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From: Pollux on 10 Jun 2010 09:32
(6/9/10 11:47 PM), eric gisse wrote:
> 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.
Look up wikipedia.

>
>>
>>>> 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.
Missing the point. The point was: why work with mixed tensors when you
can raise/lower indices to get only covariant or only contravariant tensors.

>
>>
>>>> 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.
What is an "input" index?

Pollux

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