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0-rank tensor vs vector in 1D



The Next CEO of Stack OverflowHistory of Electromagnetic Field TensorAngular displacement and the displacement vectorIn field theory, why are some symmetry transformations applied to the field values while other act on the space that the fields are defined on?Possible confusion, the inertia of something yields a tensor? (trying to understand an example)What exactly is the Parity transformation? Parity in spherical coordinatesA fundamental question about tensors and vectorsIt is correct to say that a tensor is simply a multidimensional array of related quantities? But what about a tensor as a transformation?4-Vector DefinitionDoubts on covariant and contravariant vectors and on double tensorsZero order Tensor










7












$begingroup$


What is the difference between zero-rank tensor $x$ (scalar) and vector $[x]$ in 1D?



As far as I understand tensor is anything which can be measured and different measures can be transformed into each other. That is, there are different basises for looking at one object.



Is length a scalar (zero rank tensor)?
I think it is not.
ex.:



  • physical parameter: writing pen's length

  • tensor: $l$

  • length in inches: $[5.511811023622]$

  • length in centimeters: $[14]$

  • transformation law: 1cm = 2.54inch

so $l$ is a scalar, but on the other hand it's a tensor of rank 1 since "physical parameter of length is invariant, only it's measures (in different units) are".



The same example can be made with classical example of temperature (which is used as a primer of zero rank tensor most in any book) in C and K units. I'm confused.










share|cite|improve this question











$endgroup$
















    7












    $begingroup$


    What is the difference between zero-rank tensor $x$ (scalar) and vector $[x]$ in 1D?



    As far as I understand tensor is anything which can be measured and different measures can be transformed into each other. That is, there are different basises for looking at one object.



    Is length a scalar (zero rank tensor)?
    I think it is not.
    ex.:



    • physical parameter: writing pen's length

    • tensor: $l$

    • length in inches: $[5.511811023622]$

    • length in centimeters: $[14]$

    • transformation law: 1cm = 2.54inch

    so $l$ is a scalar, but on the other hand it's a tensor of rank 1 since "physical parameter of length is invariant, only it's measures (in different units) are".



    The same example can be made with classical example of temperature (which is used as a primer of zero rank tensor most in any book) in C and K units. I'm confused.










    share|cite|improve this question











    $endgroup$














      7












      7








      7





      $begingroup$


      What is the difference between zero-rank tensor $x$ (scalar) and vector $[x]$ in 1D?



      As far as I understand tensor is anything which can be measured and different measures can be transformed into each other. That is, there are different basises for looking at one object.



      Is length a scalar (zero rank tensor)?
      I think it is not.
      ex.:



      • physical parameter: writing pen's length

      • tensor: $l$

      • length in inches: $[5.511811023622]$

      • length in centimeters: $[14]$

      • transformation law: 1cm = 2.54inch

      so $l$ is a scalar, but on the other hand it's a tensor of rank 1 since "physical parameter of length is invariant, only it's measures (in different units) are".



      The same example can be made with classical example of temperature (which is used as a primer of zero rank tensor most in any book) in C and K units. I'm confused.










      share|cite|improve this question











      $endgroup$




      What is the difference between zero-rank tensor $x$ (scalar) and vector $[x]$ in 1D?



      As far as I understand tensor is anything which can be measured and different measures can be transformed into each other. That is, there are different basises for looking at one object.



      Is length a scalar (zero rank tensor)?
      I think it is not.
      ex.:



      • physical parameter: writing pen's length

      • tensor: $l$

      • length in inches: $[5.511811023622]$

      • length in centimeters: $[14]$

      • transformation law: 1cm = 2.54inch

      so $l$ is a scalar, but on the other hand it's a tensor of rank 1 since "physical parameter of length is invariant, only it's measures (in different units) are".



      The same example can be made with classical example of temperature (which is used as a primer of zero rank tensor most in any book) in C and K units. I'm confused.







      vectors coordinate-systems tensor-calculus linear-algebra






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 21 mins ago









      Qmechanic

      107k121981229




      107k121981229










      asked 10 hours ago









      coobitcoobit

      365110




      365110




















          2 Answers
          2






          active

          oldest

          votes


















          12












          $begingroup$

          “Scalar”, “vector”, and “tensor” have no meaning without specifying the group of transformations. In physics we focus on groups such as rotations, Galilean transformations, Lorentz transformations, Poincaire transformations, and gauge transformations because these are symmetries of various physical theories, built in to reflect symmetries of the natural world.



          The length of a writing pen is a scalar under rotations and Galilean transformations. This is a significant physical fact about our world.



          But the fact that you can measure its length in various units is not significant, because units are inventions of humans, not of Nature. Physicists never say that the length of a writing pen “transforms” because you can choose to measure it in different length units. Different units such as inches and centimeters for a particular physical quantity like length do not have any physical significance at all.



          Going back to your original question, the difference between a scalar and a vector under rotations should now be obvious: a scalar is a single number that stays the same under a rotation, while a vector is a directed quantity that requires three numbers to describe it, and under rotations these numbers transform into linear combinations of each other, as specified by the relevant rotation matrix.



