . . . . . . . . 479 26.2 Area And Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 27 The Integral On Two Dimensional Surfaces In R 3 27-28 Nov. 485 27.1 Parametrically Defined Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 485 27.2 The Two Dimensional Area In R 3 . . . . . . . . . . . . . . . . . . . . . . . . . 487 27.2.1 Surfaces Of The Form z = f (x, y) . . . . . . . . . . . . . . . . . . . . 494 27.3 Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496 27.3.1 Exercises With Answers . . . . . . . . . . . . . . . . . . . . . . . . . . 496 XII Divergence Theorem 501 28 The Divergence Theorem 29-30 Nov. 505 28.1 Divergence Of A Vector Field . . . . . . . . . . . . . . . . . . . . . . . . 505 28.2 The Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506 28.2.1 Coordinate Free Concept Of Divergence, Flux Density . . . . . . . . . 510 28.3 The Weak Maximum Principle ∗ . . . . . . . . . . . . . . . . . . . . . . . 510 28.4 Some Applications Of The Divergence Theorem ∗ . . . . . . . . . . . . 511 28.4.1 Hydrostatic Pressure ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . 511 28.4.2 Archimedes Law Of Buoyancy ∗ . . . . . . . . . . . . . . . . . . . . . . 512 28.4.3 Equations Of Heat And Diffusion ∗ . . . . . . . . . . . . . . . . . . . . 512 CONTENTS 9 28.4.4 Balance Of Mass ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 28.4.5 Balance Of Momentum ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . 514 28.4.6 Bernoulli’s Principle ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 28.4.7 The Wave Equation ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 28.4.8 A Negative Observation ∗ . . . . . . . . . . . . . . . . . . . . . . . . . 521 28.4.9 Electrostatics ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 XIII Stoke’s Theorem 523 29 Stoke’s Theorem 4-5 Dec. 527 29.1 Curl Of A Vector Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 29.2 Green’s Theorem, A Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 29.3 Stoke’s Theorem From Green’s Theorem . . . . . . . . . . . . . . . . . . . . . 529 29.3.1 Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 29.3.2 Conservative Vector Fields And Stoke’s Theorem . . . . . . . . . . . . 533 29.3.3 Some Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534 29.3.4 Vector Identities ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534 29.3.5 Vector Potentials ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536 29.3.6 Maxwell’s Equations And The Wave Equation ∗ . . . . . . . . . . . . . 536 XIV Some Iterative Techniques For Linear Algebra 539 30 Iterative Methods For Linear Systems 541 30.1 Jacobi Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 30.2 Gauss Seidel Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 31 Iterative Methods For Finding Eigenvalues 551 31.1 The Power Method For Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . 551 31.1.1 Rayleigh Quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 31.2 The Shifted Inverse Power Method . . . . . . . . . . . . . . . . . . . . . . . . 556 XV The Correct Version Of The Riemann Integral ∗ 563 A The Theory Of The Riemann Integral ∗∗ 565 A.1 An Important Warning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565 A.2 The Definition Of The Riemann Integral . . . . . . . . . . . . . . . . . . . . . 565 A.3 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568 A.4 Iterated Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581 A.5 The Change Of Variables Formula . . . . . . . . . . . . . . . . . . . . . . . . 584 A.6 Some Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591 Copyright c  2005, 10 CONTENTS [...]... of a parallelogram ii the area or a triangle iii physical quantities such as moment of force and angular velocity G Find the volume of a parallelepiped using the scalar triple product H Recall, apply and derive the algebraic properties of the dot and cross products Reading: Multivariable Calculus 1.2-3, Linear Algebra 1.2 Outcome Mapping: A 1,2bd,3,7 B 3 C 2egi D 2kmp,7dgh E 4 F 5,15,B5 G 6,B6 H 8,17,B1,B2,B3,B4... the line and the direction of the line or (b) two points contained in the line (c) the direction cosines of the line C Determine the direction of a line given its parameterization D Find the angle between two lines E Determine a point of intersection between a line and a surface 17 Reading: Multivariable Calculus 1.5, Linear Algebra 1.3 Outcome Mapping: A 3,4 B 3,4 C 1 D 2 E 11,14 18 Vectors And Points... vector addition, scalar multiplication and norm geometrically F Recall, apply and verify the basic properties of vector addition, scalar multiplication and norm G Model and solve application problems using vectors Reading: Multivariable Calculus 1.1, Linear Algebra 1.1 Outcome Mapping: A 1,2,4 B A1,A2 C 8,9,11,13,14 D 9,11,12,13 E 8,10 F 17,A3,A4 G A5 Vector Products A Evaluate a dot product from the angle... century I will often not bother to draw a distinction between the point in n dimensional space and its Cartesian coordinates 2.2 Vectors And Algebra In Rn There are two algebraic operations done with points of Rn One is addition and the other is multiplication by numbers, called scalars Definition 2.2.1 If x ∈ Rn and a is a number, also called a scalar, then ax ∈ Rn is defined by ax = a (x1 , · · ·, xn )... deformation gradient is a derivative.) and Newton’s method for solving nonlinear systems of equations.(The entire method involves looking at the derivative and its inverse.) If you don’t want to learn anything more than what you will be tested on, then you can omit these sections This is up to you It is your choice A word about notation might help Most of the linear algebra works in any field Examples are... + x1 , · · ·, an ) and from there to the point (a1 + x1 , a2 + x2 , a3 · ··, an ) and then to (a1 + x1 , a2 + x2 , a3 + x3 , · · ·, an ) and continue this way until you obtain the point (a1 + x1 , a2 + x2 , · · ·, an + xn ) The arrow having its tail at (a1 , a2 , · · ·, an ) and its point at (a1 + x1 , a2 + x2 , · · ·, an + xn ) looks just like the arrow which has its tail at 0 and its point at (x1... which is usually followed, especially in R2 and R3 is to denote the first component of a point in R2 by x and the second component by y In R3 it is customary to denote the first and second components as just described while the third component is called z Example 2.5.3 Describe the points which are at the same distance between (1, 2, 3) and (0, 1, 2) VECTORS AND POINTS IN RN 5 SEPT 26 Let (x, y, z) be... have the same direction and magnitude as (αu1 , αu2 , αu3 ) I         u   +v    ¢  u ¢ ¢   ¢   I  ¢   ¢ v   The following example is art which illustrates these definitions and conventions Exercise 2.6.1 Here is a picture of two vectors, u and v         u   rr rr v r j Sketch a picture of u + v, u − v, and u+2v First here is a picture of u + v You first draw u and then at the point of... is standard position A vector is in standard position if the tail is placed at the origin It is customary to identify the point in Rn with its position vector VECTORS AND POINTS IN RN 5 SEPT 34 → The magnitude of a vector determined by a directed line segment − is just the distance pq between the point p and the point q By the distance formula this equals 1/2 n (qk − pk ) 2 = |p − q| k=1 n k=1 and. .. diagonal of a parallelogram determined from the two vectors u and v and that identifying u + v with the directed diagonal of the parallelogram determined by the vectors u and v amounts to the same thing as the above procedure An item of notation should be mentioned here In the case of Rn where n ≤ 3, it is standard notation to use i for e1 , j for e2 , and k for e3 Now here are some applications of vector