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| {{for|an introduction to the nature and significance of tensors in a broad context|Tensor}}
| | Calories appear to have gotten a bad standing. Just how regularly are we going to hear a individual state anything advantageous regarding calories? Poor little guys, imagine what it would appear like to be blamed with regard to anything from putting on fat to the taste of food. In cases where you have ever noticed oneself thinking: Sure this tastes great, but it involves a ton of calories, we already recognize what I mean. Calories can definitely turn out to be our neighbors, and I wish to be capable to demonstrate how to know and look at them in a different manner.<br><br>Obesity is not a single problem. There are several effects of weight. One of the first plus foremost effects is inability to carry out day-to-day activities. An obese individual is usually inactive plus feels fatigued nearly all of the time. Obesity provides birth to different wellness problems. Some obese people develop Pickwick syndrome in that a individual feels drowsy plus color of his face remains reddish. Pain in joints, bones, plus lower back portion of the body is frequently experienced by over-weight folks. Another chronic wellness issue that 1 can is susceptible to is diabetes that is incurable.<br><br>Don't worry regarding the non-integer exponent. Any off-the-shelf scientific calculator will handle it. I would not use the LI for scientific purposes--even though it's more fair than BMI.<br><br>A [http://safedietplansforwomen.com/bmi-calculator bmi calculator women] is important tool to have especially for women because they are more prone to get fat deposits than guys. Hormonal changes are the most significant reason. Women are moreover proven to emotionally devour food than guy. Whenever women are depressed, happy or tired they tend to look for comfort foods. Women are structurally different than guy to accommodate the growing baby inside the abdominal area. Plus a healthy girl (21-36%) would have twice more fat compared to a healthy male (8-25%). Because women start off with a better fat percentage than men, then it's not surprising that they are more liable to be overweight.<br><br>For woman with low activity that has a fat at 149lbs plus below, must try a 1,200 calorie diet to help their fat reduction. Women 150lbs to 164 lbs should have 1,400 calories; 165 to 184lbs 1,600 calories; and finally women over 185lbs must have 1,800 calories.<br><br>Do you have a extended neck or a short neck? Do you have a big head? Necks are light, heads are heavy! All these factors fluctuate. It's difficult to find information, nevertheless they make hats in sizes for men with heads from 20 inches to 25 inches (http://www.countrycalendar.com/Country_Store/hats/hat_sizes.htm). Per the website above, the average human head weighs regarding 12 pounds. However a head which is 25" inside circumference might have virtually double the volume of one which is 20" ... so, does a head weigh 5 or 10 pounds? That's about another point of BMI.<br><br>It is in your interest to stay in the perfect weight Hangchow you choose to get (omit) a business. However as lengthy because you have a healthy diet, we exercise frequently, plus you're treating the body like the temple it is then it's not important to reside and die by a weight chart. |
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| In [[mathematics]], the modern [[component-free]] approach to the theory of a '''tensor''' views a tensor as an [[abstract object]], expressing some definite type of multi-linear concept. Their well-known properties can be derived from their definitions, as linear maps or more generally; and the rules for manipulations of tensors arise as an extension of [[linear algebra]] to [[multilinear algebra]].
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| In [[differential geometry]] an intrinsic geometric statement may be described by a [[tensor field]] on a [[manifold]], and then doesn't need to make reference to coordinates at all. The same is true in [[general relativity]], of tensor fields describing a [[physical property]]. The component-free approach is also used heavily in [[abstract algebra]] and [[homological algebra]], where tensors arise naturally.
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| :'''''Note:''' This article assumes an understanding of the [[tensor product]] of [[vector space]]s without chosen [[Basis (linear algebra)|bases]]. An overview of the subject can be found in the main [[tensor]] article.
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| ==Definition via tensor products of vector spaces==
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| Given a finite set { ''V''<sub>1</sub>, ..., ''V''<sub>''n''</sub> } of [[vector space]]s over a common [[Field (mathematics)|field]] ''F'', one may form their [[Tensor product#Tensor product of vector spaces|tensor product]] ''V''<sub>1</sub> ⊗ ... ⊗ ''V''<sub>n</sub>, an element of which is termed a '''tensor'''.
