Calculus III - Vector Functions

<!DOCTYPE html> <html> <head><meta charset="utf-8" /><meta name="viewport" content="width=device-width, initial-scale=1, user-scalable=yes" /><meta http-equiv="X-UA-Compatible" content="IE=edge" />  <meta name="google-site-verification" content="uLoA31CJfOhIVMJWBjCmQL8xNMmmLybZU3LRKavy9WQ" /><title> Calculus III - Vector Functions </title>  <script async src="https://www.googletagmanager.com/gtag/js?id=G-9SCXJM7BEJ"></script> <script> window.dataLayer = window.dataLayer || []; function gtag() { dataLayer.push(arguments); } gtag('js', new Date()); gtag('config', 'G-9SCXJM7BEJ'); </script> <link type="text/css" href="/css/jquery.mmenu.all.css" rel="stylesheet" /><link type="text/css" href="/css/jquery.dropdown.css" rel="stylesheet" /><link href="/FA/css/all.min.css" rel="stylesheet" /><link type="text/css" href="/css/notes-all.css" rel="stylesheet" /><link type="text/css" href="/css/notes-google.css" rel="stylesheet" /><link type="text/css" href="/css/notes-mmenu.css" rel="stylesheet" /><link type="text/css" href="/css/notes-dropdown.css" rel="stylesheet" /> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ TeX: { equationNumbers: { autoNumber: "AMS" } } }); </script> <script type="text/javascript" async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/MathJax.js?config=TeX-AMS_CHTML-full"></script> <script type="text/javascript" src="/js/jquery_on.js"></script> <script type="text/javascript" src="/js/jquery.mmenu.all.js"></script> <script type="text/javascript" src="/js/jquery.dropdown.js"></script> <script type="text/javascript" src="/js/notes-all.js"></script> <script> (function () { var cx = '001004262401526223570:11yv6vpcqvy'; var gcse = document.createElement('script'); gcse.type = 'text/javascript'; gcse.async = true; gcse.src = 'https://cse.google.com/cse.js?cx=' + cx; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(gcse, s); })(); </script> <meta http-equiv="keywords" name="keywords" content="vector functions, graph vector functions" /><meta http-equiv="description" name="description" content="In this section we introduce the concept of vector functions concentrating primarily on curves in three dimensional space. We will however, touch briefly on surfaces as well. We will illustrate how to find the domain of a vector function and how to graph a vector function. We will also show a simple relationship between vector functions and parametric equations that will be very useful at times." /></head> <body onload="init({Notes: 'NoteMobile;8/21/2018;true'})"> <div id="page"> <div class="header"> <div class="header-row">  <div id="side-menu-icon" class="header-side-menu-icon"><a href="#menu"><span class="fas fa-bars fa-lg" aria-hidden="true" title="Main Menu - Change between topics, chapters and sections as well as a few extra pages."></span></a></div> <span class="header-title"><a href="/" class="header-title-link">Paul's Online Notes</a></span> <div class="header-spacer"></div> <div id="content-top-menu" class="top-menu"> <button id="content-type-menu" class="top-menu-button" data-jq-dropdown="#jq-dropdown-type" title="View (Notes, Practice Problems or Assignment Problems, Show/Hide Solutions and/or Steps) Menu"> <span id="tab_top_menu_notes" class="top-menu-item-title">Notes</span> <span class="far fa-eye fa-lg" aria-hidden="true"></span> 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Graphing and Functions</a> <ul> <li><a href="/Classes/Alg/Graphing.aspx" class="mm-link">3.1 Graphing</a></li> <li><a href="/Classes/Alg/Lines.aspx" class="mm-link">3.2 Lines</a></li> <li><a href="/Classes/Alg/Circles.aspx" class="mm-link">3.3 Circles</a></li> <li><a href="/Classes/Alg/FunctionDefn.aspx" class="mm-link">3.4 The Definition of a Function</a></li> <li><a href="/Classes/Alg/GraphFunctions.aspx" class="mm-link">3.5 Graphing Functions</a></li> <li><a href="/Classes/Alg/CombineFunctions.aspx" class="mm-link">3.6 Combining Functions</a></li> <li><a href="/Classes/Alg/InverseFunctions.aspx" class="mm-link">3.7 Inverse Functions</a></li> </ul> </li> <li><a href="/Classes/Alg/CommonGraphs.aspx" class="mm-link">4. Common Graphs</a> <ul> <li><a href="/Classes/Alg/Lines_Circles_PWF.aspx" class="mm-link">4.1 Lines, Circles and Piecewise Functions</a></li> <li><a href="/Classes/Alg/Parabolas.aspx" class="mm-link">4.2 Parabolas</a></li> <li><a href="/Classes/Alg/Ellipses.aspx" class="mm-link">4.3 Ellipses</a></li> <li><a href="/Classes/Alg/Hyperbolas.aspx" class="mm-link">4.4 Hyperbolas</a></li> <li><a href="/Classes/Alg/MiscFunctions.aspx" class="mm-link">4.5 Miscellaneous Functions</a></li> <li><a href="/Classes/Alg/Transformations.aspx" class="mm-link">4.6 Transformations</a></li> <li><a href="/Classes/Alg/Symmetry.aspx" class="mm-link">4.7 Symmetry</a></li> <li><a href="/Classes/Alg/GraphRationalFcns.aspx" class="mm-link">4.8 Rational Functions</a></li> </ul> </li> <li><a href="/Classes/Alg/PolynomialFunctions.aspx" class="mm-link">5. Polynomial Functions</a> <ul> <li><a href="/Classes/Alg/DividingPolynomials.aspx" class="mm-link">5.1 Dividing Polynomials</a></li> <li><a href="/Classes/Alg/ZeroesOfPolynomials.aspx" class="mm-link">5.2 Zeroes/Roots of Polynomials</a></li> <li><a href="/Classes/Alg/GraphingPolynomials.aspx" class="mm-link">5.3 Graphing Polynomials</a></li> <li><a href="/Classes/Alg/FindingZeroesOfPolynomials.aspx" class="mm-link">5.4 Finding Zeroes of Polynomials</a></li> <li><a href="/Classes/Alg/PartialFractions.aspx" class="mm-link">5.5 Partial Fractions</a></li> </ul> </li> <li><a href="/Classes/Alg/ExpAndLog.aspx" class="mm-link">6. Exponential and Logarithm Functions</a> <ul> <li><a href="/Classes/Alg/ExpFunctions.aspx" class="mm-link">6.1 Exponential Functions</a></li> <li><a href="/Classes/Alg/LogFunctions.aspx" class="mm-link">6.2 Logarithm Functions</a></li> <li><a href="/Classes/Alg/SolveExpEqns.aspx" class="mm-link">6.3 Solving Exponential Equations</a></li> <li><a href="/Classes/Alg/SolveLogEqns.aspx" class="mm-link">6.4 Solving Logarithm Equations</a></li> <li><a href="/Classes/Alg/ExpLogApplications.aspx" class="mm-link">6.5 Applications</a></li> </ul> </li> <li><a href="/Classes/Alg/Systems.aspx" class="mm-link">7. Systems of Equations</a> <ul> <li><a href="/Classes/Alg/SystemsTwoVrble.aspx" class="mm-link">7.1 Linear Systems with Two Variables</a></li> <li><a href="/Classes/Alg/SystemsThreeVrble.aspx" class="mm-link">7.2 Linear Systems with Three Variables</a></li> <li><a href="/Classes/Alg/AugmentedMatrix.aspx" class="mm-link">7.3 Augmented Matrices</a></li> <li><a href="/Classes/Alg/AugmentedMatrixII.aspx" class="mm-link">7.4 More on the Augmented Matrix</a></li> <li><a href="/Classes/Alg/NonlinearSystems.aspx" class="mm-link">7.5 Nonlinear Systems</a></li> </ul> </li> </ul> </li> <li><a href="/Classes/CalcI/CalcI.aspx" class="mm-link">Calculus I</a> <ul> <li><a href="/Classes/CalcI/ReviewIntro.aspx" class="mm-link">1. Review</a> <ul> <li><a href="/Classes/CalcI/Functions.aspx" class="mm-link">1.1 Functions</a></li> <li><a href="/Classes/CalcI/InverseFunctions.aspx" class="mm-link">1.2 Inverse Functions</a></li> <li><a href="/Classes/CalcI/TrigFcns.aspx" class="mm-link">1.3 Trig Functions</a></li> <li><a href="/Classes/CalcI/TrigEquations.aspx" class="mm-link">1.4 Solving Trig Equations</a></li> <li><a href="/Classes/CalcI/TrigEquations_CalcI.aspx" class="mm-link">1.5 Trig Equations with Calculators, Part I</a></li> <li><a href="/Classes/CalcI/TrigEquations_CalcII.aspx" class="mm-link">1.6 Trig Equations with Calculators, Part II</a></li> <li><a href="/Classes/CalcI/ExpFunctions.aspx" class="mm-link">1.7 Exponential Functions</a></li> <li><a href="/Classes/CalcI/LogFcns.aspx" class="mm-link">1.8 Logarithm Functions</a></li> <li><a href="/Classes/CalcI/ExpLogEqns.aspx" class="mm-link">1.9 Exponential and Logarithm Equations</a></li> <li><a href="/Classes/CalcI/CommonGraphs.aspx" class="mm-link">1.10 Common Graphs</a></li> </ul> </li> <li><a href="/Classes/CalcI/limitsIntro.aspx" class="mm-link">2. Limits</a> <ul> <li><a href="/Classes/CalcI/Tangents_Rates.aspx" class="mm-link">2.1 Tangent Lines and Rates of Change</a></li> <li><a href="/Classes/CalcI/TheLimit.aspx" class="mm-link">2.2 The Limit</a></li> <li><a href="/Classes/CalcI/OneSidedLimits.aspx" class="mm-link">2.3 One-Sided Limits</a></li> <li><a href="/Classes/CalcI/LimitsProperties.aspx" class="mm-link">2.4 Limit Properties</a></li> <li><a href="/Classes/CalcI/ComputingLimits.aspx" class="mm-link">2.5 Computing Limits</a></li> <li><a href="/Classes/CalcI/InfiniteLimits.aspx" class="mm-link">2.6 Infinite Limits</a></li> <li><a href="/Classes/CalcI/LimitsAtInfinityI.aspx" class="mm-link">2.7 Limits At Infinity, Part I</a></li> <li><a href="/Classes/CalcI/LimitsAtInfinityII.aspx" class="mm-link">2.8 Limits At Infinity, Part II</a></li> <li><a href="/Classes/CalcI/Continuity.aspx" class="mm-link">2.9 Continuity</a></li> <li><a href="/Classes/CalcI/DefnOfLimit.aspx" class="mm-link">2.10 The Definition of the Limit</a></li> </ul> </li> <li><a href="/Classes/CalcI/DerivativeIntro.aspx" class="mm-link">3. Derivatives</a> <ul> <li><a href="/Classes/CalcI/DefnOfDerivative.aspx" class="mm-link">3.1 The Definition of the Derivative</a></li> <li><a href="/Classes/CalcI/DerivativeInterp.aspx" class="mm-link">3.2 Interpretation of the Derivative</a></li> <li><a href="/Classes/CalcI/DiffFormulas.aspx" class="mm-link">3.3 Differentiation Formulas</a></li> <li><a href="/Classes/CalcI/ProductQuotientRule.aspx" class="mm-link">3.4 Product and Quotient Rule</a></li> <li><a href="/Classes/CalcI/DiffTrigFcns.aspx" class="mm-link">3.5 Derivatives of Trig Functions</a></li> <li><a href="/Classes/CalcI/DiffExpLogFcns.aspx" class="mm-link">3.6 Derivatives of Exponential and Logarithm Functions</a></li> <li><a href="/Classes/CalcI/DiffInvTrigFcns.aspx" class="mm-link">3.7 Derivatives of Inverse Trig Functions</a></li> <li><a href="/Classes/CalcI/DiffHyperFcns.aspx" class="mm-link">3.8 Derivatives of Hyperbolic Functions</a></li> <li><a href="/Classes/CalcI/ChainRule.aspx" class="mm-link">3.9 Chain Rule</a></li> <li><a href="/Classes/CalcI/ImplicitDIff.aspx" class="mm-link">3.10 Implicit Differentiation</a></li> <li><a href="/Classes/CalcI/RelatedRates.aspx" class="mm-link">3.11 Related Rates</a></li> <li><a href="/Classes/CalcI/HigherOrderDerivatives.aspx" class="mm-link">3.12 Higher Order Derivatives</a></li> <li><a href="/Classes/CalcI/LogDiff.aspx" class="mm-link">3.13 Logarithmic Differentiation</a></li> </ul> </li> <li><a href="/Classes/CalcI/DerivAppsIntro.aspx" class="mm-link">4. Applications of Derivatives</a> <ul> <li><a href="/Classes/CalcI/RateOfChange.aspx" class="mm-link">4.1 Rates of Change</a></li> <li><a href="/Classes/CalcI/CriticalPoints.aspx" class="mm-link">4.2 Critical Points</a></li> <li><a href="/Classes/CalcI/MinMaxValues.aspx" class="mm-link">4.3 Minimum and Maximum Values</a></li> <li><a href="/Classes/CalcI/AbsExtrema.aspx" class="mm-link">4.4 Finding Absolute Extrema</a></li> <li><a href="/Classes/CalcI/ShapeofGraphPtI.aspx" class="mm-link">4.5 The Shape of a Graph, Part I</a></li> <li><a href="/Classes/CalcI/ShapeofGraphPtII.aspx" class="mm-link">4.6 The Shape of a Graph, Part II</a></li> <li><a href="/Classes/CalcI/MeanValueTheorem.aspx" class="mm-link">4.7 The Mean Value Theorem</a></li> <li><a href="/Classes/CalcI/Optimization.aspx" class="mm-link">4.8 Optimization</a></li> <li><a href="/Classes/CalcI/MoreOptimization.aspx" class="mm-link">4.9 More Optimization Problems</a></li> <li><a href="/Classes/CalcI/LHospitalsRule.aspx" class="mm-link">4.10 L'Hospital's Rule and Indeterminate Forms</a></li> <li><a href="/Classes/CalcI/LinearApproximations.aspx" class="mm-link">4.11 Linear Approximations</a></li> <li><a href="/Classes/CalcI/Differentials.aspx" class="mm-link">4.12 Differentials</a></li> <li><a href="/Classes/CalcI/NewtonsMethod.aspx" class="mm-link">4.13 Newton's Method</a></li> <li><a href="/Classes/CalcI/BusinessApps.aspx" class="mm-link">4.14 Business Applications</a></li> </ul> </li> <li><a href="/Classes/CalcI/IntegralsIntro.aspx" class="mm-link">5. Integrals</a> <ul> <li><a href="/Classes/CalcI/IndefiniteIntegrals.aspx" class="mm-link">5.1 Indefinite Integrals</a></li> <li><a href="/Classes/CalcI/ComputingIndefiniteIntegrals.aspx" class="mm-link">5.2 Computing Indefinite Integrals</a></li> <li><a href="/Classes/CalcI/SubstitutionRuleIndefinite.aspx" class="mm-link">5.3 Substitution Rule for Indefinite Integrals</a></li> <li><a href="/Classes/CalcI/SubstitutionRuleIndefinitePtII.aspx" class="mm-link">5.4 More Substitution Rule</a></li> <li><a href="/Classes/CalcI/AreaProblem.aspx" class="mm-link">5.5 Area Problem</a></li> <li><a href="/Classes/CalcI/DefnOfDefiniteIntegral.aspx" class="mm-link">5.6 Definition of the Definite Integral</a></li> <li><a href="/Classes/CalcI/ComputingDefiniteIntegrals.aspx" class="mm-link">5.7 Computing Definite Integrals</a></li> <li><a href="/Classes/CalcI/SubstitutionRuleDefinite.aspx" class="mm-link">5.8 Substitution Rule for Definite Integrals</a></li> </ul> </li> <li><a href="/Classes/CalcI/IntAppsIntro.aspx" class="mm-link">6. Applications of Integrals</a> <ul> <li><a href="/Classes/CalcI/AvgFcnValue.aspx" class="mm-link">6.1 Average Function Value</a></li> <li><a href="/Classes/CalcI/AreaBetweenCurves.aspx" class="mm-link">6.2 Area Between Curves</a></li> <li><a href="/Classes/CalcI/VolumeWithRings.aspx" class="mm-link">6.3 Volumes of Solids of Revolution / Method of Rings</a></li> <li><a href="/Classes/CalcI/VolumeWithCylinder.aspx" class="mm-link">6.4 Volumes of Solids of Revolution/Method of Cylinders</a></li> <li><a href="/Classes/CalcI/MoreVolume.aspx" class="mm-link">6.5 More Volume Problems</a></li> <li><a href="/Classes/CalcI/Work.aspx" class="mm-link">6.6 Work</a></li> </ul> </li> <li><a href="/Classes/CalcI/ExtrasIntro.aspx" class="mm-link">Appendix A. Extras</a> <ul> <li><a href="/Classes/CalcI/LimitProofs.aspx" class="mm-link">A.1 Proof of Various Limit Properties</a></li> <li><a href="/Classes/CalcI/DerivativeProofs.aspx" class="mm-link">A.2 Proof of Various Derivative Properties</a></li> <li><a href="/Classes/CalcI/ProofTrigDeriv.aspx" class="mm-link">A.3 Proof of Trig Limits</a></li> <li><a href="/Classes/CalcI/DerivativeAppsProofs.aspx" class="mm-link">A.4 Proofs of Derivative Applications Facts</a></li> <li><a href="/Classes/CalcI/ProofIntProp.aspx" class="mm-link">A.5 Proof of Various Integral Properties </a></li> <li><a href="/Classes/CalcI/Area_Volume_Formulas.aspx" class="mm-link">A.6 Area and Volume Formulas</a></li> <li><a href="/Classes/CalcI/TypesOfInfinity.aspx" class="mm-link">A.7 Types of Infinity</a></li> <li><a href="/Classes/CalcI/SummationNotation.aspx" class="mm-link">A.8 Summation Notation</a></li> <li><a href="/Classes/CalcI/ConstantofIntegration.aspx" class="mm-link">A.9 Constant of Integration</a></li> </ul> </li> </ul> </li> <li><a href="/Classes/CalcII/CalcII.aspx" class="mm-link">Calculus II</a> <ul> <li><a href="/Classes/CalcII/IntTechIntro.aspx" class="mm-link">7. Integration Techniques</a> <ul> <li><a href="/Classes/CalcII/IntegrationByParts.aspx" class="mm-link">7.1 Integration by Parts</a></li> <li><a href="/Classes/CalcII/IntegralsWithTrig.aspx" class="mm-link">7.2 Integrals Involving Trig Functions</a></li> <li><a href="/Classes/CalcII/TrigSubstitutions.aspx" class="mm-link">7.3 Trig Substitutions</a></li> <li><a href="/Classes/CalcII/PartialFractions.aspx" class="mm-link">7.4 Partial Fractions</a></li> <li><a href="/Classes/CalcII/IntegralsWithRoots.aspx" class="mm-link">7.5 Integrals Involving Roots</a></li> <li><a href="/Classes/CalcII/IntegralsWithQuadratics.aspx" class="mm-link">7.6 Integrals Involving Quadratics</a></li> <li><a href="/Classes/CalcII/IntegrationStrategy.aspx" class="mm-link">7.7 Integration Strategy</a></li> <li><a href="/Classes/CalcII/ImproperIntegrals.aspx" class="mm-link">7.8 Improper Integrals</a></li> <li><a href="/Classes/CalcII/ImproperIntegralsCompTest.aspx" class="mm-link">7.9 Comparison Test for Improper Integrals</a></li> <li><a href="/Classes/CalcII/ApproximatingDefIntegrals.aspx" class="mm-link">7.10 Approximating Definite Integrals</a></li> </ul> </li> <li><a href="/Classes/CalcII/IntAppsIntro.aspx" class="mm-link">8. Applications of Integrals</a> <ul> <li><a href="/Classes/CalcII/ArcLength.aspx" class="mm-link">8.1 Arc Length</a></li> <li><a href="/Classes/CalcII/SurfaceArea.aspx" class="mm-link">8.2 Surface Area</a></li> <li><a href="/Classes/CalcII/CenterOfMass.aspx" class="mm-link">8.3 Center of Mass</a></li> <li><a href="/Classes/CalcII/HydrostaticPressure.aspx" class="mm-link">8.4 Hydrostatic Pressure</a></li> <li><a href="/Classes/CalcII/Probability.aspx" class="mm-link">8.5 