# Visualizing Commutator of Two Vector Fields

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?

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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. –  Willie Wong Jun 26 '12 at 11:04