Calculus III - Stokes' Theorem

<!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 - Stokes' Theorem </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="stokes theorem" /><meta http-equiv="description" name="description" content="In this section we will discuss Stokes’ Theorem." /></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> </button> <button id="quicknav-menu" class="top-menu-button" data-jq-dropdown="#jq-dropdown-quicknav" title="Quick Navigation (Previous/Next Sections and Problems and Full Problem List) Menu"> <span class="top-menu-item-title">Quick Nav</span> <span class="fas fa-exchange fa-lg" aria-hidden="true"></span> </button> <button id="download-menu" class="top-menu-button" data-jq-dropdown="#jq-dropdown-download" title="Download pdf Menu"> <span class="top-menu-item-title">Download</span> <span class="far fa-download fa-lg" aria-hidden="true"></span> </button> <button id="print-menu" class="top-menu-button top-menu-button-icon-only" data-jq-dropdown="#jq-print-download" title="Print Menu"> <span class="far fa-print fa-lg" aria-hidden="true"></span> </button> </div> <div id="header-google-search" class="header-search"> <gcse:search></gcse:search> </div> <div id="header-search-icon" title="Site Search" class="header-menu-icon"><span id="search-icon" class="fas fa-search" aria-hidden="true"></span></div> </div> </div> <div id="jq-dropdown-type" class="jq-dropdown jq-dropdown-tip"> <ul class="jq-dropdown-menu"> <li id="li_type_menu_goto" class="top-menu-nav-title">Go To</li> <li id="li_type_menu_notes"> <span id="type_menu_notes_span" title="Viewing the Notes for the current topic." class="top-menu-current">Notes</span> </li> <li id="li_type_menu_practice"> <a href="/Problems/CalcIII/StokesTheorem.aspx" id="type_menu_practice_link" title="Go to Practice Problems for current topic.">Practice Problems</a> </li> <li id="li_type_menu_asgn"> <a href="/ProblemsNS/CalcIII/StokesTheorem.aspx" id="type_menu_asgn_link" title="Go to Assignment Problems for current topic.">Assignment Problems</a> </li> <li id="li_type_menu_sh" class="top-menu-nav-title">Show/Hide</li> <li id="li_type_menu_show" title="Show any hidden solutions and/or steps that may be present on the page."><a href="javascript:SHPrintPage(1,0)" id="view_menu_show">Show all Solutions/Steps/<em>etc.</em></a></li> <li id="li_type_menu_hide" title="Hide any visible solutions and/or steps that may be present on the page."><a href="javascript:SHPrintPage(0,0)" id="view_menu_hide">Hide all Solutions/Steps/<em>etc.</em></a></li> </ul> </div> <div id="jq-dropdown-quicknav" class="jq-dropdown jq-dropdown-tip"> <ul class="jq-dropdown-menu"> <li id="li_nav_menu_sections" class="top-menu-nav-title">Sections</li> <li id="li_nav_menu_prev_section"><a href="/Classes/CalcIII/SurfIntVectorField.aspx" id="a_nav_menu_prev_section" class="top-menu-nav-link" title="Previous Section : Surface Integrals of Vector Fields"><span class="top-menu-prev fas fa-chevron-left"></span> Surface Integrals of Vector Fields</a></li> <li id="li_nav_menu_next_section"><a href="/Classes/CalcIII/DivergenceTheorem.aspx" id="a_nav_menu_next_section" class="top-menu-nav-link" title="Next Section : Divergence Theorem"><span class="top-menu-prev-hidden fas fa-chevron-left"></span> Divergence Theorem <span class="top-menu-next fas fa-chevron-right"></span></a></li> <li id="li_nav_menu_chapters" class="top-menu-nav-title">Chapters</li> <li