Consider two vector fields X and Y and an arbitrary smooth scalar function f(x). The Lie derivative of f along X is given by Lx(f) = X(f) = X"af. Similarly, in the lectures we saw that the Lie derivative of Y along X, written as CxY, is given by the commutator [X,Y]. From the definition of [X, Y] acting on f as in the notes, [X,Y](f) = X(Y(f)) – Y(X(ƒ)), (a) Show that the components of the vector field [X, Y] are [X,Y|@ = X@Y@ _Y9Xa. (b) Show that [X,Y]a transforms as a vector under general coordinate transformations. (c) Consider a one form wa. Using that the Lie derivative satisfies the Leibniz rule and recalling that way is a scalar, show that Lx (w)a = Xbwa+w₁daxb. (d) Is Lx(w), a one form?
Consider two vector fields X and Y and an arbitrary smooth scalar function f(x). The Lie derivative of f along X is given by Lx(f) = X(f) = X"af. Similarly, in the lectures we saw that the Lie derivative of Y along X, written as CxY, is given by the commutator [X,Y]. From the definition of [X, Y] acting on f as in the notes, [X,Y](f) = X(Y(f)) – Y(X(ƒ)), (a) Show that the components of the vector field [X, Y] are [X,Y|@ = X@Y@ _Y9Xa. (b) Show that [X,Y]a transforms as a vector under general coordinate transformations. (c) Consider a one form wa. Using that the Lie derivative satisfies the Leibniz rule and recalling that way is a scalar, show that Lx (w)a = Xbwa+w₁daxb. (d) Is Lx(w), a one form?
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![Consider two vector fields X and Y and an arbitrary
smooth scalar function f(x). The Lie derivative of f along X is given by
Lx(f) = X(f) = X"af. Similarly, in the lectures we saw that the Lie derivative of Y
along X, written as CxY, is given by the commutator [X,Y]. From the definition of
[X, Y] acting on f as in the notes,
[X,Y](f) = X(Y(f)) – Y(X(ƒ)),
(a) Show that the components of the vector field [X, Y] are
[X,Y|@ = X@Y@ _Y9Xa.
(b) Show that [X,Y]a transforms as a vector under general coordinate
transformations.
(c) Consider a one form wa. Using that the Lie derivative satisfies the Leibniz rule
and recalling that way is a scalar, show that
Lx (w)a = Xbwa+w₁daxb.
(d) Is Lx(w), a one form?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0de9773e-39c1-4df6-a7d6-864501c7f552%2F1112a3d6-1c2f-4799-88e5-0c8b8ebaa631%2F4gmgzz_processed.png&w=3840&q=75)
Transcribed Image Text:Consider two vector fields X and Y and an arbitrary
smooth scalar function f(x). The Lie derivative of f along X is given by
Lx(f) = X(f) = X"af. Similarly, in the lectures we saw that the Lie derivative of Y
along X, written as CxY, is given by the commutator [X,Y]. From the definition of
[X, Y] acting on f as in the notes,
[X,Y](f) = X(Y(f)) – Y(X(ƒ)),
(a) Show that the components of the vector field [X, Y] are
[X,Y|@ = X@Y@ _Y9Xa.
(b) Show that [X,Y]a transforms as a vector under general coordinate
transformations.
(c) Consider a one form wa. Using that the Lie derivative satisfies the Leibniz rule
and recalling that way is a scalar, show that
Lx (w)a = Xbwa+w₁daxb.
(d) Is Lx(w), a one form?
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