| Topic: |
Science > Physics |
| User: |
"" |
| Date: |
02 Mar 2006 09:00:54 PM |
| Object: |
meaning of contravariant/covariant |
Would somebody out there please illuminate for me what the relationship
is between the terms
contravariant/covariant
when they on the one hand refer to
1) the way a tensor transforms
and on the other hand refer to
2) different conventions for interpreting the components of a tensor
with respect to a given coordinate system (paralles or perpendicular to
the coordinate axes).
I sort of understand the basic idea, but I still somtimes can't see the
forest for the trees.
Thanks,
Wolfgang,
Santa Barbara, CA
.
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| User: "Ken S. Tucker" |
|
| Title: Re: meaning of contravariant/covariant |
03 Mar 2006 01:03:42 PM |
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Ok...Bruce Scott is of course correct, however
that's standardized text stuff.
mot12345@alexandria.ucsb.edu wrote:
Would somebody out there please illuminate for me what the relationship
is between the terms
contravariant/covariant
when they on the one hand refer to
1) the way a tensor transforms
and on the other hand refer to
2) different conventions for interpreting the components of a tensor
with respect to a given coordinate system (paralles or perpendicular to
the coordinate axes).
That's true too at a basic level.
I sort of understand the basic idea, but I still somtimes can't see the
forest for the trees.
The "covariant" components form "invariants"
but the contravariant form "relations". Let me
demo that...using (& is a partial),
A'^u = ( &x'^u /&x^v ) A^v
A'_u = ( &x^v / &x'^u ) A_v
then find using differential products dx^v and dx'^u
A'^u dx^v = dx'^u A^v (relational)
A'_u dx'^u = A_v dx^v (covariant and invariant).
The relational equation depends upon values
obtained from relational Coordinate Systems,
K' and K, but the covariant tensor transform
establishes an invariant " A_v dx^v ", true in
all *proper* CS's.
As an excersize transform between a CS
using "centimeters" and a CS using "inches".
Once you do that, you can fold those results
in the metric tensor "g_uv". Let me know how
that goes...
Regards
Ken S. Tucker
Thanks,
Wolfgang,
Santa Barbara, CA
.
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| User: "Bruce Scott TOK" |
|
| Title: Re: meaning of contravariant/covariant |
03 Mar 2006 06:21:49 AM |
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Wolfgang wrote:
Would somebody out there please illuminate for me what the relationship
is between the terms
contravariant/covariant
when they on the one hand refer to
1) the way a tensor transforms
and on the other hand refer to
2) different conventions for interpreting the components of a tensor
with respect to a given coordinate system (paralles or perpendicular to
the coordinate axes).
I sort of understand the basic idea, but I still somtimes can't see the
forest for the trees.
Basically, these are different representations. Given a coordinate
basis (x^a), you can form the gradients of the coordinates to get a set
of basis vectors, grad x^a. Now you can try either way to represent any
given vector V, as a set of dot products:
V^a = V dot gradx^a
or as a set of coefficients (sum of repeated indiced implied):
V = V_a grad x^a
The former are contravariant (varying against the coordinate gradients)
and the latter are covariant (varying with the coordinate gradients).
The transformation rules then basically follow from application of the
chain rule on the transformation matrix dy^b/dx^a between coordinate
systems x and y. A vector is a vector if you then get
V = V_a grad x^a = V'_b grad y^b
i.e., the same _form_ (not necessarily the same _values_) in any
coordinate system.
The rest is a grunge of algebra... going through that as a student is
mandatory if you want the understanding. Of course, building serious
computations using this stuff helps :-)
--
ciao,
Bruce
drift wave turbulence: http://www.rzg.mpg.de/~bds/
.
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| User: "FrediFizzx" |
|
| Title: Re: meaning of contravariant/covariant |
03 Mar 2006 12:35:31 PM |
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|
"Bruce Scott TOK" <Use-Author-Supplied-Address-Header@[127.1]> wrote in
message news:200603031221.k23CLn4v015984@ipp.mpg.de...
| Wolfgang wrote:
|
| >Would somebody out there please illuminate for me what the
relationship
| >is between the terms
| >contravariant/covariant
| >when they on the one hand refer to
| >
| >1) the way a tensor transforms
| >
| > and on the other hand refer to
| >
| >2) different conventions for interpreting the components of a tensor
| >with respect to a given coordinate system (paralles or perpendicular
to
| >the coordinate axes).
| >
| >I sort of understand the basic idea, but I still somtimes can't see
the
| >forest for the trees.
|
| Basically, these are different representations. Given a coordinate
| basis (x^a), you can form the gradients of the coordinates to get a
set
| of basis vectors, grad x^a. Now you can try either way to represent
any
| given vector V, as a set of dot products:
|
| V^a = V dot gradx^a
|
| or as a set of coefficients (sum of repeated indiced implied):
|
| V = V_a grad x^a
|
| The former are contravariant (varying against the coordinate
gradients)
| and the latter are covariant (varying with the coordinate gradients).
