Extended Green's Theorem | MIT 18.02SC Multivariable Calculus, Fall 2010
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it when asking about or citing the extended version of Green's theorem and the definite-integral method for finding potential functions.
Recommended by 1 people across 1 different months, 2016 to 2017.
What it is
A 23-minute MIT OpenCourseWare lecture from the 18.02SC Multivariable Calculus course, taught by Christine Breiner. It covers the extended version of Green's theorem, applying it to regions with holes and to finding potential functions for conservative vector fields.
What people said
Recommends Twenty-six colours · 2017-06 · Mathematics Stack Exchange · on “Finding a potential function using integration (vector fields)”
For the method of finding a potential function of a conservative vector field, usually I would let $\nabla \phi$ be equal to the vector field, and do some integration (indefinite) and differentiating whilst equating it to other components of $\nabla \phi$. However, in this youtube video [this video] (at 11:47), the technique was to do a definite integral twice, over some random path. Where could I learn this method and why does this work? The question is to find the potential scalar field for the conservative vector field: $$F = \begin{pmatrix} x \\ y \end{pmatrix} r^{n},\qquad \text{with $r = \sqrt{x^2 + y^2}$}$$ The method in the video found the potential vector field by evaluating the followin
Cites YoTengoUnLCD · 2016-02 · Mathematics Stack Exchange · on “References for the "extended" Green and Stokes' theorem.”
I was watching these videos from MIT's series: Green [this video] , Stokes [LINK] . And I didn't understand the justification: their "extended version" of the theorems. I looked up on google and couldn't find many references (or they explained things with differential forms, which haven't yet studied). Does anyone know some references about these versions? I'm not looking for very rigorous/proofy style notes, just some explanation and examples.
Cites user21820 · 2017-04 · Mathematics Stack Exchange · on “What is the difference between a generalization and an extension?”
I agree with Joonas that "extension" can be used to refer to objects, such as functions extended to a larger domain while preserving some properties, or in field extensions (a larger field containing the original one). However, we do often use the same word to refer to generalizations of a theorem as well, namely we could say something like "We can easily extend this result to an arbitrary metric space.". Hence you see things like " extended mean-value theorem [LINK] " and " extended Bayes theorem [LINK] " and " extended Green's theorem [this video] ". Also, the Wikipedia article about the binomial theorem [LINK] even uses both words together: "The generalized binomial theorem can be extended to