- #1
JuanC97
- 48
- 0
Hello guys, I have a question regarding commutators of vector fields and its pushforwards.
Let me define a clockwise rotation in the plane [itex] \,\phi:\mathbb{R}^2\rightarrow\mathbb{R}^2 \,.\; [\,\partial_x\,,\,\partial_y\,]=0 \,[/itex], [itex] \;(\phi_{*}\partial_x) = \partial_r [/itex] and [itex] \,(\phi_{*}\partial_y) = \partial_\theta [/itex]
I'd like to compute [itex] [\,
\frac{x\partial_x+\,y\partial_y}{\sqrt{x^2+y^2}}
\,,\,
\frac{x\partial_y-\,y\partial_x}{\sqrt{x^2+y^2}}
\,][/itex], i.e: [itex] [\,
(\cos\hspace{-0.025cm}\theta)\partial_x+(\sin\hspace{-0.025cm}\theta)\partial_y
\,,\,
-(\sin\hspace{-0.025cm}\theta)\partial_x+(\cos\hspace{-0.025cm}\theta)\partial_y
\,][/itex].
We can identify this result with [itex][\,
\partial_r
\,,\,
\partial_\theta
\,]
=
[\,
\phi_{*}\partial_x
\,,\,
\phi_{*}\partial_y
\,] [/itex] but we know that [itex] [\,
\phi_{*}\partial_x
\,,\,
\phi_{*}\partial_y
\,] = \phi_{*}[\, \partial_x \,,\, \partial_y \,] [/itex]
hence [itex][\,
\partial_r
\,,\,
\partial_\theta
\,] = 0 [/itex]. But... I don't know, this seems to contradict some ideas I had before about holonomic bases.
As far as I knew, if one defines a new basis [itex]\{e_{\mu'}\}[/itex] in terms of an holonomic basis [itex]\{\partial_{\mu}\}[/itex] such that [itex] e_{\mu'} = A_{\mu'}^{\mu}\partial_\mu [/itex], one should not expect this new basis to be holonomic but, following the reasoning I just showed you above, one could always identify the action of the matrix A with the action of the pushforward of some map, ending up again with a situation like [itex] [\, e_{\mu'} \,,\, e_{\nu'} \,] = \phi_{*}[\, \partial_\mu \,,\, \partial_\nu \,] = 0[/itex] which would explicitly contradict the original idea about what to expect (or not) about this kind of bases.
Could anyone, please, help me to understand where is the problem in this reasoning (if there's one)?.
Let me define a clockwise rotation in the plane [itex] \,\phi:\mathbb{R}^2\rightarrow\mathbb{R}^2 \,.\; [\,\partial_x\,,\,\partial_y\,]=0 \,[/itex], [itex] \;(\phi_{*}\partial_x) = \partial_r [/itex] and [itex] \,(\phi_{*}\partial_y) = \partial_\theta [/itex]
I'd like to compute [itex] [\,
\frac{x\partial_x+\,y\partial_y}{\sqrt{x^2+y^2}}
\,,\,
\frac{x\partial_y-\,y\partial_x}{\sqrt{x^2+y^2}}
\,][/itex], i.e: [itex] [\,
(\cos\hspace{-0.025cm}\theta)\partial_x+(\sin\hspace{-0.025cm}\theta)\partial_y
\,,\,
-(\sin\hspace{-0.025cm}\theta)\partial_x+(\cos\hspace{-0.025cm}\theta)\partial_y
\,][/itex].
We can identify this result with [itex][\,
\partial_r
\,,\,
\partial_\theta
\,]
=
[\,
\phi_{*}\partial_x
\,,\,
\phi_{*}\partial_y
\,] [/itex] but we know that [itex] [\,
\phi_{*}\partial_x
\,,\,
\phi_{*}\partial_y
\,] = \phi_{*}[\, \partial_x \,,\, \partial_y \,] [/itex]
hence [itex][\,
\partial_r
\,,\,
\partial_\theta
\,] = 0 [/itex]. But... I don't know, this seems to contradict some ideas I had before about holonomic bases.
As far as I knew, if one defines a new basis [itex]\{e_{\mu'}\}[/itex] in terms of an holonomic basis [itex]\{\partial_{\mu}\}[/itex] such that [itex] e_{\mu'} = A_{\mu'}^{\mu}\partial_\mu [/itex], one should not expect this new basis to be holonomic but, following the reasoning I just showed you above, one could always identify the action of the matrix A with the action of the pushforward of some map, ending up again with a situation like [itex] [\, e_{\mu'} \,,\, e_{\nu'} \,] = \phi_{*}[\, \partial_\mu \,,\, \partial_\nu \,] = 0[/itex] which would explicitly contradict the original idea about what to expect (or not) about this kind of bases.
Could anyone, please, help me to understand where is the problem in this reasoning (if there's one)?.
Last edited: