Let omega=x dx+y dy+z dz be a 1-form

[carbis]

Here an excercise I found between old exams:
Let omega=x dx+y dy+z dz be a 1-form on R^3 and X=(x,y,z) a vectorfield.
Compute L_X(omega) as derivative of the pullback of omega under the flow generated by X.
I think the flow generated by X is (e^t,e^t,e^t), but I don't know how to proceed (computing the pullback, inserting omega en differentating at t=0.
Anyone?

 

[Reb]

Here an excercise I found between old exams:
Let omega=x dx+y dy+z dz be a 1-form on R^3 and X=(x,y,z) a vectorfield.
Compute L_X(omega) as derivative of the pullback of omega under the flow generated by X.
I think the flow generated by X is (e^t,e^t,e^t), but I don't know how to proceed (computing the pullback, inserting omega en differentating at t=0.
Anyone?

Apply Cartan's formula: $$L_X(\omega)=i_X d\omega+d(i_X\omega)$$, $$d\omega$$ being the exterior derivative and $$i_X(\omega)$$ the antiderivation.

 

[George Jones]

While this is a nice answer, the thread is over four years old, and it has been more than two years since the original poster lat signed in.

 

[Reb]

While this is a nice answer, the thread is over four years old, and it has been more than two years since the original poster lat signed in.

Yeap, I guess Carbis won't be needing it for his homework anymore.


Anyhow, I enjoy digging up interesting questions that no-one ever bothered to answer...
It's something like dating a middle-aged virgin.
(Will this get me banned? damn...:yuck:)

 

[John Creighto]

Yeap, I guess Carbis won't be needing it for his homework anymore.


Anyhow, I enjoy digging up interesting questions that no-one ever bothered to answer...
It's something like dating a middle-aged virgin.
(Will this get me banned? damn...:yuck:)

I think answering old questions is good because when someone Googles the question, then they'll be more likely to find the answer.