MULTIVARIABLE CALCLABS WITH MAPLE for Stewart’s CALCULUS,FOURTH EDITION Arthur Belmonte and Philip B. Yasskin DRAFT April 22, 1999 c 1999 Brooks/Cole Publishing Company ii Contents Contents iii Dedication viii Introduction ix 1 The Geometry of Rn 1 1.1VectorAlgebra......................................... 1 1.1.1 ScalarsAreNumbers;PointsandVectorsAreLists.................. 1 1.1.2 Addition, Scalar Multiplication and Simplification .................. 3 1.1.3 The Dot Product . ................................. 4 1.1.4 The Cross Product . ................................. 8 1.2Coordinates........................................... 9 1.2.1 Polar Coordinates in R2 ................................ 9 1.2.2 Cylindrical and Spherical Coordinates in R3 ..................... 10 1.3 Curves and Surfaces . ..................................... 12 1.3.1 LinesandPlanes.................................... 12 1.3.2 Quadric Curves and Quadric Surfaces . ...................... 16 1.3.3 Parametric Curves and Parametric Surfaces ...................... 21 1.4Exercises............................................ 23 2 Vector Functions of One Variable: Analysis of Curves 25 2.1VectorFunctionsofOneVariable................................ 25 2.1.1 Definition . ..................................... 25 2.1.2 Limits, Derivatives and Integrals and the map Command............... 27 2.2FrenetAnalysisofCurves................................... 29 2.2.1 Position and Plot . ................................. 29 2.2.2 Velocity, Acceleration and Jerk ............................. 31 2.2.3 Speed,ArcLengthandArcLengthParameter..................... 32 2.2.4 UnitTangent,UnitPrincipalNormal,UnitBinormal................. 34 2.2.5 CurvatureandTorsion................................. 36 2.2.6 Tangential and Normal Components of Acceleration . .............. 37 2.3Exercises............................................ 38 iii iv CONTENTS 3 Partial Derivatives 40 3.1ScalarFunctionsofSeveralVariables............................. 40 3.1.1 Definition . ..................................... 40 3.1.2 Plots.......................................... 41 3.1.3 PartialDerivatives................................... 44 3.1.4 GradientandHessian.................................. 46 3.2Applications........................................... 47 3.2.1 TangentPlanetoaGraph................................ 47 3.2.2 DifferentialsandtheLinearApproximation...................... 48 3.2.3 Taylor Polynomial Approximations . ...................... 52 3.2.4 Chainrule........................................ 55 3.2.5 DerivativesalongaCurveandDirectionalDerivatives................ 60 3.2.6 InterpretationoftheGradient.............................. 63 3.2.7 TangentPlanetoaLevelSurface............................ 66 3.3Exercises............................................ 67 4 Max-Min Problems 69 4.1UnconstrainedMax-MinProblems............................... 70 4.1.1 FindingCriticalPoints................................. 70 4.1.2 ClassifyingCriticalPointsbytheSecondDerivativeTest............... 75 4.2ConstrainedMax-MinProblems................................ 78 4.2.1 EliminatingaVariable................................. 79 4.2.2 ParametrizingtheConstraint.............................. 80 4.2.3 Lagrange Multipliers . ................................. 82 4.2.4 TwoorMoreConstraints................................ 84 4.3Exercises............................................ 87 5 Multiple Integrals 89 5.1 Multiple Integrals in Rectangular Coordinates . ...................... 89 5.1.1 Computation...................................... 89 5.1.2 Applications...................................... 92 5.2 Multiple Integrals in Standard Curvilinear Coordinates . .................. 95 5.2.1 PolarCoordinates.................................... 95 5.2.2 CylindricalCoordinates................................ 96 5.2.3 SphericalCoordinates................................. 98 5.2.4 Applications...................................... 100 5.3 Multiple Integrals in General Curvilinear Coordinates . .................. 103 5.3.1 General Curvilinear Coordinates . ...................... 103 5.3.2 Multiple Integrals . ................................. 107 5.4Exercises............................................ 113 6 Line and Surface Integrals 116 6.1ParametrizedCurves...................................... 116 6.1.1 LineIntegralsofScalars................................ 116 6.1.2 Mass,CenterofMassandMomentofInertia..................... 118 6.1.3 LineIntegralsofVectors................................ 120 CONTENTS v 6.1.4 WorkandCirculation.................................. 122 6.2 Parametrized Surfaces . ..................................... 125 6.2.1 TangentandNormalVectors.............................. 126 6.2.2 SurfaceArea...................................... 