proof of dα(X,Y)=X(α(Y)) - Y(α(X)) - α([X,Y]) (local coordinates)

Since this result is local (in other words, the identityPlanetmathPlanetmathPlanetmath holds on the whole manifold if and only if its restrictionPlanetmathPlanetmathPlanetmathPlanetmath to every coordinate patch of the manifold holds), it suffices to demonstrate it in a local coordinate system. To do this, we shall compute coordinate expressions for the various terms and verify that the sum of terms on the right-hand side equals the left-hand side.


Upon combining the right-hand sides of the last three equations and cancelling common terms, we obtain


Upon renaming dummy indices (switching i with j), the second and fourth terms cancel. What remains is exactly the right-hand side of the first equation. Hence, we have

Title proof of dα(X,Y)=X(α(Y)) - Y(α(X)) - α([X,Y]) (local coordinates)
Canonical name ProofOfDalphaXYXalphaYYalphaXalphaXYlocalCoordinates
Date of creation 2013-03-22 15:34:01
Last modified on 2013-03-22 15:34:01
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 8
Author rspuzio (6075)
Entry type Proof
Classification msc 53-00