Probability</a></li> </ul> </li> <li><a href="/Classes/CalcII/ParametricIntro.aspx" class="mm-link">9. Parametric Equations and Polar Coordinates</a> <ul> <li><a href="/Classes/CalcII/ParametricEqn.aspx" class="mm-link">9.1 Parametric Equations and Curves</a></li> <li><a href="/Classes/CalcII/ParaTangent.aspx" class="mm-link">9.2 Tangents with Parametric Equations</a></li> <li><a href="/Classes/CalcII/ParaArea.aspx" class="mm-link">9.3 Area with Parametric Equations</a></li> <li><a href="/Classes/CalcII/ParaArcLength.aspx" class="mm-link">9.4 Arc Length with Parametric Equations</a></li> <li><a href="/Classes/CalcII/ParaSurfaceArea.aspx" class="mm-link">9.5 Surface Area with Parametric Equations</a></li> <li><a href="/Classes/CalcII/PolarCoordinates.aspx" class="mm-link">9.6 Polar Coordinates</a></li> <li><a href="/Classes/CalcII/PolarTangents.aspx" class="mm-link">9.7 Tangents with Polar Coordinates</a></li> <li><a href="/Classes/CalcII/PolarArea.aspx" class="mm-link">9.8 Area with Polar Coordinates</a></li> <li><a href="/Classes/CalcII/PolarArcLength.aspx" class="mm-link">9.9 Arc Length with Polar Coordinates</a></li> <li><a href="/Classes/CalcII/PolarSurfaceArea.aspx" class="mm-link">9.10 Surface Area with Polar Coordinates</a></li> <li><a href="/Classes/CalcII/ArcLength_SurfaceArea.aspx" class="mm-link">9.11 Arc Length and Surface Area Revisited</a></li> </ul> </li> <li><a href="/Classes/CalcII/SeriesIntro.aspx" class="mm-link">10. Series & Sequences</a> <ul> <li><a href="/Classes/CalcII/Sequences.aspx" class="mm-link">10.1 Sequences</a></li> <li><a href="/Classes/CalcII/MoreSequences.aspx" class="mm-link">10.2 More on Sequences</a></li> <li><a href="/Classes/CalcII/Series_Basics.aspx" class="mm-link">10.3 Series - The Basics</a></li> <li><a href="/Classes/CalcII/ConvergenceOfSeries.aspx" class="mm-link">10.4 Convergence/Divergence of Series</a></li> <li><a href="/Classes/CalcII/Series_Special.aspx" class="mm-link">10.5 Special Series</a></li> <li><a href="/Classes/CalcII/IntegralTest.aspx" class="mm-link">10.6 Integral Test</a></li> <li><a href="/Classes/CalcII/SeriesCompTest.aspx" class="mm-link">10.7 Comparison Test/Limit Comparison Test</a></li> <li><a href="/Classes/CalcII/AlternatingSeries.aspx" class="mm-link">10.8 Alternating Series Test</a></li> <li><a href="/Classes/CalcII/AbsoluteConvergence.aspx" class="mm-link">10.9 Absolute Convergence</a></li> <li><a href="/Classes/CalcII/RatioTest.aspx" class="mm-link">10.10 Ratio Test</a></li> <li><a href="/Classes/CalcII/RootTest.aspx" class="mm-link">10.11 Root Test</a></li> <li><a href="/Classes/CalcII/SeriesStrategy.aspx" class="mm-link">10.12 Strategy for Series</a></li> <li><a href="/Classes/CalcII/EstimatingSeries.aspx" class="mm-link">10.13 Estimating the Value of a Series</a></li> <li><a href="/Classes/CalcII/PowerSeries.aspx" class="mm-link">10.14 Power Series</a></li> <li><a href="/Classes/CalcII/PowerSeriesandFunctions.aspx" class="mm-link">10.15 Power Series and Functions</a></li> <li><a href="/Classes/CalcII/TaylorSeries.aspx" class="mm-link">10.16 Taylor Series</a></li> <li><a href="/Classes/CalcII/TaylorSeriesApps.aspx" class="mm-link">10.17 Applications of Series</a></li> <li><a href="/Classes/CalcII/BinomialSeries.aspx" class="mm-link">10.18 Binomial Series</a></li> </ul> </li> <li><a href="/Classes/CalcII/VectorsIntro.aspx" class="mm-link">11. Vectors</a> <ul> <li><a href="/Classes/CalcII/Vectors_Basics.aspx" class="mm-link">11.1 Vectors - The Basics</a></li> <li><a href="/Classes/CalcII/VectorArithmetic.aspx" class="mm-link">11.2 Vector Arithmetic</a></li> <li><a href="/Classes/CalcII/DotProduct.aspx" class="mm-link">11.3 Dot Product</a></li> <li><a href="/Classes/CalcII/CrossProduct.aspx" class="mm-link">11.4 Cross Product</a></li> </ul> </li> <li><a href="/Classes/CalcII/3DSpace.aspx" class="mm-link">12. 3-Dimensional Space</a> <ul> <li><a href="/Classes/CalcII/3DCoords.aspx" class="mm-link">12.1 The 3-D Coordinate System</a></li> <li><a href="/Classes/CalcII/EqnsOfLines.aspx" class="mm-link">12.2 Equations of Lines</a></li> <li><a href="/Classes/CalcII/EqnsOfPlanes.aspx" class="mm-link">12.3 Equations of Planes</a></li> <li><a href="/Classes/CalcII/QuadricSurfaces.aspx" class="mm-link">12.4 Quadric Surfaces</a></li> <li><a href="/Classes/CalcII/MultiVrbleFcns.aspx" class="mm-link">12.5 Functions of Several Variables</a></li> <li><a href="/Classes/CalcII/VectorFunctions.aspx" class="mm-link">12.6 Vector Functions</a></li> <li><a href="/Classes/CalcII/VectorFcnsCalculus.aspx" class="mm-link">12.7 Calculus with Vector Functions</a></li> <li><a href="/Classes/CalcII/TangentNormalVectors.aspx" class="mm-link">12.8 Tangent, Normal and Binormal Vectors</a></li> <li><a href="/Classes/CalcII/VectorArcLength.aspx" class="mm-link">12.9 Arc Length with Vector Functions</a></li> <li><a href="/Classes/CalcII/Curvature.aspx" class="mm-link">12.10 Curvature</a></li> <li><a href="/Classes/CalcII/Velocity_Acceleration.aspx" class="mm-link">12.11 Velocity and Acceleration</a></li> <li><a href="/Classes/CalcII/CylindricalCoords.aspx" class="mm-link">12.12 Cylindrical Coordinates</a></li> <li><a href="/Classes/CalcII/SphericalCoords.aspx" class="mm-link">12.13 Spherical Coordinates</a></li> </ul> </li> </ul> </li> <li><a href="/Classes/CalcIII/CalcIII.aspx" class="mm-link">Calculus III</a> <ul> <li><a href="/Classes/CalcIII/3DSpace.aspx" class="mm-link">12. 3-Dimensional Space</a> <ul> <li><a href="/Classes/CalcIII/3DCoords.aspx" class="mm-link">12.1 The 3-D Coordinate System</a></li> <li><a href="/Classes/CalcIII/EqnsOfLines.aspx" class="mm-link">12.2 Equations of Lines</a></li> <li><a href="/Classes/CalcIII/EqnsOfPlanes.aspx" class="mm-link">12.3 Equations of Planes</a></li> <li><a href="/Classes/CalcIII/QuadricSurfaces.aspx" class="mm-link">12.4 Quadric Surfaces</a></li> <li><a href="/Classes/CalcIII/MultiVrbleFcns.aspx" class="mm-link">12.5 Functions of Several Variables</a></li> <li><a href="/Classes/CalcIII/VectorFunctions.aspx" class="mm-link">12.6 Vector Functions</a></li> <li><a href="/Classes/CalcIII/VectorFcnsCalculus.aspx" class="mm-link">12.7 Calculus with Vector Functions</a></li> <li><a href="/Classes/CalcIII/TangentNormalVectors.aspx" class="mm-link">12.8 Tangent, Normal and Binormal Vectors</a></li> <li><a href="/Classes/CalcIII/VectorArcLength.aspx" class="mm-link">12.9 Arc Length with Vector Functions</a></li> <li><a href="/Classes/CalcIII/Curvature.aspx" class="mm-link">12.10 Curvature</a></li> <li><a href="/Classes/CalcIII/Velocity_Acceleration.aspx" class="mm-link">12.11 Velocity and Acceleration</a></li> <li><a href="/Classes/CalcIII/CylindricalCoords.aspx" class="mm-link">12.12 Cylindrical Coordinates</a></li> <li><a href="/Classes/CalcIII/SphericalCoords.aspx" class="mm-link">12.13 Spherical Coordinates</a></li> </ul> </li> <li><a href="/Classes/CalcIII/PartialDerivsIntro.aspx" class="mm-link">13. Partial Derivatives</a> <ul> <li><a href="/Classes/CalcIII/Limits.aspx" class="mm-link">13.1 Limits</a></li> <li><a href="/Classes/CalcIII/PartialDerivatives.aspx" class="mm-link">13.2 Partial Derivatives</a></li> <li><a href="/Classes/CalcIII/PartialDerivInterp.aspx" class="mm-link">13.3 Interpretations of Partial Derivatives</a></li> <li><a href="/Classes/CalcIII/HighOrderPartialDerivs.aspx" class="mm-link">13.4 Higher Order Partial Derivatives</a></li> <li><a href="/Classes/CalcIII/Differentials.aspx" class="mm-link">13.5 Differentials</a></li> <li><a href="/Classes/CalcIII/ChainRule.aspx" class="mm-link">13.6 Chain Rule</a></li> <li><a href="/Classes/CalcIII/DirectionalDeriv.aspx" class="mm-link">13.7 Directional Derivatives</a></li> </ul> </li> <li><a href="/Classes/CalcIII/PartialDerivAppsIntro.aspx" class="mm-link">14. Applications of Partial Derivatives</a> <ul> <li><a href="/Classes/CalcIII/TangentPlanes.aspx" class="mm-link">14.1 Tangent Planes and Linear Approximations</a></li> <li><a href="/Classes/CalcIII/GradientVectorTangentPlane.aspx" class="mm-link">14.2 Gradient Vector, Tangent Planes and Normal Lines</a></li> <li><a href="/Classes/CalcIII/RelativeExtrema.aspx" class="mm-link">14.3 Relative Minimums and Maximums</a></li> <li><a href="/Classes/CalcIII/AbsoluteExtrema.aspx" class="mm-link">14.4 Absolute Minimums and Maximums</a></li> <li><a href="/Classes/CalcIII/LagrangeMultipliers.aspx" class="mm-link">14.5 Lagrange