127 6.2.3 SurfaceIntegralsofScalars............................... 128 6.2.4 Mass,CenterofMassandMomentofInertia..................... 128 6.2.5 SurfaceIntegralsofVectors.............................. 131 6.2.6 FluxandExpansion.................................. 133 6.3Exercises............................................ 136 7 Vector Differential Operators 139 7.1TheDelOperatorandtheGradient............................... 139 7.2Divergence........................................... 139 7.2.1 Computation...................................... 139 7.2.2 Applications...................................... 142 7.3Curl............................................... 144 7.3.1 Computation...................................... 144 7.3.2 Applications...................................... 145 7.4 Higher Order Differential Operators and Identities . ...................... 146 7.4.1 LaplacianofaScalar.................................. 146 7.4.2 LaplacianofaVector.................................. 147 7.4.3 HessianofaScalar................................... 147 7.4.4 HigherOrderGradientsofScalars........................... 148 7.4.5 CurlofaGradient................................... 148 7.4.6 DivergenceofaCurl.................................. 149 7.4.7 Differential Identities . ................................. 150 7.5FindingPotentials........................................ 151 7.5.1 ScalarPotentials.................................... 152 7.5.2 VectorPotentials.................................... 153 7.6Exercises............................................ 154 8 Fundamental Theorems of Vector Calculus 156 8.1 Generalizing the Fundamental Theorem of Calculus ...................... 156 8.2 Fundamental Theorem of Calculus for Curves . ...................... 157 8.2.1 Verification....................................... 157 8.2.2 Applications...................................... 158 8.3Green’sTheorem........................................ 161 8.3.1 Verification....................................... 161 8.3.2 Applications...................................... 165 8.4Stokes’Theorem(TheCurlTheorem)............................. 167 8.4.1 Verification....................................... 167 8.4.2 Applications...................................... 168 8.5Gauss’Theorem(TheDivergenceTheorem).......................... 174 8.5.1 Verification....................................... 174 8.5.2 Applications...................................... 176 8.6RelatedLine,SurfaceandVolumeIntegrals.......................... 179 vi CONTENTS 8.6.1 RelatedLineandSurfaceIntegrals........................... 179 8.6.2 RelatedSurfaceandVolumeIntegrals......................... 182 8.7Exercises............................................ 185 9Labs 188 9.1Orienteering........................................... 189 9.2 Dot and Cross Products ..................................... 191 9.3 Lines, Planes, Quadric Curves and Quadric Surfaces . .................. 192 9.4ParametricCurves........................................ 193 9.5FrenetAnalysisofCurves................................... 195 9.6LinearandQuadraticApproximations............................. 197 9.7MultivariableMax-MinProblems............................... 198 9.8AVolumeofDesserts...................................... 200 9.9InterpretationoftheDivergence................................ 203 9.10InterpretationoftheCurl.................................... 204 9.11Gauss’Law........................................... 206 9.12Ampere’sLaw.......................................... 208 10 Projects 210 Projects on Vectors and Multivariable Differentiation ..................... 211 10.1TotalingGravitationalForces.................................. 211 10.2AnimateaCurve........................................ 212 10.3Newton’sMethodin2Dimensions............................... 212 10.4GradientMethodofFindingExtrema.............................. 213 10.5TheTrashDumpster...................................... 215 10.6LocatinganApartment..................................... 216 Projects on Multivariable Integration .............................. 216 10.7 p-Normed Spaceballs: The Area of a Unit p-NormedCircle.................. 216 10.8TheVolumeBetweenaSurfaceandItsTangentPlane..................... 218 10.9 Hyper-Spaceballs: The Hypervolume of a Hypersphere . .................. 218 10.10TheCenterofMassofPlanetX................................ 220 10.11 The Skimpy Donut . ..................................... 222 10.12 Steradian Measure . ..................................... 223 AThevec calc Package 225 A.1Acknowledgments....................................... 225 A.2DescriptionofthePackage................................... 225 A.3 Obtaining and Installing the Files . ............................