Lec 22: Green's theorem | MIT 18.02 Multivariable Calculus, Fall 2007

This paragraph introduces the concept of Green's theorem in the context of vector calculus, specifically for computing line integrals along closed curves. It discusses the curl of a vector field, which measures its deviation from being conservative. If the curl is zero everywhere, the field is conservative, and the line integral over any closed path will be zero. The paragraph also introduces the two methods of calculating line integrals: direct computation and using Green's theorem, which is applicable when the vector field is defined and differentiable everywhere, including inside the enclosed region.

The speaker provides a practical example to illustrate the application of Green's theorem. They describe a scenario where a line integral over a circle is to be calculated using the theorem, avoiding the direct computation method. The example involves a circle with a specific radius and center, and a vector field with components that, when applied through Green's theorem, simplify the calculation significantly. The explanation includes a brief discussion on parameterization and the transformation of a line integral into a double integral.

This section delves deeper into the simplification of complex line integrals through the application of Green's theorem. It explains how the theorem can transform a challenging line integral into a more manageable double integral. The speaker provides a method to compute the double integral by relating it to the geometric center of mass of the region, offering a clever way to avoid direct integration and instead use symmetry and geometric properties to find the result.

The speaker discusses the conditions under which Green's theorem and the concept of conservative vector fields hold true. They highlight the importance of the curl being zero for a vector field to be conservative and emphasize that this must be true everywhere in the plane. The paragraph also addresses the limitations of Green's theorem, noting that it cannot be applied to non-closed curves or vector fields that are not defined everywhere, such as those with singularities at certain points.

The paragraph presents a counterexample to demonstrate the limitations of Green's theorem. It describes a vector field with zero curl everywhere except at the origin, where it is undefined. This example shows that even if a vector field has zero curl, it may not be conservative if it is not defined over the entire plane. The discussion also touches on the implications of the vector field not being defined at a point, which affects the applicability of Green's theorem.

This section outlines the process of proving Green's theorem. The speaker breaks down the proof into simpler cases, starting with scenarios where the vector field has only an x-component and the region is vertically simple. The explanation involves setting up line integrals for each part of the boundary and comparing them to the double integral over the region. The paragraph also discusses the strategy of decomposing a complex region into simpler ones to facilitate the proof.

The speaker completes the proof of Green's theorem by considering the case of a vertically simple region and then generalizing it to any region. They demonstrate how the line integral along the boundary of the region is equivalent to the double integral over the region, leading to the conclusion of the theorem. The explanation includes the process of integrating over the boundary segments and summing these integrals to match the double integral.

The final paragraph discusses a historical application of Green's theorem in the form of a device called a planimeter, which was used to measure areas under curves on graph paper. The speaker explains how the planimeter works by computing a line integral along the curve, which, according to Green's theorem, equates to the area of the region enclosed by the curve. The paragraph reflects on the obsolescence of such devices due to modern computing technology.