Multipliers</a></li> </ul> </li> <li><a href="/Classes/CalcIII/MultipleIntegralsIntro.aspx" class="mm-link">15. Multiple Integrals</a> <ul> <li><a href="/Classes/CalcIII/DoubleIntegrals.aspx" class="mm-link">15.1 Double Integrals</a></li> <li><a href="/Classes/CalcIII/IteratedIntegrals.aspx" class="mm-link">15.2 Iterated Integrals</a></li> <li><a href="/Classes/CalcIII/DIGeneralRegion.aspx" class="mm-link">15.3 Double Integrals over General Regions</a></li> <li><a href="/Classes/CalcIII/DIPolarCoords.aspx" class="mm-link">15.4 Double Integrals in Polar Coordinates</a></li> <li><a href="/Classes/CalcIII/TripleIntegrals.aspx" class="mm-link">15.5 Triple Integrals</a></li> <li><a href="/Classes/CalcIII/TICylindricalCoords.aspx" class="mm-link">15.6 Triple Integrals in Cylindrical Coordinates</a></li> <li><a href="/Classes/CalcIII/TISphericalCoords.aspx" class="mm-link">15.7 Triple Integrals in Spherical Coordinates</a></li> <li><a href="/Classes/CalcIII/ChangeOfVariables.aspx" class="mm-link">15.8 Change of Variables</a></li> <li><a href="/Classes/CalcIII/SurfaceArea.aspx" class="mm-link">15.9 Surface Area</a></li> <li><a href="/Classes/CalcIII/Area_Volume.aspx" class="mm-link">15.10 Area and Volume Revisited</a></li> </ul> </li> <li><a href="/Classes/CalcIII/LineIntegralsIntro.aspx" class="mm-link">16. Line Integrals</a> <ul> <li><a href="/Classes/CalcIII/VectorFields.aspx" class="mm-link">16.1 Vector Fields</a></li> <li><a href="/Classes/CalcIII/LineIntegralsPtI.aspx" class="mm-link">16.2 Line Integrals - Part I</a></li> <li><a href="/Classes/CalcIII/LineIntegralsPtII.aspx" class="mm-link">16.3 Line Integrals - Part II</a></li> <li><a href="/Classes/CalcIII/LineIntegralsVectorFields.aspx" class="mm-link">16.4 Line Integrals of Vector Fields</a></li> <li><a href="/Classes/CalcIII/FundThmLineIntegrals.aspx" class="mm-link">16.5 Fundamental Theorem for Line Integrals</a></li> <li><a href="/Classes/CalcIII/ConservativeVectorField.aspx" class="mm-link">16.6 Conservative Vector Fields</a></li> <li><a href="/Classes/CalcIII/GreensTheorem.aspx" class="mm-link">16.7 Green's Theorem</a></li> </ul> </li> <li><a href="/Classes/CalcIII/SurfaceIntegralsIntro.aspx" class="mm-link">17.Surface Integrals</a> <ul> <li><a href="/Classes/CalcIII/CurlDivergence.aspx" class="mm-link">17.1 Curl and Divergence</a></li> <li><a href="/Classes/CalcIII/ParametricSurfaces.aspx" class="mm-link">17.2 Parametric Surfaces</a></li> <li><a href="/Classes/CalcIII/SurfaceIntegrals.aspx" class="mm-link">17.3 Surface Integrals</a></li> <li><a href="/Classes/CalcIII/SurfIntVectorField.aspx" class="mm-link">17.4 Surface Integrals of Vector Fields</a></li> <li><a href="/Classes/CalcIII/StokesTheorem.aspx" class="mm-link">17.5 Stokes' Theorem</a></li> <li><a href="/Classes/CalcIII/DivergenceTheorem.aspx" class="mm-link">17.6 Divergence Theorem</a></li> </ul> </li> </ul> </li> <li><a href="/Classes/DE/DE.aspx" class="mm-link">Differential Equations</a> <ul> <li><a href="/Classes/DE/IntroBasic.aspx" class="mm-link">1. Basic Concepts</a> <ul> <li><a href="/Classes/DE/Definitions.aspx" class="mm-link">1.1 Definitions</a></li> <li><a href="/Classes/DE/DirectionFields.aspx" class="mm-link">1.2 Direction Fields</a></li> <li><a href="/Classes/DE/FinalThoughts.aspx" class="mm-link">1.3 Final Thoughts</a></li> </ul> </li> <li><a href="/Classes/DE/IntroFirstOrder.aspx" class="mm-link">2. First Order DE's</a> <ul> <li><a href="/Classes/DE/Linear.aspx" class="mm-link">2.1 Linear Equations</a></li> <li><a href="/Classes/DE/Separable.aspx" class="mm-link">2.2 Separable Equations</a></li> <li><a href="/Classes/DE/Exact.aspx" class="mm-link">2.3 Exact Equations</a></li> <li><a href="/Classes/DE/Bernoulli.aspx" class="mm-link">2.4 Bernoulli Differential Equations</a></li> <li><a href="/Classes/DE/Substitutions.aspx" class="mm-link">2.5 Substitutions</a></li> <li><a href="/Classes/DE/IoV.aspx" class="mm-link">2.6 Intervals of Validity</a></li> <li><a href="/Classes/DE/Modeling.aspx" class="mm-link">2.7 Modeling with First Order DE's</a></li> <li><a href="/Classes/DE/EquilibriumSolutions.aspx" class="mm-link">2.8 Equilibrium Solutions</a></li> <li><a href="/Classes/DE/EulersMethod.aspx" class="mm-link">2.9 Euler's Method</a></li> </ul> </li> <li><a href="/Classes/DE/IntroSecondOrder.aspx" class="mm-link">3. Second Order DE's</a> <ul> <li><a href="/Classes/DE/SecondOrderConcepts.aspx" class="mm-link">3.1 Basic Concepts</a></li> <li><a href="/Classes/DE/RealRoots.aspx" class="mm-link">3.2 Real & Distinct Roots</a></li> <li><a href="/Classes/DE/ComplexRoots.aspx" class="mm-link">3.3 Complex Roots</a></li> <li><a href="/Classes/DE/RepeatedRoots.aspx" class="mm-link">3.4 Repeated Roots</a></li> <li><a href="/Classes/DE/ReductionofOrder.aspx" class="mm-link">3.5 Reduction of Order</a></li> <li><a href="/Classes/DE/FundamentalSetsofSolutions.aspx" class="mm-link">3.6 Fundamental Sets of Solutions</a></li> <li><a href="/Classes/DE/Wronskian.aspx" class="mm-link">3.7 More on the Wronskian</a></li> <li><a href="/Classes/DE/NonhomogeneousDE.aspx" class="mm-link">3.8 Nonhomogeneous Differential Equations</a></li> <li><a href="/Classes/DE/UndeterminedCoefficients.aspx" class="mm-link">3.9 Undetermined Coefficients</a></li> <li><a href="/Classes/DE/VariationofParameters.aspx" class="mm-link">3.10 Variation of Parameters</a></li> <li><a href="/Classes/DE/Vibrations.aspx" class="mm-link">3.11 Mechanical Vibrations</a></li> </ul> </li> <li><a href="/Classes/DE/LaplaceIntro.aspx" class="mm-link">4. Laplace Transforms</a> <ul> <li><a href="/Classes/DE/LaplaceDefinition.aspx" class="mm-link">4.1 The Definition</a></li> <li><a href="/Classes/DE/LaplaceTransforms.aspx" class="mm-link">4.2 Laplace Transforms</a></li> <li><a href="/Classes/DE/InverseTransforms.aspx" class="mm-link">4.3 Inverse Laplace Transforms</a></li> <li><a href="/Classes/DE/StepFunctions.aspx" class="mm-link">4.4 Step Functions</a></li> <li><a href="/Classes/DE/IVPWithLaplace.aspx" class="mm-link">4.5 Solving IVP's with Laplace Transforms</a></li> <li><a href="/Classes/DE/IVPWithNonConstantCoefficient.aspx" class="mm-link">4.6 Nonconstant Coefficient IVP's</a></li> <li><a href="/Classes/DE/IVPWithStepFunction.aspx" class="mm-link">4.7 IVP's With Step Functions</a></li> <li><a href="/Classes/DE/DiracDeltaFunction.aspx" class="mm-link">4.8 Dirac Delta Function</a></li> <li><a href="/Classes/DE/ConvolutionIntegrals.aspx" class="mm-link">4.9 Convolution Integrals</a></li> <li><a href="/Classes/DE/Laplace_Table.aspx" class="mm-link">4.10 Table Of Laplace Transforms</a></li> </ul> </li> <li><a href="/Classes/DE/SystemsIntro.aspx" class="mm-link">5. Systems of DE's</a> <ul> <li><a href="/Classes/DE/LA_Systems.aspx" class="mm-link">5.1 Review : Systems of Equations</a></li> <li><a href="/Classes/DE/LA_Matrix.aspx" class="mm-link">5.2 Review : Matrices & Vectors</a></li> <li><a href="/Classes/DE/LA_Eigen.aspx" class="mm-link">5.3 Review : Eigenvalues & Eigenvectors</a></li> <li><a href="/Classes/DE/SystemsDE.aspx" class="mm-link">5.4 Systems of Differential Equations</a></li> <li><a href="/Classes/DE/SolutionsToSystems.aspx" class="mm-link">5.5 Solutions to Systems</a></li> <li><a href="/Classes/DE/PhasePlane.aspx" class="mm-link">5.6 Phase Plane</a></li> <li><a href="/Classes/DE/RealEigenvalues.aspx" class="mm-link">5.7 Real Eigenvalues</a></li> <li><a href="/Classes/DE/ComplexEigenvalues.aspx" class="mm-link">5.8 Complex Eigenvalues</a></li> <li><a href="/Classes/DE/RepeatedEigenvalues.aspx" class="mm-link">5.9 Repeated Eigenvalues</a></li> <li><a href="/Classes/DE/NonhomogeneousSystems.aspx" class="mm-link">5.10 Nonhomogeneous Systems</a></li> <li><a href="/Classes/DE/SystemsLaplace.aspx" class="mm-link">5.11 Laplace Transforms</a></li> <li><a href="/Classes/DE/SystemsModeling.aspx" class="mm-link">5.12 Modeling</a></li> </ul> </li> <li><a href="/Classes/DE/SeriesIntro.aspx" class="mm-link">6. Series Solutions to DE's</a> <ul> <li><a href="/Classes/DE/PowerSeries.aspx" class="mm-link">6.1 Review : Power Series</a></li> <li><a href="/Classes/DE/TaylorSeries.aspx" class="mm-link">6.2 Review : Taylor Series</a></li> <li><a href="/Classes/DE/SeriesSolutions.aspx" class="mm-link">6.3 Series Solutions</a></li> <li><a href="/Classes/DE/EulerEquations.aspx" class="mm-link">6.4 Euler Equations</a></li> </ul> </li> <li><a href="/Classes/DE/IntroHigherOrder.aspx" class="mm-link">7. Higher