Walk ε along v then w — vs w then v.
The gap closes to ε2·[v,w] as ε → 0
— shown both in the chart (left) and lifted to a curved manifold (right).
v then w (endpoint A)
w then v (endpoint B)
gap A − B
Step size
0.40
Field pair
Two paths, one starting point. Pick a base point p
(drag it on the canvas) and a step size ε. The
green path walks ε along
v, then ε along w; the
blue path walks ε along
w, then ε along v. In a flat space with
v, w commuting, both paths land at the same point.
In general the endpoints differ by
A − B = ε2 [v, w]|p + O(ε3),
with the Lie bracket
[v,w]j = vi ∂iwj − wi ∂ivj.
Drag the ε slider down: the gap shrinks like ε2.
Now turn on Rescale by 1/ε2: the rescaled
gap arrow stops shrinking and locks onto the
analytic [v,w] vector
— the bracket emerges as the ε → 0 limit of the
parallelogram defect.