CalcIII_Complete - CALCULUS III Paul Dawkins Calculus III...

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CALCULUS III Paul Dawkins
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Calculus III © 2007 Paul Dawkins i http://tutorial.math.lamar.edu/terms.aspx Table of Contents Preface ........................................................................................................................................... iii Outline ........................................................................................................................................... iv Three Dimensional Space .............................................................................................................. 1 Introduction ................................................................................................................................................ 1 The 3-D Coordinate System ....................................................................................................................... 3 Equations of Lines ..................................................................................................................................... 9 Equations of Planes .................................................................................................................................. 15 Quadric Surfaces ...................................................................................................................................... 18 Functions of Several Variables ................................................................................................................ 24 Vector Functions ...................................................................................................................................... 31 Calculus with Vector Functions ............................................................................................................... 40 Tangent, Normal and Binormal Vectors .................................................................................................. 43 Arc Length with Vector Functions ........................................................................................................... 46 Curvature .................................................................................................................................................. 49 Velocity and Acceleration ........................................................................................................................ 51 Cylindrical Coordinates ........................................................................................................................... 54 Spherical Coordinates .............................................................................................................................. 56 Partial Derivatives ....................................................................................................................... 62 Introduction .............................................................................................................................................. 62 Limits ....................................................................................................................................................... 63 Partial Derivatives .................................................................................................................................... 68 Interpretations of Partial Derivatives ....................................................................................................... 77 Higher Order Partial Derivatives .............................................................................................................. 81 Differentials ............................................................................................................................................. 85 Chain Rule ............................................................................................................................................... 86 Directional Derivatives ............................................................................................................................ 96 Applications of Partial Derivatives .......................................................................................... 105 Introduction ............................................................................................................................................. 105 Tangent Planes and Linear Approximations ........................................................................................... 106 Gradient Vector, Tangent Planes and Normal Lines ............................................................................... 110 Relative Minimums and Maximums ....................................................................................................... 112 Absolute Minimums and Maximums ...................................................................................................... 121 Lagrange Multipliers ............................................................................................................................... 129 Multiple Integrals ...................................................................................................................... 139 Introduction ............................................................................................................................................. 139 Double Integrals ...................................................................................................................................... 140 Iterated Integrals ..................................................................................................................................... 144 Double Integrals Over General Regions ................................................................................................. 151 Double Integrals in Polar Coordinates .................................................................................................... 162 Triple Integrals ........................................................................................................................................ 173 Triple Integrals in Cylindrical Coordinates ............................................................................................. 181 Triple Integrals in Spherical Coordinates ................................................................................................ 184 Change of Variables ................................................................................................................................ 188 Surface Area ............................................................................................................................................ 197 Area and Volume Revisited .................................................................................................................... 200 Line Integrals ............................................................................................................................. 201 Introduction ............................................................................................................................................. 201 Vector Fields ........................................................................................................................................... 202 Line Integrals – Part I .............................................................................................................................. 207 Line Integrals – Part II ............................................................................................................................ 218 Line Integrals of Vector Fields ................................................................................................................ 221 Fundamental Theorem for Line Integrals ................................................................................................ 224 Conservative Vector Fields ..................................................................................................................... 228
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