We present a theorem that says that a vector field has path-independent line integrals if and only if it is a gradient field.
Textbook: "Vector Calculus" by Susan J. Colley and Santiago Cañez
Canada link: https://www.amazon.ca/dp/B09M8DL4TJ/&tag=veccalc06-20
USA link: https://www.amazon.com/dp/B09M8DL4TJ/&tag=veccalc-20
Vector Calculus playlist: https://www.youtube.com/playlist?list=PLOAf1ViVP13haWs-MkyL9u_r8pMgFoWT6
Previous lecture: https://youtu.be/THx7Il1fQh8
Blank course notes (lectures 20-23): https://njohnston.ca/vector_calculus/week6.pdf
Annotated course notes (lectures 20-23): https://njohnston.ca/vector_calculus/week6_annotated.pdf
Desmos graphs used in this video:
Line segment and parabolic arc example: https://www.desmos.com/calculator/mt6i9cz2hw
A non-path-independent example: https://www.desmos.com/calculator/uqlykxc8sg
Please leave a comment below if you have any questions, comments, or corrections.
Timestamps:
00:00 - Introduction
00:46 - The main theorem
08:18 - Our first example
10:58 - A non-conservative example
#vector_calculus #line_integral #path_independence
