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From: eric gisse on 9 Jun 2010 21:06
Pol Lux wrote:

> On Jun 7, 1:30 am, eric gisse <jowr.pi.nos...(a)gmail.com> wrote:
>> Pollux wrote:
>> > I'm looking for examples of simple contravariant and covariant
>> > quantities in physics. I understand that position is contravariant, and
>> > a gradient covariant (please correct if I'm mistaken). Any other
>> > quantities? How about mixed contravariant/covariant quantities? What
>> > would be the simplest example?
>>
>> > Thanks,
>>
>> > Pollux
>>
>> > --- news://freenews.netfront.net/ - complaints: n...(a)netfront.net ---
>>
>> Vector - covariant.
>> Vector in frequency domain [Fourier transform] - contravariant.
>> Riemann tensor R^a_bcd - mixed
>> T^a_a [trace of tensor] - mixed.
>
> OK. So for a very trivial example, the slope of a mountain path, in
> units of "meters per meters" would be covariant, because 10 m/m go to
> 10000 km/km in "kilometers per kilometers" (slope is also a gradient,
> right?).

Gradient is contravariant.

>
> Can I ask why we bother with mixed tensors?

Because. That's why.

> You can use the metric to
> change mixed tensors to all contravariant or all covariant tensors,
> right? Is it easier to work with mixed tensors?

No, it is equally easy. It is also required usually.

>
> Pollux

From: Pol Lux on 9 Jun 2010 21:41
On Jun 9, 6:06 pm, eric gisse <jowr.pi.nos...(a)gmail.com> wrote:
> Pol Lux wrote:
> > On Jun 7, 1:30 am, eric gisse <jowr.pi.nos...(a)gmail.com> wrote:
> >> Pollux wrote:
> >> > I'm looking for examples of simple contravariant and covariant
> >> > quantities in physics. I understand that position is contravariant, and
> >> > a gradient covariant (please correct if I'm mistaken). Any other
> >> > quantities? How about mixed contravariant/covariant quantities? What
> >> > would be the simplest example?
>
> >> > Thanks,
>
> >> > Pollux
>
> >> > --- news://freenews.netfront.net/ - complaints: n...(a)netfront.net ---
>
> >> Vector - covariant.
> >> Vector in frequency domain [Fourier transform] - contravariant.
> >> Riemann tensor R^a_bcd - mixed
> >> T^a_a [trace of tensor] - mixed.
>
> > OK. So for a very trivial example, the slope of a mountain path, in
> > units of "meters per meters" would be covariant, because 10 m/m go to
> > 10000 km/km in "kilometers per kilometers" (slope is also a gradient,
> > right?).
>
> Gradient is contravariant.
You think so?

> > Can I ask why we bother with mixed tensors?
>
> Because. That's why.
Very smart.

> > You can use the metric to
> > change mixed tensors to all contravariant or all covariant tensors,
> > right? Is it easier to work with mixed tensors?
>
> No, it is equally easy. It is also required usually.
Required?

Pollux
From: eric gisse on 10 Jun 2010 02:47
Pol Lux wrote:

> On Jun 9, 6:06 pm, eric gisse <jowr.pi.nos...(a)gmail.com> wrote:
>> Pol Lux wrote:
>> > On Jun 7, 1:30 am, eric gisse <jowr.pi.nos...(a)gmail.com> wrote:
>> >> Pollux wrote:
>> >> > I'm looking for examples of simple contravariant and covariant
>> >> > quantities in physics. I understand that position is contravariant,
>> >> > and a gradient covariant (please correct if I'm mistaken). Any other
>> >> > quantities? How about mixed contravariant/covariant quantities? What
>> >> > would be the simplest example?
>>
>> >> > Thanks,
>>
>> >> > Pollux
>>
>> >> > --- news://freenews.netfront.net/ - complaints: n...(a)netfront.net
>> >> > ---
>>
>> >> Vector - covariant.
>> >> Vector in frequency domain [Fourier transform] - contravariant.
>> >> Riemann tensor R^a_bcd - mixed
>> >> T^a_a [trace of tensor] - mixed.
>>
>> > OK. So for a very trivial example, the slope of a mountain path, in
>> > units of "meters per meters" would be covariant, because 10 m/m go to
>> > 10000 km/km in "kilometers per kilometers" (slope is also a gradient,
>> > right?).
>>
>> Gradient is contravariant.
> You think so?

I think one-forms are contravariant.

>
>> > Can I ask why we bother with mixed tensors?
>>
>> Because. That's why.
> Very smart.

What do you expect? You are asking the tensorial equivalent of 'why do we
bother with vectors'? They are a handy mathematical construct.

>
>> > You can use the metric to
>> > change mixed tensors to all contravariant or all covariant tensors,
>> > right? Is it easier to work with mixed tensors?
>>
>> No, it is equally easy. It is also required usually.
> Required?

Because the input index can't always be arbitrary? Besides, raise/lower the
index with the metric - learn index gymnastics. It is way simpler than you
think.

>
> Pollux

From: eric gisse on 10 Jun 2010 03:02
Rock Brentwood wrote:

> Pollux wrote:
>> I'm looking for examples of simple contravariant and covariant
>> quantities in physics.
>
> On Jun 7, 3:30 am, eric gisse <jowr.pi.nos...(a)gmail.com> wrote:
>> Vector - covariant.
>> Vector in frequency domain [Fourier transform] - contravariant.
>
> Ignoring the untutored replies from people self-confident in their
> ignorance [...]

If you think its' wrong, say its' wrong. I'm a grownup.

The passive aggressive routine is sad.
From: Pollux on 10 Jun 2010 09:31
(6/10/10 12:02 AM), eric gisse wrote:
> Rock Brentwood wrote:
>
>> Pollux wrote:
>>> I'm looking for examples of simple contravariant and covariant
>>> quantities in physics.
>>
>> On Jun 7, 3:30 am, eric gisse<jowr.pi.nos...(a)gmail.com> wrote:
>>> Vector - covariant.
>>> Vector in frequency domain [Fourier transform] - contravariant.
>>
>> Ignoring the untutored replies from people self-confident in their
>> ignorance [...]
>
> If you think its' wrong, say its' wrong. I'm a grownup.
>
> The passive aggressive routine is sad.
Eric, look up contravariant/covariant on wikipedia.

Pollux

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