Order Differential Equations</a> <ul> <li><a href="/Classes/DE/HOBasicConcepts.aspx" class="mm-link">7.1 Basic Concepts for <em>n</em><sup>th</sup> Order Linear Equations</a></li> <li><a href="/Classes/DE/HOHomogeneousDE.aspx" class="mm-link">7.2 Linear Homogeneous Differential Equations</a></li> <li><a href="/Classes/DE/HOUndeterminedCoeff.aspx" class="mm-link">7.3 Undetermined Coefficients</a></li> <li><a href="/Classes/DE/HOVariationOfParam.aspx" class="mm-link">7.4 Variation of Parameters</a></li> <li><a href="/Classes/DE/HOLaplaceTransforms.aspx" class="mm-link">7.5 Laplace Transforms</a></li> <li><a href="/Classes/DE/HOSystems.aspx" class="mm-link">7.6 Systems of Differential Equations</a></li> <li><a href="/Classes/DE/HOSeries.aspx" class="mm-link">7.7 Series Solutions</a></li> </ul> </li> <li><a href="/Classes/DE/IntroBVP.aspx" class="mm-link">8. Boundary Value Problems & Fourier Series</a> <ul> <li><a href="/Classes/DE/BoundaryValueProblem.aspx" class="mm-link">8.1 Boundary Value Problems</a></li> <li><a href="/Classes/DE/BVPEvals.aspx" class="mm-link">8.2 Eigenvalues and Eigenfunctions</a></li> <li><a href="/Classes/DE/PeriodicOrthogonal.aspx" class="mm-link">8.3 Periodic Functions & Orthogonal Functions</a></li> <li><a href="/Classes/DE/FourierSineSeries.aspx" class="mm-link">8.4 Fourier Sine Series</a></li> <li><a href="/Classes/DE/FourierCosineSeries.aspx" class="mm-link">8.5 Fourier Cosine Series</a></li> <li><a href="/Classes/DE/FourierSeries.aspx" class="mm-link">8.6 Fourier Series</a></li> <li><a href="/Classes/DE/ConvergenceFourierSeries.aspx" class="mm-link">8.7 Convergence of Fourier Series</a></li> </ul> </li> <li><a href="/Classes/DE/IntroPDE.aspx" class="mm-link">9. Partial Differential Equations </a> <ul> <li><a href="/Classes/DE/TheHeatEquation.aspx" class="mm-link">9.1 The Heat Equation</a></li> <li><a href="/Classes/DE/TheWaveEquation.aspx" class="mm-link">9.2 The Wave Equation</a></li> <li><a href="/Classes/DE/PDETerminology.aspx" class="mm-link">9.3 Terminology</a></li> <li><a href="/Classes/DE/SeparationofVariables.aspx" class="mm-link">9.4 Separation of Variables</a></li> <li><a href="/Classes/DE/SolvingHeatEquation.aspx" class="mm-link">9.5 Solving the Heat Equation</a></li> <li><a href="/Classes/DE/HeatEqnNonZero.aspx" class="mm-link">9.6 Heat Equation with Non-Zero Temperature Boundaries</a></li> <li><a href="/Classes/DE/LaplacesEqn.aspx" class="mm-link">9.7 Laplace's Equation</a></li> <li><a href="/Classes/DE/VibratingString.aspx" class="mm-link">9.8 Vibrating String</a></li> <li><a href="/Classes/DE/PDESummary.aspx" class="mm-link">9.9 Summary of Separation of Variables</a></li> </ul> </li> </ul> </li> <li><span>Extras</span></li> <li><a href="/Extras/AlgebraTrigReview/AlgebraTrig.aspx" class="mm-link">Algebra & Trig Review</a> <ul> <li><a href="/Extras/AlgebraTrigReview/AlgebraIntro.aspx" class="mm-link">1. Algebra</a> <ul> <li><a href="/Extras/AlgebraTrigReview/Exponents.aspx" class="mm-link">1.1 Exponents </a></li> <li><a href="/Extras/AlgebraTrigReview/AbsoluteValue.aspx" class="mm-link">1.2 Absolute Value</a></li> <li><a href="/Extras/AlgebraTrigReview/Radicals.aspx" class="mm-link">1.3 Radicals</a></li> <li><a href="/Extras/AlgebraTrigReview/Rationalizing.aspx" class="mm-link">1.4 Rationalizing </a></li> <li><a href="/Extras/AlgebraTrigReview/Functions.aspx" class="mm-link">1.5 Functions </a></li> <li><a href="/Extras/AlgebraTrigReview/MultPoly.aspx" class="mm-link">1.6 Multiplying Polynomials</a></li> <li><a href="/Extras/AlgebraTrigReview/Factoring.aspx" class="mm-link">1.7 Factoring</a></li> <li><a href="/Extras/AlgebraTrigReview/SimpRatExp.aspx" class="mm-link">1.8 Simplifying Rational Expressions</a></li> <li><a href="/Extras/AlgebraTrigReview/Graphing.aspx" class="mm-link">1.9 Graphing and Common Graphs</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveEqnPtI.aspx" class="mm-link">1.10 Solving Equations, Part I</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveEqnPtII.aspx" class="mm-link">1.11 Solving Equations, Part II</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveSystems.aspx" class="mm-link">1.12 Solving Systems of Equations</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveIneq.aspx" class="mm-link">1.13 Solving Inequalities</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveAbsValue.aspx" class="mm-link">1.14 Absolute Value Equations and Inequalities</a></li> </ul> </li> <li><a href="/Extras/AlgebraTrigReview/TrigIntro.aspx" class="mm-link">2. Trigonometry</a> <ul> <li><a href="/Extras/AlgebraTrigReview/TrigFunctions.aspx" class="mm-link">2.1 Trig Function Evaluation</a></li> <li><a href="/Extras/AlgebraTrigReview/TrigGraphs.aspx" class="mm-link">2.2 Graphs of Trig Functions</a></li> <li><a href="/Extras/AlgebraTrigReview/TrigFormulas.aspx" class="mm-link">2.3 Trig Formulas</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveTrigEqn.aspx" class="mm-link">2.4 Solving Trig Equations</a></li> <li><a href="/Extras/AlgebraTrigReview/InverseTrig.aspx" class="mm-link">2.5 Inverse Trig Functions</a></li> </ul> </li> <li><a href="/Extras/AlgebraTrigReview/ExpLogIntro.aspx" class="mm-link">3. Exponentials & Logarithms</a> <ul> <li><a href="/Extras/AlgebraTrigReview/ExponentialFcns.aspx" class="mm-link">3.1 Basic Exponential Functions</a></li> <li><a href="/Extras/AlgebraTrigReview/LogarithmFcns.aspx" class="mm-link">3.2 Basic Logarithm Functions</a></li> <li><a href="/Extras/AlgebraTrigReview/LogProperties.aspx" class="mm-link">3.3 Logarithm Properties</a></li> <li><a href="/Extras/AlgebraTrigReview/SimpLogs.aspx" class="mm-link">3.4 Simplifying Logarithms</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveExpEqn.aspx" class="mm-link">3.5 Solving Exponential Equations</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveLogEqn.aspx" class="mm-link">3.6 Solving Logarithm Equations</a></li> </ul> </li> </ul> </li> <li><a href="/Extras/CommonErrors/CommonMathErrors.aspx" class="mm-link">Common Math Errors</a> <ul> <li><a href="/Extras/CommonErrors/GeneralErrors.aspx" class="mm-link">1. General Errors</a> </li> <li><a href="/Extras/CommonErrors/AlgebraErrors.aspx" class="mm-link">2. Algebra Errors</a> </li> <li><a href="/Extras/CommonErrors/TrigErrors.aspx" class="mm-link">3. Trig Errors</a> </li> <li><a href="/Extras/CommonErrors/CommonErrors.aspx" class="mm-link">4. Common Errors</a> </li> <li><a href="/Extras/CommonErrors/CalculusErrors.aspx" class="mm-link">5. Calculus Errors</a> </li> </ul> </li> <li><a href="/Extras/ComplexPrimer/ComplexNumbers.aspx" class="mm-link">Complex Number Primer</a> <ul> <li><a href="/Extras/ComplexPrimer/Definition.aspx" class="mm-link">1. The Definition</a> </li> <li><a href="/Extras/ComplexPrimer/Arithmetic.aspx" class="mm-link">2. Arithmetic</a> </li> <li><a href="/Extras/ComplexPrimer/ConjugateModulus.aspx" class="mm-link">3. Conjugate and Modulus</a> </li> <li><a href="/Extras/ComplexPrimer/Forms.aspx" class="mm-link">4. Polar and Exponential Forms</a> </li> <li><a href="/Extras/ComplexPrimer/Roots.aspx" class="mm-link">5. Powers and Roots</a> </li> </ul> </li> <li><a href="/Extras/StudyMath/HowToStudyMath.aspx" class="mm-link">How To Study Math</a> <ul> <li><a href="/Extras/StudyMath/GeneralTips.aspx" class="mm-link">1. General Tips</a> </li> <li><a href="/Extras/StudyMath/TakingNotes.aspx" class="mm-link">2. Taking Notes</a> </li> <li><a href="/Extras/StudyMath/GettingHelp.aspx" class="mm-link">3. Getting Help</a> </li> <li><a href="/Extras/StudyMath/Homework.aspx" class="mm-link">4. Doing Homework</a> </li> <li><a href="/Extras/StudyMath/ProblemSolving.aspx" class="mm-link">5. Problem Solving</a> </li> <li><a href="/Extras/StudyMath/StudyForExam.aspx" class="mm-link">6. Studying For an Exam</a> </li> <li><a href="/Extras/StudyMath/TakingExam.aspx" class="mm-link">7. Taking an Exam</a> </li> <li><a href="/Extras/StudyMath/Errors.aspx" class="mm-link">8. 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In that section we talked about them because we wrote down the equation of a line in ${\mathbb{R}^3}$ in terms of a <em>vector function</em> (sometimes called a <em>vector-valued function</em>). In this section we want to look a little closer at them and we also want to look at some vector functions in ${\mathbb{R}^3}$other than lines.