Lie derivative question

[Daniel Volinski]

Hi All,

Please take a look at the attached file. My question refers to the last result: I get zero for the lie derivative.
It should depend on the values of v1, v2, v3, not zero. What I am missing?

Thanks,

Daniel

[Eberhard Schruefer]

Thanks for the report. You are of course right that the result shouldn't be zero.
Way, way back I made the assumption that when you are using 'natural' vectors, i.e. vectors of the form '@ x', then every thing you will be manipulating is explicitly expressed in these coordiantes. I see that this assumption is not a good one.

If you would have expressed your vector vr by
tvector x1,x2,x3;
vr := v1 * x1 + v2 * x2 + v3 * x3;
then you would have gotten the result you expected

29: vr |_ f2;

d(x1 _| f2)*v1 + d(x2 _| f2)*v2 + d(x3 _| f2)*v3 + x1 _| d f2*v1 + x2 _| d f2*v2

 + x3 _| d f2*v3 - x1 _| f2^d v1 - x2 _| f2^d v2 - x3 _| f2^d v3

I'll most likely remove the above said assumption from the code in the very near future.

Eberhard

[Daniel Volinski]

Hi Eberhard,

Actually I was looking for the following result:
(-diff(v3,z)*x-diff(v2,y)*x-v1)*dy^dz-(diff(v2,x))*x*dx^dz+(diff(v3,x))*x*dx^dy;
Daniel

[Eberhard Schruefer]

There must be something missing. You must have assigned something to f2 otherwise f2 would show up in the result you are looking for.

Eberhard

[Daniel Volinski]

Hi Eberhard,

You are right, of course; my bad.
I forgot to include the following line before the Lie derivative:
f2 := d z ^ f1;
Then, I get the result I was expecting.
Daniel

[Daniel Volinski]

Now I would like to ask you about the style, how the result is presented:

Example
f1 := x ^ d y + z ^ d z;
I get as result:
f1 := d y x + d z z
I would like the result to be shown as:
f1 := x d y + z d z;
as in books on the subject

Another example:
f2 := d z ^ f1;
is shown as:
f2 := - d y ^ d z x
I would like the result to be shown as:
f2 := - x d y ^ d z
again, as in books on the subject

vr |_ f2;
I would like to be presented as
(coeff1) d x ^ d y + (coeff2) d y ^ d z + (coeff3) d z ^ d x
that is, all the terms containing the same basic 2-form to be gathered together.

Is that possible?

Thanks,

Daniel