I'm reading a book on calculus, the part about vector fields on manifolds. It's a nice book, but with a severe drawback --- it has no pictures.

I like how vectors are treated algebraically, as derivatives over a local ring (ring of germs). But I still want to use "geometrical" view on vector fields.

The problem is I can't imagine vector field "multiplication" as a composition of derivatives. And thus I can't picture commutator of two vector fields.

Has anybody here got pictures too help me?


In Gauge fields, knots and gravity by J. Baez and J. P. Muniain authors got the following two pictures to visualize commutator:
enter image description here enter image description here

I hope it helps you.

  • 5
    $\begingroup$ Nice find! It would help a bit if separately they'd also shown what the vector field $w$ and $v$ are like individually, so it is clear that the $w$ at the base of the $v$ and the $w$ at the head of the $v$ are not parallel. $\endgroup$ – Willie Wong Jun 26 '12 at 11:04

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