The most important theorem about vector fields, Stokes' theorem, which relates the surface integral of the curl with the line integral over the boundary. Here orientation matters, which I'll explain as well.
Old Stokes Theorem Video https://youtu.be/bDILtddFKxw
Vector Calculus Playlist: https://youtube.com/playlist?list=PLJb1qAQIrmmAU22cfpgxqSR5iB7jxh933
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