id="li_nav_menu_prev_chapter"><a href="/Classes/CalcIII/LineIntegralsIntro.aspx" id="a_nav_menu_prev_chapter" class="top-menu-nav-link" title="Previous Chapter : Line Integrals"><span class="top-menu-prev fas fa-chevron-left"></span><span class="top-menu-prev fas fa-chevron-left"></span> Line Integrals</a></li> <li id="li_nav_menu_classes" class="top-menu-nav-title">Classes</li> <li> <a href="/Classes/Alg/Alg.aspx" id="nav_menu_alg_link" title="Go To Algebra Notes">Algebra</a> </li> <li> <a href="/Classes/CalcI/CalcI.aspx" id="nav_menu_calci_link" title="Go To Calculus I Notes">Calculus I</a> </li> <li> <a href="/Classes/CalcII/CalcII.aspx" id="nav_menu_calcii_link" title="Go To Calculus II Notes">Calculus II</a> </li> <li> <span id="nav_menu_calciii_span" title="Currently Viewing Calculus III Material" class="top-menu-current">Calculus III</span> </li> <li> <a href="/Classes/DE/DE.aspx" id="nav_menu_de_link" title="Go To Differential Equations Notes">Differential Equations</a> </li> <li id="li_nav_menu_extras" class="top-menu-nav-title">Extras</li> <li> <a href="/Extras/AlgebraTrigReview/AlgebraTrig.aspx" id="nav_menu_algtrig_link" title="Go To Algebra & Trig Review">Algebra & Trig Review</a> </li> <li> <a href="/Extras/CommonErrors/CommonMathErrors.aspx" id="nav_menu_commonerrors_link" title="Go To Common Math Errors">Common Math Errors</a> </li> <li> <a href="/Extras/ComplexPrimer/ComplexNumbers.aspx" id="nav_menu_complexnumbers_link" title="Go To Complex Numbers Primer">Complex Number Primer</a> </li> <li> <a href="/Extras/StudyMath/HowToStudyMath.aspx" id="nav_menu_studymath_link" title="Go To How To Study Math">How To Study Math</a> </li> <li> <a href="/Extras/CheatSheets_Tables.aspx" id="nav_menu_cheattables_link" title="Go To List of Cheat Sheets and Tables">Cheat Sheets & Tables</a> </li> <li id="li_nav_menu_misc" class="top-menu-nav-title">Misc</li> <li><a href="/contact.aspx" id="nav_menu_contact" title="Contact Me!">Contact Me</a></li> <li><a href="/mathjax.aspx" id="nav_menu_mathjax" title="Info on MathJax and MathJax Configuration Menu">MathJax Help and Configuration</a></li> </ul> </div> <div id="jq-dropdown-download" class="jq-dropdown jq-dropdown-anchor-right jq-dropdown-tip"> <ul class="jq-dropdown-menu"> <li id="li_download_menu_notes" class="top-menu-nav-title">Notes Downloads</li> <li id="li_download_menu_notes_book"><a href="/GetFile.aspx?file=B,20,N" id="download_menu_notes_book" data-PDF="Download - Menu$Notes - Book$Calculus$/Downloads/Calculus/Notes/Complete.pdf">Complete Book</a></li> <li id="li_download_menu_practice" class="top-menu-nav-title">Practice Problems Downloads</li> <li id="li_download_menu_practice_book"><a href="/GetFile.aspx?file=B,20,P" id="download_menu_practice_book" data-PDF="Download - Menu$Practice Problems - Book$Calculus$/Downloads/Calculus/Practice Problems/Complete.pdf">Complete Book - Problems Only</a></li> <li id="li_download_menu_solutions_book"><a href="/GetFile.aspx?file=B,20,S" id="download_menu_solutions_book" data-PDF="Download - Menu$Practice Problems Solutions - Book$Calculus$/Downloads/Calculus/Practice Problems Solutions/Complete.pdf">Complete Book - Solutions</a></li> <li id="li_download_menu_asgn" class="top-menu-nav-title">Assignment Problems Downloads</li> <li id="li_download_menu_asgn_book"><a href="/GetFile.aspx?file=B,20,A" id="download_menu_asgn_book" data-PDF="Download - Menu$Assignment Problems - Book$Calculus$/Downloads/Calculus/Assignment