4C: Line Integrals and Stokes' Theorem

Captured On [2020-02-06 Thu 17:22] Source Session 88: Line Integrals in Space | Part C: Line Integrals and Stokes’ Theorem | 4. Triple Integrals and Surface Integrals in 3-Space | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Line Integrals in Space Figure 2: Evaluation of Line Integrals Figure 3: Example of Line Integrals Figure 4: Example, same \(\vb{F}\) changing path Figure 5: Evaluation of example 2...

Captured On [2020-02-06 Thu 17:39] Source Session 89: Gradient Fields and Potential Functions | Part C: Line Integrals and Stokes’ Theorem | 4. Triple Integrals and Surface Integrals in 3-Space | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Test for Gradient Fields Figure 2: Criterion for Gradient Field Figure 3: Example for criterion gradient field Figure 4: Find the Potential Figure 5: FTC and Path method...

Captured On [2020-02-06 Thu 17:41] Source Session 90: Curl in 3D | Part C: Line Integrals and Stokes’ Theorem | 4. Triple Integrals and Surface Integrals in 3-Space | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Curl in 3D Figure 2: \(\vb{F}\) in simply-connected region Figure 3: Del notation Figure 4: Mnemotecnic formula for curl \(\vb{F}\) Figure 5: Geometric meaning of curl 2 Is the \(\text{curl}\vb{F}\) a scalar function?...

Captured On [2020-02-06 Thu 17:42] Source Session 91: Stokes’ Theorem | Part C: Line Integrals and Stokes’ Theorem | 4. Triple Integrals and Surface Integrals in 3-Space | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Recall \(\text{curl}\vb{F}\) Figure 2: Curl examples Figure 3: Stokes’ Theorem Figure 4: Orientation of \(S\) and \(C\) Figure 5: Right-hand rule for \(S\) and \(C\) orientation Figure 6: Comparing Stokes with Green...

Captured On [2020-02-06 Thu 17:43] Source Session 92: Proof of Stokes’ Theorem | Part C: Line Integrals and Stokes’ Theorem | 4. Triple Integrals and Surface Integrals in 3-Space | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Why Stokes is true? Figure 2: Strategy for proofing Stokes’ Theorem 2 Why Stokes is true? 2.1 Front Why Stokes is true? 2.2 Back Know it for \(C\), \(S\) in \(xy\text{-plane}\) (Green’s Theorem) also for \(C\), \(S\) in any plane using that work, flux, curl make sense independently of the coordinate system Strategy of proofing Given any \(S\): decompose it into tiny, almost flat pieces Sum of work around each piece = work along \(C\) Similar to Green’s Theorem Sum of flux through each piece = flux through \(S\)

Captured On [2020-02-06 Thu 17:43] Source Session 93: Example | Part C: Line Integrals and Stokes’ Theorem | 4. Triple Integrals and Surface Integrals in 3-Space | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Example of Stokes’ Theorem Figure 2: Computing the surface integral Figure 3: Same result with Stokes’ Theorem

Captured On [2020-02-06 Thu 17:44] Source Session 94: Simply Connected Regions; Topology | Part C: Line Integrals and Stokes’ Theorem | 4. Triple Integrals and Surface Integrals in 3-Space | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Stokes and Path Independence Figure 2: Examples of simply-connected spaces Figure 3: Theorem for path independence of line integrals Figure 4: Proof of path independence of line integrals of conservative \(\vb{F}\)...

Captured On [2020-02-06 Thu 17:45] Source Session 95: Stokes’ Theorem and Surface Independence | Part C: Line Integrals and Stokes’ Theorem | 4. Triple Integrals and Surface Integrals in 3-Space | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Stokes and surface independence Figure 2: Check with divergence theorem Figure 3: Divergence of curl of vector fiels is 0 Figure 4: For real vectors 2 Why for Stokes’ Theorem we can use any surface (surface independence)?...