This is one of the best explanations I have ever seen! Why don't they
just say what you said in the text books?
FrediFizzx
| The transformation rules then basically follow from application of the
| chain rule on the transformation matrix dy^b/dx^a between coordinate
| systems x and y. A vector is a vector if you then get
|
| V = V_a grad x^a = V'_b grad y^b
|
| i.e., the same _form_ (not necessarily the same _values_) in any
| coordinate system.
|
| The rest is a grunge of algebra... going through that as a student is
| mandatory if you want the understanding. Of course, building serious
| computations using this stuff helps :-)
|
| --
| ciao,
| Bruce
|
| drift wave turbulence: http://www.rzg.mpg.de/~bds/
|
.
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|
| User: "PD" |
|
| Title: Re: meaning of contravariant/covariant |
03 Mar 2006 10:58:19 AM |
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|
Bruce Scott TOK wrote:
Wolfgang wrote:
Would somebody out there please illuminate for me what the relationship
is between the terms
contravariant/covariant
when they on the one hand refer to
1) the way a tensor transforms
and on the other hand refer to
2) different conventions for interpreting the components of a tensor
with respect to a given coordinate system (paralles or perpendicular to
the coordinate axes).
I sort of understand the basic idea, but I still somtimes can't see the
forest for the trees.
Basically, these are different representations. Given a coordinate
basis (x^a), you can form the gradients of the coordinates to get a set
of basis vectors, grad x^a. Now you can try either way to represent any
given vector V, as a set of dot products:
V^a = V dot gradx^a
or as a set of coefficients (sum of repeated indiced implied):
V = V_a grad x^a
The former are contravariant (varying against the coordinate gradients)
and the latter are covariant (varying with the coordinate gradients).
This is very useful, Bruce, and gives a nice little extra handle on
what a 1-form is.
The transformation rules then basically follow from application of the
chain rule on the transformation matrix dy^b/dx^a between coordinate
systems x and y. A vector is a vector if you then get
V = V_a grad x^a = V'_b grad y^b
i.e., the same _form_ (not necessarily the same _values_) in any
coordinate system.
The rest is a grunge of algebra... going through that as a student is
mandatory if you want the understanding. Of course, building serious
computations using this stuff helps :-)
--
ciao,
Bruce
drift wave turbulence: http://www.rzg.mpg.de/~bds/
.
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| User: "" |
|
| Title: Re: meaning of contravariant/covariant |
03 Mar 2006 06:44:21 AM |
|
|
In article <200603031221.k23CLn4v015984@ipp.mpg.de>,
Bruce Scott TOK <Use-Author-Supplied-Address-Header@[127.1]> wrote:
Wolfgang wrote:
Would somebody out there please illuminate for me what the relationship
is between the terms
contravariant/covariant
when they on the one hand refer to
1) the way a tensor transforms
and on the other hand refer to
2) different conventions for interpreting the components of a tensor
with respect to a given coordinate system (paralles or perpendicular to
the coordinate axes).
I sort of understand the basic idea, but I still somtimes can't see the
forest for the trees.
Basically, these are different representations. Given a coordinate
basis (x^a), you can form the gradients of the coordinates to get a set
of basis vectors, grad x^a. Now you can try either way to represent any
given vector V, as a set of dot products:
V^a = V dot gradx^a
or as a set of coefficients (sum of repeated indiced implied):
V = V_a grad x^a
The former are contravariant (varying against the coordinate gradients)
and the latter are covariant (varying with the coordinate gradients).
The transformation rules then basically follow from application of the
chain rule on the transformation matrix dy^b/dx^a between coordinate
systems x and y. A vector is a vector if you then get
V = V_a grad x^a = V'_b grad y^b
i.e., the same _form_ (not necessarily the same _values_) in any
coordinate system.
The rest is a grunge of algebra... going through that as a student is
mandatory if you want the understanding. Of course, building serious
computations using this stuff helps :-)
About 8, maybe 10, years ago, John Baez, Ed Green, and
[names I can't recall] had a pretty lengthy discussion
here about these two concepts. Perhaps the thread is
still available, or somebody squirreled it away on a disk
that didn't die last summer.
/BAH
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