Problems/Complete.pdf">Complete Book</a></li> <li id="li_download_menu_other" class="top-menu-nav-title">Other Items</li> <li id="li_download_menu_urls"> <a href="/DownloadURLs.aspx?bi=20" id="download_menu_urls">Get URL's for Download Items</a> </li> </ul> </div> <div id="jq-print-download" class="jq-dropdown jq-dropdown-anchor-right jq-dropdown-tip"> <ul class="jq-dropdown-menu"> <li id="li_print_menu_default"><a href="javascript:SHPrintPage()" id="print_menu_default">Print Page in Current Form (Default)</a></li> <li id="li_print_menu_show"><a href="javascript:SHPrintPage(1,1)" id="print_menu_show">Show all Solutions/Steps and Print Page</a></li> <li id="li_print_menu_hide"><a href="javascript:SHPrintPage(0,1)" id="print_menu_hide">Hide all Solutions/Steps and Print Page</a></li> </ul> </div> <nav id="menu" class="notes-nav"> <ul> <li><a href="/" class="mm-link">Home</a></li> <li><span>Classes</span></li> <li><a href="/Classes/Alg/Alg.aspx" class="mm-link">Algebra</a> <ul> <li><a href="/Classes/Alg/Preliminaries.aspx" class="mm-link">1. Preliminaries</a> <ul> <li><a href="/Classes/Alg/IntegerExponents.aspx" class="mm-link">1.1 Integer Exponents</a></li> <li><a href="/Classes/Alg/RationalExponents.aspx" class="mm-link">1.2 Rational Exponents</a></li> <li><a href="/Classes/Alg/Radicals.aspx" class="mm-link">1.3 Radicals</a></li> <li><a href="/Classes/Alg/Polynomials.aspx" class="mm-link">1.4 Polynomials</a></li> <li><a href="/Classes/Alg/Factoring.aspx" class="mm-link">1.5 Factoring Polynomials</a></li> <li><a href="/Classes/Alg/RationalExpressions.aspx" class="mm-link">1.6 Rational Expressions</a></li> <li><a href="/Classes/Alg/ComplexNumbers.aspx" class="mm-link">1.7 Complex Numbers</a></li> </ul> </li> <li><a href="/Classes/Alg/Solving.aspx" class="mm-link">2. Solving Equations and Inequalities</a> <ul> <li><a href="/Classes/Alg/SolutionSets.aspx" class="mm-link">2.1 Solutions and Solution Sets</a></li> <li><a href="/Classes/Alg/SolveLinearEqns.aspx" class="mm-link">2.2 Linear Equations</a></li> <li><a href="/Classes/Alg/LinearApps.aspx" class="mm-link">2.3 Applications of Linear Equations</a></li> <li><a href="/Classes/Alg/SolveMultiVariable.aspx" class="mm-link">2.4 Equations With More Than One Variable</a></li> <li><a href="/Classes/Alg/SolveQuadraticEqnsI.aspx" class="mm-link">2.5 Quadratic Equations - Part I</a></li> <li><a href="/Classes/Alg/SolveQuadraticEqnsII.aspx" class="mm-link">2.6 Quadratic Equations - Part II</a></li> <li><a href="/Classes/Alg/SolveQuadraticEqnSummary.aspx" class="mm-link">2.7 Quadratic Equations : A Summary</a></li> <li><a href="/Classes/Alg/QuadraticApps.aspx" class="mm-link">2.8 Applications of Quadratic Equations</a></li> <li><a href="/Classes/Alg/ReducibleToQuadratic.aspx" class="mm-link">2.9 Equations Reducible to Quadratic in Form</a></li> <li><a href="/Classes/Alg/SolveRadicalEqns.aspx" class="mm-link">2.10 Equations with Radicals</a></li> <li><a href="/Classes/Alg/SolveLinearInequalities.aspx" class="mm-link">2.11 Linear Inequalities</a></li> <li><a href="/Classes/Alg/SolvePolyInequalities.aspx" class="mm-link">2.12 Polynomial Inequalities</a></li> <li><a href="/Classes/Alg/SolveRationalInequalities.aspx" class="mm-link">2.13 Rational Inequalities</a></li> <li><a href="/Classes/Alg/SolveAbsValueEqns.aspx" class="mm-link">2.14 Absolute Value Equations</a></li> <li><a href="/Classes/Alg/SolveAbsValueIneq.aspx" class="mm-link">2.15 Absolute Value Inequalities</a></li> </ul> </li> <li><a href="/Classes/Alg/Graphing_Functions.aspx" class="mm-link">3. 