Vector Calculus - Lecture 16: Introduction to Green's Theorem

Описание: We introduce Green's theorem, which shows that double integrals can be computed via vector line integrals and vice-versa. We also discuss what the "orientation" of a curve refers to, and why it matters when computing vector line integrals (but not scalar line integrals).

Textbook: "Vector Calculus" by Susan J. Colley and Santiago Cañez
Canada link: https://www.amazon.ca/dp/B09M8DL4TJ/&...
USA link: https://www.amazon.com/dp/B09M8DL4TJ/...

Vector Calculus playlist:    • Vector Calculus  
Previous lecture:    • Vector Calculus - Lecture 15: Examples and...  
Next lecture:    • Vector Calculus - Lecture 17: Making Use o...  

Blank course notes (lectures 16-19): https://njohnston.ca/vector_calculus/...
Annotated course notes (lectures 16-19): https://njohnston.ca/vector_calculus/...

Desmos graphs used in this video:
Scalar line integrals are path-independent: https://www.desmos.com/calculator/5ls...
Vector line integrals are path-independent (except for orientation): https://www.desmos.com/calculator/oql...
Green's theorem example as a vector line integral: https://www.desmos.com/calculator/0c0...
Green's theorem example as a double integral: https://www.desmos.com/3d/h9nbjcev65

Please leave a comment below if you have any questions, comments, or corrections.

Timestamps:
00:00 - Introduction and statement of theorem
02:50 - Discussing notation and hypotheses of the theorem
09:20 - Orientation of a curve and why it matters
13:51 - Example: verifying Green's theorem

#vector_calculus #greens_theorem #line_integral