</p> <p>A vector function is a function that takes one or more variables and returns a vector. We’ll spend most of this section looking at vector functions of a single variable as most of the places where vector functions show up here will be vector functions of single variables. We will however briefly look at vector functions of two variables at the end of this section.</p> <p>A vector functions of a single variable in ${\mathbb{R}^2}$ and ${\mathbb{R}^3}$ have the form,</p> \[\vec r\left( t \right) = \left\langle {f\left( t \right),g\left( t \right)} \right\rangle \hspace{0.25in}\hspace{0.25in}\hspace{0.25in}\hspace{0.25in}\vec r\left( t \right) = \left\langle {f\left( t \right),g\left( t \right),h\left( t \right)} \right\rangle \] <p>respectively, where $f\left( t \right)$,$g\left( t \right)$ and $h\left( t \right)$ are called the <em>component functions</em>.</p> <p>The main idea that we want to discuss in this section is that of graphing and identifying the graph given by a vector function. Before we do that however, we should talk briefly about the domain of a vector function. The <em>domain</em> of a vector function is the set of all $t$’s for which all the component functions are defined.</p> <a class="anchor" name="ThreeD_VecFcn_Ex1"></a> <div class="example"> <span class="example-title">Example 1</span> Determine the domain of the following function. \[\vec r\left( t \right) = \left\langle {\cos t,\ln \left( {4 - t} \right),\sqrt {t + 1} } \right\rangle \] <div class="example-content"> <span id="SHLink_Soln1" class="SH-Link soln-title">Show Solution</span> <span id="SHImg_Soln1" class="fas fa-caret-right" aria-hidden="true"></span> <div id="SHObj_Soln1" class="soln-content"> <p>The first component is defined for all $t$’s. The second component is only defined for $t < 4$. The third component is only defined for $t \ge - 1$. Putting all of these together gives the following domain.</p> \[\left[ { - 1,4} \right)\] <p>This is the largest possible interval for which all three components are defined.</p> </div> </div> </div> <p>Let’s now move into looking at the graph of vector functions. In order to graph a vector function all we do is think of the vector returned by the vector function as a position vector for points on the graph. Recall that a position vector, say $\vec v = \left\langle {a,b,c} \right\rangle $, is a vector that starts at the origin and ends at the point $\left( {a,b,c} \right)$.</p> <p>So, in order to sketch the graph of a vector function all we need to do is plug in some values of $t$ and then plot points that correspond to the resulting position vector we get out of the vector function.</p> <p>Because it is a little easier to visualize things we’ll start off by looking at graphs of vector functions in ${\mathbb{R}^2}$.</p> <a class="anchor" name="ThreeD_VecFcn_Ex2"></a> <div class="example"> <span class="example-title">Example 2</span> Sketch the graph of each of the following vector functions. <ol class="example_parts_list"> <li>$\vec r\left( t \right) = \left\langle {t,1} \right\rangle $</li> <li>$\vec r\left( t \right) = \left\langle {t,{t^3} - 10t + 7} \right\rangle $</li> </ol> <span id="SHALink_S_Soln2" class="SH-Link SH-All">Show All Solutions</span> <span id="SHALink_H_Soln2" class="SH-Link SH-All">Hide All Solutions</span> <div class="example-content"> <span class="soln-list-item soln-list-subitem">a</span> $\vec r\left( t \right) = \left\langle {t,1} \right\rangle $ <span id="SHLink_Soln2a" class="SH-Link soln-title">Show Solution</span> <span id="SHImg_Soln2a" class="fas fa-caret-right" aria-hidden="true"></span> <div id="SHObj_Soln2a" class="soln-content"> <p>Okay, the first thing that we need to do is plug in a few values of $t$ and get some position vectors. Here are a few,</p> \[\vec r\left( { - 3} \right) = \left\langle { - 3,1} \right\rangle \hspace{0.25in}\hspace{0.25in}\vec r\left( { - 1} \right) = \left\langle { - 1,1} \right\rangle \hspace{0.25in}\hspace{0.25in}\vec r\left( 2 \right) = \left\langle {2,1} \right\rangle \hspace{0.25in}\hspace{0.25in}\vec r\left( 5 \right) = \left\langle {5,1} \right\rangle \] <p>So, what this tells us is that the following points are all on the graph of this vector function.</p> \[\left( { - 3,1} \right)\hspace{0.25in}\hspace{0.25in}\left( { - 1,1} \right)\hspace{0.25in}\hspace{0.25in}\left( {2,1} \right)\hspace{0.25in}\hspace{0.25in}\left( {5,1} \right)\] <p>Here is a sketch of this vector function.</p> <div class="center-div"><img alt="This is a horizontal line at y=1. There are also a series of dashed vectors going form the origin to the point (x,1) for x’s ranging from -5 to 5 representing the position vectors that were used to plot out the graph." border="0" height="253" src="VectorFunctions_Files/image001.png" width="360" /></div> <p>In this sketch we’ve included many more evaluations than just those above. Also note that we’ve put in the position vectors (in gray and dashed) so you can see how all this is working. Note however, that in practice the position vectors are generally not included in the sketch.</p> <p>In this case it looks like we’ve got the graph of the line $y = 1$.</p> </div> <br /> <span class="soln-list-item soln-list-subitem">b</span> $\vec r\left( t \right) = \left\langle {t,{t^3} - 10t + 7} \right\rangle $ <span id="SHLink_Soln2b" class="SH-Link soln-title">Show Solution</span> <span id="SHImg_Soln2b" class="fas fa-caret-right" aria-hidden="true"></span> <div id="SHObj_Soln2b" class="soln-content"> <p>Here are a couple of evaluations for this vector function.</p> \[\vec r\left( { - 3} \right) = \left\langle { - 3,10} \right\rangle \hspace{0.25in}\vec r\left( { - 1} \right) = \left\langle { - 1,16} \right\rangle \hspace{0.25in}\vec r\left( 1 \right) = \left\langle {1, - 2} \right\rangle \hspace{0.25in}\vec r\left( 3 \right) = \left\langle {3,4} \right\rangle \] <p>So, we’ve got a few points on the graph of this function. However, unlike the first part this isn’t really going to be enough points to get a good idea of this graph. In general, it can take quite a few function evaluations to get an idea of what the graph is and it’s usually easier to use a computer to do the graphing.</p> <p>Here is a sketch of this graph. We’ve put in a few vectors/evaluations to illustrate them, but the reality is that we did have to use a computer to get a good sketch here.</p> <div class="center-div"><img alt="This is a graph that starts at approximately (-3.5,0) and increases to a peak at approximately (-2,19). The graph then decreases through the y-axis at (0,7) and continues to decrease until a valley at approximately (2,5). It then increases for the rest of the graph until approximately (3.5,15). There are also a series of dashed vectors starting at the origin and ending on the graph at the points at x=-3, -2, -1, 0, 1, 2, 3 representing the position vectors used to graph the curve." border="0" height="240" src="VectorFunctions_Files/image002.png" width="360" /></div> </div> </div> </div> <p>Both of the vector functions in the above example were in the form,</p> \[\vec r\left( t \right) = \left\langle {t,g\left( t \right)} \right\rangle \] <p>and what we were really sketching is the graph of $y = g\left( x \right)$ as you probably caught onto. Let’s graph a couple of other vector functions that do not fall into this pattern.</p> <a class="anchor" name="ThreeD_VecFcn_Ex3"></a> <div class="example"> <span class="example-title">Example 3</span> Sketch the graph of each of the following vector functions. <ol class="example_parts_list"> <li>$\vec r\left( t \right) = \left\langle {6\cos t,3\sin t} \right\rangle $</li> <li>$\vec r\left( t \right) = \left\langle {t - 2\sin t,{t^2}} \right\rangle $</li> </ol> <span id="SHALink_S_Soln3" class="SH-Link SH-All">Show All Solutions</span> <span id="SHALink_H_Soln3" class="SH-Link SH-All">Hide All Solutions</span> <div class="example-content"> <span id="SHLink_DSoln3" class="SH-Link soln-title">Show Discussion</span> <span id="SHImg_DSoln3" class="fas fa-caret-right" aria-hidden="true"></span> <div id="SHObj_DSoln3" class="soln-content"> <p>As we saw in the last part of the previous example it can really take quite a few function evaluations to really be able to sketch the graph of a vector function. Because of that we’ll be skipping all the function evaluations here and just giving the graph. The main point behind this set of examples is to not get you too locked into the form we were looking at above. The first part will also lead to an important idea that we’ll discuss after this example.