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. Learn From Your Errors</a> </li> </ul> </li> <li><span>Misc Links</span></li> <li><a href="/contact.aspx" class="mm-link">Contact Me</a></li> <li><a href="/links.aspx" class="mm-link">Links</a></li> <li><a href="/mathjax.aspx" class="mm-link">MathJax Help and Configuration</a></li> <li><a href="/privacy.aspx" class="mm-link">Privacy Statement</a></li> <li><a href="/help.aspx" class="mm-link">Site Help & FAQ</a></li> <li><a href="/terms.aspx" class="mm-link">Terms of Use</a></li> </ul> </nav> <div class="top-info-bar"> <span id="mobile-title" class="mobile-header-title header-title">Paul's Online Notes</span> <br /> <span class="top-menu-breadcrumb"> <a href="/" id="breadcrumb_home_link" title="Go to Main Page">Home</a> <span id="breadcrumb_h_b_sep_span">/ </span> <a href="/Classes/CalcIII/CalcIII.aspx" id="breadcrumb_book_link" title="Go to Book Introduction">Calculus III</a> <span id="breadcrumb_b_c_sep_span">/ </span> <a href="/Classes/CalcIII/SurfaceIntegralsIntro.aspx" id="breadcrumb_chapter_link" title="Go to Chapter Introduction">Surface Integrals</a> <span id="breadcrumb_section_span"> / Stokes' Theorem</span> </span> </div> <div id="nav_links" class="top-nav-bar"> <a href="/Classes/CalcIII/SurfIntVectorField.aspx" id="nav_links_prev_section" title="Goto Previous Section : Surface Integrals of Vector Fields"><span class="top-menu-prev fas fa-chevron-left"></span><span class="nav_desktop_extra_pn"> Prev. Section</span></a> <div class="top-nav-bar-link-spacer"></div> <span id="nav_current_notes">Notes</span> <a href="/Problems/CalcIII/StokesTheorem.aspx" id="nav_links_practice" title="Go to Practice Problems for current topic.">Practice<span class="nav_desktop_extra"> Problems</span></a> <a href="/ProblemsNS/CalcIII/StokesTheorem.aspx" id="nav_links_asgn" title="Go to Assignment Problems for current topic.">Assignment<span class="nav_desktop_extra"> Problems</span></a> <div class="top-nav-bar-link-spacer"></div> <a href="/Classes/CalcIII/DivergenceTheorem.aspx" id="nav_links_next_section" title="Goto Next Section : Divergence Theorem"><span class="nav_desktop_extra_pn"> Next Section </span><span class="top-menu-next fas fa-chevron-right"></span></a> </div> <div class="header-divider"></div> <div class="content"> <span id="SHLink_NoteMobile" class="SH-Link content-note-link-mobile">Show Mobile Notice</span> <span id="SHImg_NoteMobile" class="fas fa-caret-right SH-padding content-note-link-mobile" aria-hidden="true"></span> <span id="SHALink_S_Note" class="SH-Link SH-Hide SH-Bracket">Show All Notes</span> <span id="SHALink_H_Note" class="SH-Link SH-Hide SH-Bracket">Hide All Notes</span> <div id="SHObj_NoteMobile" class="content-note-container content-note-container-mobile"> <div class="content-note-header">Mobile Notice</div> <div class="content-note">You appear to be on a device with a "narrow" screen width (<em>i.e.</em> you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll/swipe to see them) and some of the menu items will be cut off due to the narrow screen width.</div> </div> <form method="post" action="./StokesTheorem.aspx" id="ctl00"> <div class="aspNetHidden"> <input type="hidden" name="__VIEWSTATE" id="__VIEWSTATE" value="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" /> </div> <div class="aspNetHidden"> <input type="hidden" name="__VIEWSTATEGENERATOR" id="__VIEWSTATEGENERATOR" value="C6A4703C" /> </div> </form> <h3>Section 17.5 : Stokes' Theorem</h3> <p>In this section we are going to take a look at a theorem that is a higher dimensional version of <a href="GreensTheorem.aspx">Green’s Theorem</a>. In Green’s Theorem we related a line integral to a double integral over some region. In this section we are going to relate a line integral to a surface integral. However, before we give the theorem we first need to define the curve that we’re going to use in the line integral.</p> <p>Let’s start off with the following surface with the indicated orientation.</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 moves off the right and slightly downward. This is the upper hemisphere without a specified radius and whose base is at an unspecified point on the positive z-axis. At several places on hemisphere there are normal vectors pointing outwards from the hemisphere. Along the base of the hemisphere is a curve, labeled C, with arrow heads indicated that it is traced out in a counter clockwise motion if viewed from above." border="0" height="284" src="StokesTheorem_Files/image001.png" width="295" /></div> <p>Around the edge of this surface we have a curve $C$. This curve is called the <strong>boundary curve</strong>. The orientation of the surface $S$ will induce the <strong>positive orientation of $C$</strong>. To get the positive orientation of $C$ think of yourself as walking along the curve. While you are walking along the curve if your head is pointing in the same direction as the unit normal vectors while the surface is on the left then you are walking in the positive direction on $C$.</p> <p>Now that we have this curve definition out of the way we can give Stokes’ Theorem.</p> <h4>Stokes’ Theorem</h4> <div class="fact"> <p>Let $S$ be an oriented smooth surface that is bounded by a simple, closed, smooth boundary curve $C$ with positive orientation. Also let $\vec F$ be a vector field then,</p> \[\int\limits_{C}{{\vec F\centerdot \,d\,\vec r}} = \iint\limits_{S}{{{\mathop{\rm curl}\nolimits} \vec F\,\centerdot \,d\vec S}}\] </div> <p>In this theorem note that the surface $S$ can actually be any surface so long as its boundary curve is given by $C$. This is something that can be used to our advantage to simplify the surface integral on occasion.</p> <p>Let’s take a look at a couple of examples.</p> <a class="anchor" name="SurfInt_Stokes_Ex1"></a> <div class="example"> <span class="example-title">Example 1</span> Use Stokes’ Theorem to evaluate $\displaystyle \iint\limits_{S}{{{\mathop{\rm curl}\nolimits} \vec F\,\centerdot \,d\vec S}}$ where $\vec F = {z^2}\,\vec i - 3xy\,\vec j + {x^3}{y^3}\,\vec k$ and $S$ is the part of $z = 5 - {x^2} - {y^2}$ above the plane $z = 1$. Assume that $S$ is oriented upwards. <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>Let’s start this off with a sketch of the surface.