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In [[tensor analysis]], a '''mixed tensor''' is a [[tensor]] which is neither strictly [[Covariance and contravariance of vectors|covariant]] nor strictly [[Covariance and contravariance of vectors|contravariant]]; at least one of the indices of a mixed tensor will be a subscript (covariant) and at least one of the indices will be a superscript (contravariant).
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A mixed tensor of '''type''' or '''valence''' <math>\scriptstyle\binom{M}{N}</math>, also written "type (''M'', ''N'')", with both ''M'' > 0 and ''N'' > 0, is a tensor which has ''M'' contravariant indices and ''N'' covariant indices. Such a tensor can be defined as a [[linear operator|linear function]] which maps an (''M'' + ''N'')-tuple of ''M'' [[one-form]]s and ''N'' [[Vector (geometry)|vector]]s to a [[scalar (mathematics)|scalar]].
 
==Changing the tensor type==
{{main|raising and lowering indices}}
Consider the following octet of related tensors:
:<math> T_{\alpha \beta \gamma}, \ T_{\alpha \beta} {}^\gamma, \ T_\alpha {}^\beta {}_\gamma, \
T_\alpha {}^{\beta \gamma}, \ T^\alpha {}_{\beta \gamma}, \ T^\alpha {}_\beta {}^\gamma, \
T^{\alpha \beta} {}_\gamma, \ T^{\alpha \beta \gamma} </math>.
The first one is covariant, the last one contravariant, and the remaining ones mixed.  Notationally, these tensors differ from each other by the covariance/contravariance of their indices. A given contravariant index of a tensor can be lowered using the [[metric tensor]] ''g''<sub>μν</sub>, and a given covariant index can be raised using the inverse metric tensor ''g''<sup>μν</sup>. Thus, ''g''<sub>μν</sub> could be called the ''index lowering operator'' and ''g<sup>μν</sup>'' the ''index raising operator''.
 
Generally, the covariant metric tensor, contracted with a tensor of type (''M'', ''N''), yields a tensor of type (''M'' − 1, ''N'' + 1), whereas its contravariant inverse, contracted with a tensor of type (''M'', ''N''), yields a tensor of type (''M'' + 1, ''N'' − 1).
 
===Examples===
As an example, a mixed tensor of type (1, 2) can be obtained by raising an index of a covariant tensor of type (0, 3),
:<math> T_{\alpha \beta} {}^\lambda = T_{\alpha \beta \gamma} \, g^{\gamma \lambda} </math>,
where <math> T_{\alpha \beta} {}^\lambda </math> is the same tensor as <math> T_{\alpha \beta} {}^\gamma </math>, because
:<math>  T_{\alpha \beta} {}^\lambda \, \delta_\lambda {}^\gamma = T_{\alpha \beta} {}^\gamma </math>,
with Kronecker ''δ'' acting here like an identity matrix.
 
Likewise,
:<math> T_\alpha {}^\lambda {}_\gamma = T_{\alpha \beta \gamma} \, g^{\beta \lambda}, </math>
:<math> T_\alpha {}^{\lambda \epsilon} = T_{\alpha \beta \gamma} \, g^{\beta \lambda} \, g^{\gamma \epsilon},</math>
:<math> T^{\alpha \beta} {}_\gamma = g_{\gamma \lambda} \, T^{\alpha \beta \lambda},</math>
:<math> T^\alpha {}_{\lambda \epsilon} = g_{\lambda \beta} \, g_{\epsilon \gamma} \, T^{\alpha \beta \gamma}. </math>
 
Raising an index of the metric tensor is equivalent to contracting it with its inverse, yielding the [[Kronecker delta]],
:<math> g^{\mu \lambda} \, g_{\lambda \nu} = g^\mu {}_\nu = \delta^\mu {}_\nu </math>,
so any mixed version of the metric tensor will be equal to the Kronecker delta, which will also be mixed.
 
==See also==
* [[Covariance and contravariance of vectors]]
* [[Einstein notation]]
* [[Ricci calculus]]
* [[Tensor (intrinsic definition)]]
* [[Two-point tensor]]
 
==References==
 
* {{cite book |author=D.C. Kay| title=Tensor Calculus| publisher= Schaum’s Outlines, McGraw Hill (USA)| year=1988 | isbn=0-07-033484-6}}
* {{cite book |page=p.85-86, §3.5| author=J.A. Wheeler, C. Misner, K.S. Thorne| title=[[Gravitation (book)|Gravitation]]| publisher=W.H. Freeman & Co| year=1973 | isbn=0-7167-0344-0}}
* {{cite book |author=R. Penrose| title=[[The Road to Reality]]| publisher= Vintage books| year=2007 | isbn=0-679-77631-1}}
 
==External links==
 
* [http://mathworld.wolfram.com/IndexGymnastics.html Index Gymnastics], Wolfram Alpha
 
{{tensors}}
 
{{DEFAULTSORT:Mixed Tensor}}
[[Category:Tensors]]

Revision as of 03:46, 17 February 2014

Online calculators are internet-based. It is rather handy when you may be online and require immediate computations done. The program is quite accurate plus efficient.

Keep in mind, everyone's body is completely different, so discovering how various calories the body demands on a daily basis might have many determining factors, but a key one is how many times you workout. You could use an online Calorie Calculator to aid approximate the number of calories the body demands to keep its existing fat. You can also consider trying a online BMI index and see should you are underweight, normal, or obese for your body size.

Not consuming enough calories plus nutritients is a major health risk which could cause not only disorder, but death even. A healthy diet must be balanced and consist of 1,500 to 2,000 calories a day depending on bodily acitivity. Taking multivitamins daily can further ensure which the body's demands for nutrients are met. Avoid fasts for fat reduction or fad diets advocating the consumpion of only 1 kind of food for a certain period of time because these will also lead to malnutrition.

Additionally, guys normally have more muscle than women plus women have more fat in their body than men. For this reason, women should have their fat plus height ratio calculated from a bmi calculator females for females.

A person having a large frame and/or a great deal of muscle is flagged by Body Mass Index as being overweight--even whenever he has low body fat. This was true for disgraced cyclist Lance Armstrong at one point inside his athletic career.

The results that we receive from a body mass index calculator are nothing over indication of how heavy you need to be based on your height. The chart can tell we in the event you are over or under weight based on the results that you receive. It is though significant to keep in your mind which there are a great deal of folks who will get results that are not especially accurate. This really is specifically true when you have a great deal of muscle or in the event you are very tall.

There are a lot of free premade diet programs plus tools which let we to create a diet program online. I found the six sites listed above to have the best free tools and programs online. All of these websites contain the tools plus diet programs themselves so we do not have to click through links. I found Spark Folks to be the best free online diet program from all of the sites I viewed. The site has everything you may want in a diet program plus every aspect is built to function with 1 another. So, go ahead, pay a visit to all these sites for yourself, plus choose that is right for we. All which we have to bring is a small will force plus you are capable to mark 1 resolution, fat loss, off your New Year's goals for advantageous!

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