Could someone check if I have done this right.(adsbygoogle = window.adsbygoogle || []).push({});

[tex]R_1 = x^2\partial_3 - x^3\partial_2[/tex]

[tex]R_2 = x^3\partial_1 - x^1\partial_3[/tex]

[tex]R_3 = x^1\partial_2 - x^2\partial_1[/tex]

Where [itex]x^i[/itex] are coordinates.

I need to calculate the commutator [itex][R_1,R_2][/itex].

[tex][R_1,R_2] = x^2\partial_3x^3\partial_1 - x^3\partial_1x^2\partial_3 + x^3\partial_2x^1\partial_3 - x^1\partial_3x^3\partial_2

- (x^3\partial_2x^3\partial_1-x^3\partial_1x^3\partial_2) - (x^2\partial_3x^1\partial_3-x^1\partial_3x^2\partial_3)

[/tex]

[tex]

.\quad\quad\quad = x^2x^3\partial_3\partial_1 - x^2x^3\partial_1\partial_3 + x^1x^3\partial_2\partial_3-x^1x^3\partial_2\partial_3

- x^3x^3\partial_2\partial_1 + x^3x^3\partial_1\partial_2-x^1x^2\partial_3\partial_3+x^1x^2\partial_3\partial_3

[/tex]

[tex].\quad\quad\quad = x^2\partial_1-x^2\partial_1+x^1\partial_2-x^1\partial_2 -x^3x^3\partial_2\partial_1 + x^3x^3\partial_1\partial_2 - x^1x^2\partial_3\partial_3 + x^1x^2\partial_3\partial_3

[/tex]

[tex]

.\quad\quad\quad = 0

[/tex]

And as a result,

[tex][R_1,R_2] = [R_2,R_3] = [R_3,R_1] = 0[/tex]

by cyclically permuting the indices.

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