</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 moves off the right and slightly downward. This is the graph of the vaguely cup shaped object given by the elliptic paraboloid in the problem statement. It starts at z=5, is centered on the z-axis and opens in the negative z directions. Along the base of the hemisphere is a curve, labeled C, with arrow heads indicated that it is traced out in a counter clockwise motion if viewed from above." border="0" height="302" src="StokesTheorem_Files/image002.png" width="271" /></div> <p>In this case the boundary curve $C$ will be where the surface intersects the plane $z = 1$ and so will be the curve</p> \[\begin{align*}1 & = 5 - {x^2} - {y^2}\\ {x^2} + {y^2} & = 4\hspace{0.25in}{\mbox{at }}z = 1\end{align*}\] <p>So, the boundary curve will be the circle of radius 2 that is in the plane $z = 1$. The parameterization of this curve is,</p> \[\vec r\left( t \right) = 2\cos t\,\vec i + 2\sin t\,\vec j + \vec k,\,\,\,\,0 \le t \le 2\pi \] <p>The first two components give the circle and the third component makes sure that it is in the plane $z = 1$.</p> <p>Using Stokes’ Theorem we can write the surface integral as the following line integral.</p> \[\iint\limits_{S}{{{\mathop{\rm curl}\nolimits} \vec F\,\centerdot \,d\vec S}} = \int\limits_{C}{{\vec F\,\centerdot \,d\,\vec r}} = \int_{{\,0}}^{{\,2\pi }}{{\vec F\left( {\vec r\left( t \right)} \right)\,\centerdot \,\vec r'\left( t \right)\,dt}}\] <p>So, it looks like we need a couple of quantities before we do this integral. Let’s first get the vector field evaluated on the curve. Remember that this is simply plugging the components of the parameterization into the vector field.</p> \[\begin{align*}\vec F\left( {\vec r\left( t \right)} \right) & = {\left( 1 \right)^2}\,\vec i - 3\left( {2\cos t} \right)\left( {2\sin t} \right)\,\vec j + {\left( {2\cos t} \right)^3}{\left( {2\sin t} \right)^3}\,\vec k\\ & = \vec i - 12\cos t\sin t\,\vec j + 64{\cos ^3}t{\sin ^3}t\,\vec k\end{align*}\] <p>Next, we need the derivative of the parameterization and the dot product of this and the vector field.</p> \[\begin{align*}\vec r'\left( t \right) & = - 2\sin t\,\vec i + 2\cos t\,\vec j\\ \vec F\left( {\vec r\left( t \right)} \right)\,\centerdot \,\vec r'\left( t \right)\, & = - 2\sin t - 24\sin t{\cos ^2}t\end{align*}\] <p>We can now do the integral.</p> \[\begin{align*}\iint\limits_{S}{{{\mathop{\rm curl}\nolimits} \vec F\,\centerdot \,d\vec S}} & = \int_{{\,0}}^{{\,2\pi }}{{ - 2\sin t - 24\sin t{{\cos }^2}t\,dt}}\\ & = \left. {\left( {2\cos t + 8{{\cos }^3}t} \right)} \right|_0^{2\pi }\\ & = 0\end{align*}\] </div> </div> </div> <a class="anchor" name="SurfInt_Stokes_Ex2"></a> <div class="example"> <span class="example-title">Example 2</span> Use Stokes’ Theorem to evaluate $\displaystyle \int\limits_{C}{{\vec F\,\centerdot \,d\,\vec r}}$ where $\vec F = {z^2}\,\vec i + {y^2}\,\vec j + x\,\vec k$ and $C$ is the triangle with vertices $\left( {1,0,0} \right)$, $\left( {0,1,0} \right)$ and $\left( {0,0,1} \right)$ with counter-clockwise rotation. <div class="example-content"> <span id="SHLink_Soln2" class="SH-Link soln-title">Show Solution</span> <span id="SHImg_Soln2" class="fas fa-caret-right" aria-hidden="true"></span> <div id="SHObj_Soln2" class="soln-content"> <p>We are going to need the curl of the vector field eventually so let’s get that out of the way first.</p> \[{\mathop{\rm curl}\nolimits} \vec F = \left| {\begin{array}{*{20}{c}}{\vec i}&{\vec j}&{\vec k}\\{\frac{\partial }{{\partial x}}}&{\frac{\partial }{{\partial y}}}&{\frac{\partial }{{\partial z}}}\\{{z^2}}&{{y^2}}&x\end{array}} \right| = 2z\,\vec j - \vec j = \left( {2z - 1} \right)\vec j\] <p>Now, all we have is the boundary curve for the surface that we’ll need to use in the surface integral. However, as noted above all we need is any surface that has this as its boundary curve. So, let’s use the following plane with upwards orientation for the surface.