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Let me first show how it works in the goodold 2D Euclidean plane. Since that space is flat, you can choose Cartesian coordinates, and you don't have to worry about connection coefficients.
Now, suppose we have two different vector fields, [itex]\vec{u}[/itex] and [itex]\vec{v}[/itex]. Then consider two different paths starting from the point [itex]p_0[/itex]
In general, [itex]p_2 \neq p_4[/itex]. We can compute, to lowest order, these two locations, as follows:
[itex]p_1 = p_0 + \beta\vec{v}(p_0)[/itex]
[itex]p_2 = p_1 + \alpha \vec{u}(p_1) \approx p_0 + \beta\vec{v}(p_0) + \alpha\vec{u}(p_0) + \alpha \beta ((\vec{v} \cdot \nabla) \vec{u})_{p_0}[/itex]
[itex]p_3 = p_0 + \alpha \vec{u}(p_0)[/itex]
[itex]p_4 = p_3 + \beta \vec{v}(p_3) \approx p_0 + \beta\vec{v}(p_0) + \alpha \vec{u}(p_0) + \alpha \beta ((\vec{u} \cdot \nabla) \vec{v})_{p_0}[/itex]
So the displacement vector from [itex]p_2[/itex] to [itex]p_4[/itex] is given by:
[itex]\vec{\delta_{24}} = p_4  p_2 = \alpha \beta ((\vec{u} \cdot \nabla) \vec{v}  (\vec{v} \cdot \nabla) \vec{u})_{p_0}[/itex]
[itex] \equiv [u,v]_{p_0}[/itex]
Now, suppose we have two different vector fields, [itex]\vec{u}[/itex] and [itex]\vec{v}[/itex]. Then consider two different paths starting from the point [itex]p_0[/itex]
 Move along displacement vector [itex]\alpha \vec{u}[/itex], and then move along displacement vector [itex]\beta \vec{v}[/itex].
 Move along displacement vector [itex]\beta \vec{v}[/itex], and then move along displacement vector [itex]\alpha \vec{u}[/itex].
In general, [itex]p_2 \neq p_4[/itex]. We can compute, to lowest order, these two locations, as follows:
[itex]p_1 = p_0 + \beta\vec{v}(p_0)[/itex]
[itex]p_2 = p_1 + \alpha \vec{u}(p_1) \approx p_0 + \beta\vec{v}(p_0) + \alpha\vec{u}(p_0) + \alpha \beta ((\vec{v} \cdot \nabla) \vec{u})_{p_0}[/itex]
[itex]p_3 = p_0 + \alpha \vec{u}(p_0)[/itex]
[itex]p_4 = p_3 + \beta \vec{v}(p_3) \approx p_0 + \beta\vec{v}(p_0) + \alpha \vec{u}(p_0) + \alpha \beta ((\vec{u} \cdot \nabla) \vec{v})_{p_0}[/itex]
So the displacement vector from [itex]p_2[/itex] to [itex]p_4[/itex] is given by:
[itex]\vec{\delta_{24}} = p_4  p_2 = \alpha \beta ((\vec{u} \cdot \nabla) \vec{v}  (\vec{v} \cdot \nabla) \vec{u})_{p_0}[/itex]
[itex] \equiv [u,v]_{p_0}[/itex]
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