3C: Green's Theorem

In this part we will learn Green’s theorem, which relates line integrals over a closed path to a double integral over the region enclosed. The line integral involves a vector field and the double integral involves derivatives (either div or curl, we will learn both) of the vector field.

First we will give Green’s theorem in work form. The line integral in question is the work done by the vector field. The double integral uses the curl of the vector field. Then we will study the line integral for flux of a field across a curve. Finally we will give Green’s theorem in flux form. This relates the line integral for flux with the divergence of the vector field.

Last Modification: 2020-09-17 Thu 12:30

Captured On: [2020-01-18 Sat]
Source: Part C: Green’s Theorem | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare

Session 65: Green's Theorem

Captured On [2020-02-06 Thu 13:53] Source Session 65: Green’s Theorem | Part C: Green’s Theorem | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Line integral for closed path Figure 2: Green’s Theorem Figure 3: Green’s Theorem (II) Figure 4: Warning and example Figure 5: Computing directly Figure 6: Using Green’s Theorem Figure 7: Solving with geometry and simmetry...

Session 66: Curl(F) = 0 Implies Conservative

Captured On [2020-02-06 Thu 16:29] Source Session 66: Curl(F) = 0 Implies Conservative | Part C: Green’s Theorem | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Special case of Green Theorem Figure 2: Proof of \(\vb{F}\) conservative when curl \(\vb{F}\) is \(0\) Figure 3: Consequence of \(\text{curl} \vb{F} = 0\) Figure 4: Cannot Green theorem when \(\vb{F}\) is not defined at point inside the region...

Session 67: Proof of Green's Theorem

Captured On [2020-02-06 Thu 16:31] Source Session 67: Proof of Green’s Theorem | Part C: Green’s Theorem | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Proof of Green’s Theorem Figure 2: 2 observations Figure 3: Proof of one part Figure 4: Cut (R) “vertically simple” Figure 5: Main step of prove Figure 6: Line integrals...

Session 68: Planimeter: Green's Theorem and Area

Captured On [2020-02-06 Thu 16:32] Source Session 68: Planimeter: Green’s Theorem and Area | Part C: Green’s Theorem | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Planimeter 2 What is a planimeter? 2.1 Front What is a planimeter? Define it, and write its equation 2.2 Back It’s an instruments for measuring areas through closing perimeters. This instruments uses the Green’s Theorem, and the equation...

Session 69: Flux in 2D

Captured On [2020-02-06 Thu 16:33] Source Session 69: Flux in 2D | Part C: Green’s Theorem | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Flux presentation Figure 2: \(\vu{n}\) as clockwise from \(\vu{T}\) Figure 3: Flux and Work comparison Figure 4: Intepretation of Flux Figure 5: Visualization of Flux Figure 6: What flows across \(C\)...

Session 70: Normal Form of Green's Theorem

Captured On [2020-02-06 Thu 16:34] Source Session 70: Normal Form of Green’s Theorem | Part C: Green’s Theorem | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: What is the Green’s Theorem for flux Figure 2: Green’s Theorem for flux Figure 3: Green’s Theorem in normal form vs tangential form Figure 4: Proof of Green’s Theorem for flux...

Session 71: Extended Green's Theorem: Boundaries with Multiple Pieces

Captured On [2020-02-06 Thu 16:35] Source Session 71: Extended Green’s Theorem: Boundaries with Multiple Pieces | Part C: Green’s Theorem | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: More about validity of Green’s Theorem Figure 2: \(\vb{F}\) defined everywhere in \(R\) Figure 3: Use of Green’s Theorem Figure 4: Remove the region where \(\vb{F}\) is not defined...

Session 72: Simply Connected Regions and Conservative Fields

Captured On [2020-02-06 Thu 16:36] Source Session 72: Simply Connected Regions and Conservative Fields | Part C: Green’s Theorem | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Connected region in the plane Figure 2: Where can I apply Green’s Theorem? Figure 3: Correct definition of \(\vb{F}\) conservative 2 What is a simply-connected region? 2.1 Front What is a simply-connected region?...