</p> <p>So, with that said here are the sketches of each of these.</p> </div> <br /> <span class="soln-list-item soln-list-subitem">a</span> $\vec r\left( t \right) = \left\langle {6\cos t,3\sin t} \right\rangle $ <span id="SHLink_Soln3a" class="SH-Link soln-title">Show Solution</span> <span id="SHImg_Soln3a" class="fas fa-caret-right" aria-hidden="true"></span> <div id="SHObj_Soln3a" class="soln-content"> <div class="center-div"><img alt="This is an ellipse centered at the origin with right/left points of (5,0)/(-5,0) respectively and top/bottom points of (0,2)/(0,-2) respectively. There are also a series of dashed vectors all starting at the origin and ending at points along the ellipse defined by a rotation of $\frac{\pi}{8}$ for each new vector. They all represent the position vectors that were used to graph this equation." border="0" height="244" src="VectorFunctions_Files/image003.png" width="360" /></div> <p>So, in this case it looks like we’ve got an ellipse.</p> </div> <br /> <span class="soln-list-item soln-list-subitem">b</span> $\vec r\left( t \right) = \left\langle {t - 2\sin t,{t^2}} \right\rangle $ <span id="SHLink_Soln3b" class="SH-Link soln-title">Show Solution</span> <span id="SHImg_Soln3b" class="fas fa-caret-right" aria-hidden="true"></span> <div id="SHObj_Soln3b" class="soln-content"> <p>Here’s the sketch for this vector function.</p> <div class="center-div"><img alt="This is a graph that starts at approximately (-3, 10) and decreases until it intersects the y-axis at approximately (0, 3.5). It continues to decrease and moving right until approximately (1,1) and then starts moving to the right (and still decreasing) until it goes through the origin horizontally. The graph then comes out of the origin in the 2nd quadrant increasing and moving to the left until approximately (-1,1) and then starts moving to the right (and still increasing) and intersects the y-axis at approximately (0, 3.5) and continues to increase and move to the right ending at approximately (3,10)." border="0" height="400" src="VectorFunctions_Files/image004.png" width="250" /></div> </div> </div> </div> <p>Before we move on to vector functions in ${\mathbb{R}^3}$ let’s go back and take a quick look at the first vector function we sketched in the previous example, $\vec r\left( t \right) = \left\langle {6\cos t,3\sin t} \right\rangle $. The fact that we got an ellipse here should not come as a surprise to you. We know that the first component function gives the $x$ coordinate and the second component function gives the $y$ coordinates of the point that we graph. If we strip these out to make this clear we get,</p> \[x = 6\cos t\hspace{0.25in}\hspace{0.25in}y = 3\sin t\] <p>This should look familiar to you. Back when we were looking at <a href="/Classes/CalcII/ParametricEqn.aspx">Parametric Equations</a> we saw that this was nothing more than one of the sets of parametric equations that gave an ellipse.</p> <p>This is an important idea in the study of vector functions. Any vector function can be broken down into a set of parametric equations that represent the same graph. In general, the two dimensional vector function, $\vec r\left( t \right) = \left\langle {f\left( t \right),g\left( t \right)} \right\rangle $, can be broken down into the parametric equations,</p> \[x = f\left( t \right)\hspace{0.25in}\hspace{0.25in}\hspace{0.25in}y = g\left( t \right)\] <p>Likewise, a three dimensional vector function, $\vec r\left( t \right) = \left\langle {f\left( t \right),g\left( t \right),h\left( t \right)} \right\rangle $, can be broken down into the parametric equations,</p> \[x = f\left( t \right)\hspace{0.25in}y = g\left( t \right)\hspace{0.25in}z = h\left( t \right)\] <p>Do not get too excited about the fact that we’re now looking at parametric equations in ${\mathbb{R}^3}$. They work in exactly the same manner as parametric equations in ${\mathbb{R}^2}$ which we’re used to dealing with already. The only difference is that we now have a third component.</p> <p>Let’s take a look at a couple of graphs of vector functions.</p> <a class="anchor" name="ThreeD_VecFcn_Ex4"></a> <div class="example"> <span class="example-title">Example 4</span> Sketch the graph of the following vector function. \[\vec r\left( t \right) = \left\langle {2 - 4t, - 1 + 5t,3 + t} \right\rangle \] <div class="example-content"> <span id="SHLink_Soln4" class="SH-Link soln-title">Show Solution</span> <span id="SHImg_Soln4" class="fas fa-caret-right" aria-hidden="true"></span> <div id="SHObj_Soln4" class="soln-content"> <p>Notice that this is nothing more than a line. It might help if we rewrite it a little.</p> \[\vec r\left( t \right) = \left\langle {2, - 1,3} \right\rangle + t\left\langle { - 4,5,1} \right\rangle \] <p>In this form we can see that this is the equation of a line that goes through the point $\left( {2, - 1,3} \right)$ and is parallel to the vector $\vec v = \left\langle { - 4,5,1} \right\rangle $.</p> <p>To graph this line all that we need to do is plot the point and then sketch in the parallel vector. In order to get the sketch will assume that the vector is on the line and will start at the point in the line. To sketch in the line all we do this is extend the parallel vector into a line.</p> <p>Here is a sketch.</p> <div class="center-div"><img alt="This is a graph with the standard 3D coordinate system. The positive z-axis is straight up, the positive x-axis moves off to the left and slightly downward and positive y-axis move off the right and slightly downward. This is a 3-D line that goes through the point (2, -1, 3) and is parallel to the vector $\vec{v}=\left\langle -4,5,1 \right\rangle $." border="0" height="217" src="VectorFunctions_Files/image005.png" width="355" /></div> </div> </div> </div> <a class="anchor" name="ThreeD_VecFcn_Ex5"></a> <div class="example"> <span class="example-title">Example 5</span> Sketch the graph of the following vector function. \[\vec r\left( t \right) = \left\langle {2\cos t,2\sin t,3} \right\rangle \] <div class="example-content"> <span id="SHLink_Soln5" class="SH-Link soln-title">Show Solution</span> <span id="SHImg_Soln5" class="fas fa-caret-right" aria-hidden="true"></span> <div id="SHObj_Soln5" class="soln-content"> <p>In this case to see what we’ve got for a graph let’s get the parametric equations for the curve.</p> \[x = 2\cos t\hspace{0.25in}\hspace{0.25in}y = 2\sin t\hspace{0.25in}\hspace{0.25in}z = 3\] <p>If we ignore the $z$ equation for a bit we’ll <a href="/Classes/CalcII/ParametricEqn.aspx#ParaEqnEllipse">recall</a> (hopefully) that the parametric equations for $x$ and $y$ give a circle of radius 2 centered on the origin (or about the $z$-axis since we are in ${\mathbb{R}^3}$).</p> <p>Now, all the parametric equations here tell us is that no matter what is going on in the graph all the $z$ coordinates must be 3. So, we get a circle of radius 2 centered on the $z$-axis and at the level of $z = 3$.</p> <p>Here is a sketch.</p> <div class="center-div"><img alt="This is the graph of a circle of radius 2, centered on the z-axis and parallel to the xy-plane." border="0" height="206" src="VectorFunctions_Files/image006.png" width="250" /></div> </div> </div> </div> <p>Note that it is very easy to modify the above vector function to get a circle centered on the $x$ or $y$-axis as well. For instance,</p> \[\vec r\left( t \right) = \left\langle {10\sin t, - 3,10\cos t} \right\rangle \] <p>will be a circle of radius 10 centered on the $y$-axis and at $y = - 3$. In other words, as long as two of the terms are a sine and a cosine (with the same coefficient) and the other is a fixed number then we will have a circle that is centered on the axis that is given by the fixed number.</p> <p>Let’s take a look at a modification of this.</p> <a class="anchor" name="ThreeD_VecFcn_Ex6"></a> <div class="example"> <span class="example-title">Example 6</span> Sketch the graph of the following vector function.