</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 moves off the right and slightly downward. This is the graph of the triangle with vertices (0,0,1), (1,0,0) and (0,1,0). Along the edges of it is a curve, labeled C, with arrow heads on it that indicate that it is traced out in the counter clockwise direction if we are in front of the plane and looking in towards the origin." border="0" height="299" src="StokesTheorem_Files/image003.png" width="246" /></div> <p>Since the plane is oriented upwards this induces the positive direction on $C$ as shown. The equation of this plane is,</p> \[x + y + z = 1\hspace{0.25in}\,\,\, \Rightarrow \hspace{0.25in}\,\,\,z = g\left( {x,y} \right) = 1 - x - y\] <p>Now, let’s use Stokes’ Theorem and get the surface integral set up.</p> \[\begin{align*}\int\limits_{C}{{\vec F\,\centerdot \,d\,\vec r}} & = \iint\limits_{S}{{{\mathop{\rm curl}\nolimits} \vec F\,\centerdot \,d\vec S}}\\ & = \iint\limits_{S}{{\left( {2z - 1} \right)\vec j\centerdot d\vec S}}\\ & = \iint\limits_{D}{{\left( {2z - 1} \right)\vec j\centerdot \frac{{\nabla f}}{{\left\| {\nabla f} \right\|}}\,\,\left\| {\nabla f} \right\|\,dA}}\end{align*}\] <p>Okay, we now need to find a couple of quantities. First let’s get the gradient. Recall that this comes from the function of the surface.</p> \[\begin{align*}f\left( {x,y,z} \right) &= z - g\left( {x,y} \right) = z - 1 + x + y\\ & \nabla f = \vec i + \vec j + \vec k\end{align*}\] <p>Note as well that this also points upwards and so we have the correct direction.</p> <p>Now, $D$ is the region in the $xy$-plane shown below,</p> <div class="center-div"><img alt="This is the 2D graph on a xy axis system of a triangle with vertices (0,0), (1,0) and (0,1). The top left edge of the triangle is given by y=-x+1. The bottom of the triangle is the x-axis and the left edge is the y-axis. The triangle has been shaded in." border="0" height="204" src="StokesTheorem_Files/image004.png" width="300" /></div> <p>We get the equation of the line by plugging in $z = 0$ into the equation of the plane. So based on this the ranges that define $D$ are,</p> \[0 \le x \le 1\hspace{0.25in}0 \le y \le - x + 1\] <p>The integral is then,</p> \[\begin{align*}\int\limits_{C}{{\vec F\,\centerdot \,d\,\vec r}} & = \iint\limits_{D}{{\left( {2z - 1} \right)\vec j\centerdot \left( {\vec i + \vec j + \vec k} \right)\,dA}}\\ & = \int_{{\,0}}^{{\,1}}{{\int_{{\,0}}^{{ - x + 1}}{{2\left( {1 - x - y} \right) - 1\,dy}}\,dx}}\end{align*}\] <p>Don’t forget to plug in for $z$ since we are doing the surface integral on the plane. Finishing this out gives,</p> \[\begin{align*}\int\limits_{C}{{\vec F\,\centerdot \,d\,\vec r}} & = \int_{{\,0}}^{{\,1}}{{\int_{{\,0}}^{{ - x + 1}}{{1 - 2x - 2y\,dy}}\,dx}}\\ & = \int_{{\,0}}^{{\,1}}{{\left. {\left( {y - 2xy - {y^2}} \right)} \right|_0^{ - x + 1}\,dx}}\\ & = \int_{{\,0}}^{{\,1}}{{{x^2} - x\,dx}}\\ & = \left. {\left( {\frac{1}{3}{x^3} - \frac{1}{2}{x^2}} \right)} \right|_0^1\\ & = - \frac{1}{6}\end{align*}\] </div> </div> </div> <p>In both of these examples we were able to take an integral that would have been somewhat unpleasant to deal with and by the use of Stokes’ Theorem we were able to convert it into an integral that wasn’t too bad.</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 - 2025 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>

CINXE.COM

Calculus III - Stokes' Theorem