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| A '''tensor on the vector space''' ''V'' is then defined to be an element of (i.e., a vector in) a vector space of the form:
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| :<math>V \otimes \cdots \otimes V \otimes V^* \otimes \cdots \otimes V^*</math>
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| where ''V''* is the [[dual space]] of ''V''.
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| If there are ''m'' copies of ''V'' and ''n'' copies of ''V''* in our product, the tensor is said to be of '''type (''m'', ''n'')''' and contravariant of order ''m'' and covariant order ''n'' and total [[tensor order|order]] ''m''+''n''. The tensors of order zero are just the scalars (elements of the field ''F''), those of contravariant order 1 are the vectors in ''V'', and those of covariant order 1 are the [[linear functional|one-forms]] in ''V''* (for this reason the last two spaces are often called the contravariant and covariant vectors). The space of all tensors of type (''m'',''n'') is denoted
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| :<math> T^m_n(V) = \underbrace{ V\otimes \dots \otimes V}_{m} \otimes \underbrace{ V^*\otimes \dots \otimes V^*}_{n} .</math>
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| The (1,1) tensors
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| :<math>V \otimes V^*</math>
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| are isomorphic in a natural way to the space of [[linear transformations]] from ''V'' to ''V''. A [[bilinear form]] on a real vector space ''V''; ''V'' × ''V'' → '''R''' corresponds in a natural way to a (0,2) tensor in
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| :<math>V^* \otimes V^*</math> | |
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| termed the associated ''[[metric tensor]]'' (or sometimes misleadingly the ''[[metric (mathematics)|metric]]'' or ''[[inner product]]'') and usually denoted '''''g'''''.
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| ==Tensor rank==
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| The term '''rank of a tensor''' extends the notion of the [[rank of a matrix]] in linear algebra, although the term is also often used to mean the order (or degree) of a tensor. The rank of a matrix is the minimum number of column vectors needed to span the range of the matrix. A matrix thus has rank one if it can be written as an [[outer product]] of two nonzero vectors:
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| :<math> A = v w^{\mathrm{T}}. </math>
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| More generally, the rank of a matrix ''A'' is the smallest number of such outer products that can be summed to produce it:
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| :<math>A = v_1w_1^\mathrm{T} + \cdots + v_kw_k^\mathrm{T}.</math>
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| Similarly, a tensor of rank one (also called a '''simple tensor''') is a tensor that can be written as a tensor product of the form
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| :<math>T=a\otimes b\otimes\cdots\otimes d</math>
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| where ''a'', ''b'', ..., ''d'' are nonzero and in ''V'' or ''V''*. That is, if the tensor is nonzero and completely [[factorization|factorizable]]. In indices, a tensor of rank 1 is a tensor of the form
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| :<math>T_{ij\dots}^{k\ell\dots}=a_ib_j\cdots c^kd^\ell\cdots.</math>
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| Every tensor can be expressed as a sum of rank 1 tensors. The rank of a general tensor ''T'' is defined to be the minimum number of rank 1 tensors with which it is possible to express ''T'' as a sum {{harv|Bourbaki|1988|loc=II, §7, no. 8}}.
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| A nonzero order 1 tensor always has rank 1. The [[zero tensor]] has rank zero. The rank of a tensor of order 2 agrees with the rank when the tensor is regarded as a [[Matrix (mathematics)|matrix]] {{harv|Halmos|1974|loc=§51}}, and can be determined from [[Gaussian elimination]] for instance. The rank of an order 3 or higher tensor is however often ''very hard'' to determine, and low rank decompositions of tensors are sometimes of great practical interest {{harv|de Groote|1987}}. Computational tasks such as the efficient multiplication of matrices and the efficient evaluation of polynomials can be recast as the problem of simultaneously evaluating a set of [[bilinear form]]s
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| :<math>z_k = \sum_{ij} T_{ijk}x_iy_j\,</math>
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| for given inputs ''x''<sub>''i''</sub> and ''y''<sub>''j''</sub>. If a low-rank decomposition of the tensor ''T'' is known, then an efficient [[evaluation strategy]] is known {{harv|Knuth|1998|pp=506–508}}.