</p> \[\vec r\left( t \right) = \left\langle {4\cos t,4\sin t,t} \right\rangle \] <div class="example-content"> <span id="SHLink_Soln6" class="SH-Link soln-title">Show Solution</span> <span id="SHImg_Soln6" class="fas fa-caret-right" aria-hidden="true"></span> <div id="SHObj_Soln6" class="soln-content"> <p>If this one had a constant in the $z$ component we would have another circle. However, in this case we don’t have a constant. Instead we’ve got a $t$ and that will change the curve. However, because the $x$ and $y$ component functions are still a circle in parametric equations our curve should have a circular nature to it in some way.</p> <p>In fact, the only change is in the $z$ component and as $t$ increases the $z$ coordinate will increase. Also, as $t$ increases the $x$ and $y$ coordinates will continue to form a circle centered on the $z$-axis. Putting these two ideas together tells us that as we increase $t$ the circle that is being traced out in the $x$ and $y$ directions should also be rising.</p> <p>Here is a sketch of this curve.</p> <div class="center-div"><img alt="This is the graph of a spiral of radius 4 and centered on the z-axis." border="0" height="294" src="VectorFunctions_Files/image007.png" width="358" /></div> <p>So, we’ve got a helix (or spiral, depending on what you want to call it) here.</p> </div> </div> </div> <p>As with circles the component that has the $t$ will determine the axis that the helix rotates about. For instance,</p> \[\vec r\left( t \right) = \left\langle {t,6\cos t,6\sin t} \right\rangle \] <p>is a helix that rotates around the $x$-axis.</p> <p>Also note that if we allow the coefficients on the sine and cosine for both the circle and helix to be different we will get ellipses.</p> <p>For example,</p> \[\vec r\left( t \right) = \left\langle {9\cos t,t,2\sin t} \right\rangle \] <p>will be a helix that rotates about the $y$-axis and is in the shape of an ellipse.</p> <p>There is a nice formula that we should derive before moving onto vector functions of two variables.</p> <a class="anchor" name="ThreeD_VecFcn_Ex7"></a> <div class="example"> <span class="example-title">Example 7</span> Determine the vector equation for the line segment starting at the point $P = \left( {{x_1},{y_1},{z_1}} \right)$ and ending at the point $Q = \left( {{x_2},{y_2},{z_2}} \right)$. <div class="example-content"> <span id="SHLink_Soln7" class="SH-Link soln-title">Show Solution</span> <span id="SHImg_Soln7" class="fas fa-caret-right" aria-hidden="true"></span> <div id="SHObj_Soln7" class="soln-content"> <p>It is important to note here that we only want the equation of the line segment that starts at $P$ and ends at $Q$. We don’t want any other portion of the line and we do want the direction of the line segment preserved as we increase $t$. With all that said, let’s not worry about that and just find the vector equation of the line that passes through the two points. Once we have this we will be able to get what we’re after.</p> <p>So, we need a point on the line. We’ve got two and we will use $P$. We need a vector that is parallel to the line and since we’ve got two points we can find the vector between them. This vector will lie on the line and hence be parallel to the line. Also, let’s remember that we want to preserve the starting and ending point of the line segment so let’s construct the vector using the same “orientation”.</p> \[\vec v = \left\langle {{x_2} - {x_1},{y_2} - {y_1},{z_2} - {z_1}} \right\rangle \] <p>Using this vector and the point $P$ we get the following vector equation of the line.</p> \[\vec r\left( t \right) = \left\langle {{x_1},{y_1},{z_1}} \right\rangle + t\left\langle {{x_2} - {x_1},{y_2} - {y_1},{z_2} - {z_1}} \right\rangle \] <p>While this is the vector equation of the line, let’s rewrite the equation slightly.</p> \[\begin{align*}\vec r\left( t \right) & = \left\langle {{x_1},{y_1},{z_1}} \right\rangle + t\left\langle {{x_2},{y_2},{z_2}} \right\rangle - t\left\langle {{x_1},{y_1},{z_1}} \right\rangle \\ & = \left( {1 - t} \right)\left\langle {{x_1},{y_1},{z_1}} \right\rangle + t\left\langle {{x_2},{y_2},{z_2}} \right\rangle \end{align*}\] <p>This is the equation of the line that contains the points $P$ and $Q$. We of course just want the line segment that starts at $P$ and ends at $Q$. We can get this by simply restricting the values of $t$.</p> <p>Notice that</p> \[\vec r\left( 0 \right) = \left\langle {{x_1},{y_1},{z_1}} \right\rangle \hspace{0.25in}\hspace{0.25in}\vec r\left( 1 \right) = \left\langle {{x_2},{y_2},{z_2}} \right\rangle \] <p>So, if we restrict $t$ to be between zero and one we will cover the line segment and we will start and end at the correct point.</p> <p>So, the vector equation of the line segment that starts at $P = \left( {{x_1},{y_1},{z_1}} \right)$ and ends at $Q = \left( {{x_2},{y_2},{z_2}} \right)$ is,</p> \[\vec r\left( t \right) = \left( {1 - t} \right)\left\langle {{x_1},{y_1},{z_1}} \right\rangle + t\left\langle {{x_2},{y_2},{z_2}} \right\rangle \hspace{0.25in}\hspace{0.25in}0 \le t \le 1\] </div> </div> </div> <p>As noted briefly at the beginning of this section we can also have vector functions of two variables. In these cases the graphs of vector function of two variables are surfaces. So, to make sure that we don’t forget that let’s work an example with that as well.</p> <div class="example"> <span class="example-title">Example 8</span> Identify the surface that is described by $\vec r\left( {x,y} \right) = x\,\vec i + y\,\vec j + \left( {{x^2} + {y^2}} \right)\vec k$. <div class="example-content"> <span id="SHLink_Soln8" class="SH-Link soln-title">Show Solution</span> <span id="SHImg_Soln8" class="fas fa-caret-right" aria-hidden="true"></span> <div id="SHObj_Soln8" class="soln-content"> <p>First, notice that in this case the vector function will in fact be a function of two variables. This will always be the case when we are using vector functions to represent surfaces.</p> <p>To identify the surface let’s go back to parametric equations.</p> \[x = x\hspace{0.25in}\hspace{0.25in}y = y\hspace{0.25in}\hspace{0.25in}\hspace{0.25in}z = {x^2} + {y^2}\] <p>The first two are really only acknowledging that we are picking $x$ and $y$ for free and then determining $z$ from our choices of these two. The last equation is the one that we want. We should recognize that function from the section on <a href="QuadricSurfaces.aspx">quadric surfaces</a>. The third equation is the equation of an elliptic paraboloid and so the vector function represents an elliptic paraboloid.</p> </div> </div> </div> <a class="anchor" name="ParaEqn_Fcn"></a> <p>As a final topic for this section let’s generalize the idea from the previous example and note that given any function of one variable ($y = f\left( x \right)$ or $x = h\left( y \right)$) or any function of two variables ($z = g\left( {x,y} \right)$,$x = g\left( {y,z} \right)$, or $y = g\left( {x,z} \right)$) we can always write down a vector form of the equation.</p> <p>For a function of one variable this will be,</p> \[\vec r\left( x \right) = x\,\vec i + f\left( x \right)\,\vec j\hspace{0.25in}\hspace{0.25in}\hspace{0.25in}\vec r\left( y \right) = h\left( y \right)\,\vec i + y\,\vec j\] <p>and for a function of two variables the vector form will be,</p> \[\begin{array}{c}\vec r\left( {x,y} \right) = x\,\vec i + y{\kern 1pt} \vec j + g\left( {x,y} \right)\,\vec k\hspace{0.25in}\hspace{0.25in}\hspace{0.25in}\vec r\left( {y,z} \right) = g\left( {y,z} \right)\,\vec i + y{\kern 1pt} \vec j + z\,\vec k\\ \vec r\left( {x,z} \right) = x\,\vec i + g\left( {x,z} \right){\kern 1pt} \vec j + z\,\vec k\end{array}\] <p>depending upon the original form of the function.</p> <p>For example, the hyperbolic paraboloid $y = 2{x^2} - 5{z^2}$ can be written as the following vector function.</p> \[\vec r\left( {x,z} \right) = x\,\vec i + \left( {2{x^2} - 5{z^2}} \right)\,\vec j + z\,\vec k\] <p>This is a fairly important idea and we will be doing quite a bit of this kind of thing in Calculus III.</p> </div>  <div class="footer"> <div class="footer-links"> [<a href="/Contact.aspx">Contact Me</a>] [<a href="/Privacy.aspx">Privacy Statement</a>] [<a href="/Help.aspx">Site Help & FAQ</a>] [<a href="/Terms.aspx">Terms of Use</a>] </div> <div class="footer-dates"> <div class="footer-copyright"><span id="lblCopyRight">© 2003 - 2024 Paul Dawkins</span></div> <div class="footer-spacer"></div> <div class="footer-modified-date">Page Last Modified : <span id="lblModified">11/16/2022</span></div> </div> </div> </div>  </body> </html>

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Calculus III - Vector Functions