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 1: Why Stokes is true?

Figure 2: Strategy for proofing Stokes’ Theorem

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 \(S\)