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| ==Universal property==
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| The space <math>T^m_n(V)</math> can be characterized by a [[universal property]] in terms of [[multilinear map]]pings. Amongst the advantages of this approach are that it gives a way to show that many linear mappings are "natural" or "geometric" (in other words are independent of any choice of basis). Explicit computational information can then be written down using bases, and this order of priorities can be more convenient than proving a formula gives rise to a natural mapping. Another aspect is that tensor products are not used only for [[free module]]s, and the "universal" approach carries over more easily to more general situations.
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| A scalar-valued function on a [[Cartesian product]] (or [[Direct sum of modules|direct sum]]) of vector spaces
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| :<math>f : V_1\times V_2\times\cdots\times V_N \to \mathbf{R}</math>
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| is multilinear if it is linear in each argument. The space of all multlinear mappings from the product ''V''<sub>1</sub>×''V''<sub>2</sub>×...×''V''<sub>''N''</sub> into ''W'' is denoted
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| ''L''<sup>''N''</sup>(''V''<sub>1</sub>,''V''<sub>2</sub>,...,''V''<sub>''N''</sub>; ''W''). When ''N'' = 1, a multilinear mapping is just an ordinary linear mapping, and the space of all linear mappings from ''V'' to ''W'' is denoted ''L''(''V'';''W'').
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| The [[tensor product#Characterization by a universal property|universal characterization of the tensor product]] implies that, for each multilinear function
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| :<math>f\in L^{m+n}(\underbrace{V,V,\dots,V}_m,\underbrace{V^*,V^*,\dots,V^*}_n;W)</math>
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| there exists a unique linear function
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| :<math>T_f \in L(V\otimes\cdots\otimes V\otimes V^*\otimes\cdots\otimes V^*; W)</math>
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| such that
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| :<math>f(v_1,\dots,v_m,\alpha_1,\dots,\alpha_n) = T_f(v_1\otimes\cdots\otimes v_m\otimes\alpha_1\otimes\cdots\otimes\alpha_n)</math>
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| for all ''v''<sub>''i''</sub> ∈ ''V'' and α<sub>''i''</sub> ∈ ''V''<sup>∗</sup>.
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| Using the universal property, it follows that the space of (''m'',''n'')-tensors admits a [[natural isomorphism]]
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| :<math>T^m_n(V) \cong
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| L(V^* \otimes \dots \otimes V^* \otimes V \otimes \dots \otimes V ; \mathbb{R})
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| \cong L^{m+n}(V^*,\dots,V^*,V,\dots,V;\mathbb{R}).</math>
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| In the formula above, the roles of ''V'' and ''V''<sup>*</sup> are reversed. In particular, one has
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| :<math>T^1_0(V) \cong L(V^*;\mathbb{R}) \cong V</math>
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| and
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| :<math>T^0_1(V) \cong L(V;\mathbb{R}) = V^*</math>
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| and
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| :<math>T^1_1(V) \cong L(V;V).</math>
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| ==Tensor fields==
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| {{Main|tensor field}}
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| [[Differential geometry]], [[physics]] and [[engineering]] must often deal with [[tensor field]]s on [[smooth manifold]]s. The term ''tensor'' is sometimes used as a shorthand for ''tensor field''. A tensor field expresses the concept of a tensor that varies from point to point.
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| ==References==
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| * {{Citation
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| | last=Abraham
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| | first=Ralph
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| | author-link=Ralph Abraham
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| | last2=Marsden
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| | first2=Jerrold E.
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| | author2-link=Jerrold E. Marsden
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| | title=Foundations of Mechanics
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| | edition=2
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| | year=1985
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| | publisher=Addison-Wesley
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| | location=Reading, Mass.
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| | isbn=0-201-40840-6
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| }}.
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| * {{citation|first = Nicolas|last=Bourbaki|authorlink=Nicolas Bourbaki | title = Elements of mathematics, Algebra I| publisher = Springer-Verlag | year = 1989|isbn=3-540-64243-9}}.
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| *{{citation | last = de Groote | first = H. F. | title = Lectures on the Complexity of Bilinear Problems | series= Lecture Notes in Computer Science | volume=245 | publisher = Springer | year = 1987 | isbn = 3-540-17205-X}}.
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| * {{citation|authorlink=Paul Halmos|first=Paul|last=Halmos|title=Finite-dimensional Vector Spaces|year=1974|publisher=Springer|isbn=0-387-90093-4}}.
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| *{{Citation | last1=Jeevanjee | first1=Nadir |year=2011 | title= An Introduction to Tensors and Group Theory for Physicists |url = http://www.springer.com/new+%26+forthcoming+titles+(default)/book/978-0-8176-4714-8 | isbn=978-0-8176-4714-8 }}
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| *{{Citation | last1=Knuth | first1=Donald | author1-link=Donald Knuth | title=The Art of Computer Programming vol. 2 | origyear=1969 | edition=3rd | isbn=978-0-201-89684-8 | year=1998 | pages=145–146}}.
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| {{tensors}}
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| {{DEFAULTSORT:Tensor (Intrinsic Definition)}}
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| [[Category:Tensors]]
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Calories appear to have gotten a bad standing. Just how regularly are we going to hear a individual state anything advantageous regarding calories? Poor little guys, imagine what it would appear like to be blamed with regard to anything from putting on fat to the taste of food. In cases where you have ever noticed oneself thinking: Sure this tastes great, but it involves a ton of calories, we already recognize what I mean. Calories can definitely turn out to be our neighbors, and I wish to be capable to demonstrate how to know and look at them in a different manner.
Obesity is not a single problem. There are several effects of weight. One of the first plus foremost effects is inability to carry out day-to-day activities. An obese individual is usually inactive plus feels fatigued nearly all of the time. Obesity provides birth to different wellness problems. Some obese people develop Pickwick syndrome in that a individual feels drowsy plus color of his face remains reddish. Pain in joints, bones, plus lower back portion of the body is frequently experienced by over-weight folks. Another chronic wellness issue that 1 can is susceptible to is diabetes that is incurable.
Don't worry regarding the non-integer exponent. Any off-the-shelf scientific calculator will handle it. I would not use the LI for scientific purposes--even though it's more fair than BMI.
A bmi calculator women is important tool to have especially for women because they are more prone to get fat deposits than guys. Hormonal changes are the most significant reason. Women are moreover proven to emotionally devour food than guy. Whenever women are depressed, happy or tired they tend to look for comfort foods. Women are structurally different than guy to accommodate the growing baby inside the abdominal area. Plus a healthy girl (21-36%) would have twice more fat compared to a healthy male (8-25%). Because women start off with a better fat percentage than men, then it's not surprising that they are more liable to be overweight.
For woman with low activity that has a fat at 149lbs plus below, must try a 1,200 calorie diet to help their fat reduction. Women 150lbs to 164 lbs should have 1,400 calories; 165 to 184lbs 1,600 calories; and finally women over 185lbs must have 1,800 calories.
Do you have a extended neck or a short neck? Do you have a big head? Necks are light, heads are heavy! All these factors fluctuate. It's difficult to find information, nevertheless they make hats in sizes for men with heads from 20 inches to 25 inches (http://www.countrycalendar.com/Country_Store/hats/hat_sizes.htm). Per the website above, the average human head weighs regarding 12 pounds. However a head which is 25" inside circumference might have virtually double the volume of one which is 20" ... so, does a head weigh 5 or 10 pounds? That's about another point of BMI.
It is in your interest to stay in the perfect weight Hangchow you choose to get (omit) a business. However as lengthy because you have a healthy diet, we exercise frequently, plus you're treating the body like the temple it is then it's not important to reside and die by a weight chart.