          Under any other transformation group, the distinction between scalars and vectors is similar.






          share|cite|improve this answer











          $endgroup$












          • $begingroup$
            I'm sorry if it might sound dumb, but ... Is 1D vector invariant under rotation? I mean is there rotation in 1D space? If so how it's different from scalar?
            $endgroup$
            – coobit
            8 hours ago











          • $begingroup$
            @coobit Consider the group of reflections along that one dimension. A vector changes sign, but a scalar doesn't.
            $endgroup$
            – Chiral Anomaly
            8 hours ago










          • $begingroup$
            Whoops, I completely overlooked the fact that you were asking about 1D. (Since you had referred to scalars as rank 0, I was thinking "rank 1" , not "1D", when you said "vector".) There are no proper rotations in 1D. As @ChiralAnomaly explains, you can consider 1D reflections, and scalar and vectors transform differently under these, even though both are only a single number.
            $endgroup$
            – G. Smith
            8 hours ago











          • $begingroup$
            @coobit there's a difference in what a mathematician and a physicist mean when they say "vector". A mathematician would just use "vector" to mean "an element of a vector space", where the vector space in question could be anything depending on context. In physics, "vector" almost always means "an element of the tangent space of some manifold", where the manifold in question depends on context (often Euclidean or Minkowski space). See also this question on math stack exchange
            $endgroup$
            – Carmeister
            4 hours ago


















          0












          $begingroup$

          First of all, I'll constrain the discussion assuming:



          1) Finite-dimensional vector spaces



          2) Real Vector spaces



          3) Talking just about contravariant tensors



          4) Physics which use the standard notion of Spacetime



          $$* * *$$



          To answer your question I need to talk a little bit about Tensors.



          I) The tensor object and pure mathematics:



          The precise answer to the question "What is a tensor?" is, by far:




          A tensor is a object of a vector space called Tensor Product.




          In order to this general statement become something that have some value to you, I would like you to think a little bit about vectors and their algebra : the linear algebra.



          I.1) What truly is a Vector?



          First of all, if you look on linear algebra texts, you'll rapidly realize that the answer to the question "What are vectors after all? Matrices? Arrows? Functions?" is:




          A vector is a element of a algebraic structure called vector space.




          So after the study of the definition of a vector space you can talk with all rigour in the world that a vector isn't a arrow or a matrix, but a element of a vector space.



          I.1.1) Some facts about vectors



          Consider then a vector formed by a linear combination of basis vectors:



          $$mathbfv = sum_j = 1^n v^jmathbfe_j tag1$$



          This is well-known fact about vectors. So, there's another key point about basis vectors: the vector space is spanned by these basis vectors. You can create a "constrain machinery" to verify if a set of vectors spans a entire vector space (i.e. forms a basis):




          A set $mathcalS$ is a basis for a vector space $mathfrakV$ if:



          1) the vectors of the set $mathcalS$ are linear independent



          2) the vectors of the set $mathcalS$ spanned the vector space $mathfrakV$, i.e. $mathfrakV equiv span(mathcalS)$




          So another point of view to "form" an entire vector space is from basis vectors. The intuitive idea is that, more or less, like if the basis vectors "constructs" (they span) and entire vector space.



          Another fact is that you can change the basis $mathbfe_j$ to another set basis of basis $mathbfe'_j$. Well, when you do this the vector components suffer a change too. And then the components transforms like:



          $$v'^k = sum^n_j=1M^k_jv^jtag2$$



          but, of course, the vector object, remains the same:



          $$mathbfv = sum_j = 1^n v^jmathbfe_j = sum_j = 1^n v'^jmathbfe'_j$$



          So, a vector, truly, is a object of a vector space, which have the form of $(1)$ and their components transforms like $(2)$.



          I.1.2) The "physicist way" of definition of a Tensor



          When you're searching about tensors on physics/engeneering texts you certainly will encounter the following definition of a tensor:




          A Tensor is defined as the kind of object which transforms, under a coordinate transformation, like:



          $$T'^ij = sum^n_k=1sum^n_l=1 M^i_kM^j_l T^kl tag3$$




          This definition serves to encode the notion that a valid physical law must be independent of coordinate systems (or all that G.Smith said).
          Well, there's some interesting happening here. A vector, is a object which have a precise formulation in terms of a algebraic structure, have a precise form (that of $(1)$, which the basis vectors spans the entire $mathfrakV$) and their components have "transformation behaviour" like $(2)$. If you compare what I exposed about vector and $(3)$, you may reach the conclusion that, concerning about tensors, some information about their nature is missing.
          The fact is, the definition $(3)$ isn't a tensor, but the "transformation behaviour" of the components of a tensor $mathbfT$.



          I.2) What truly is a Tensor?



          Well, you have the transformation of components of a tensor, i.e. $(3)$, well defined. But what about their "space" and "form" (something like $(1)$)?



          So, the space is called tensor product of two vector spaces:




          $$Votimes W tag4$$




          The construction of tensor product is something beyond the scope of this answer [*]. But the mathematical considerations about tensor products are that they generalize the concept of products of vectors (remember that, in linear algebra and analytic geometry you're able to "multiply" vector just using the inner product and vector product), they construct a concept of products of vector spaces (remember that a Direct sum of vector space gives you a notion of Sum of vector spaces), and they construct the precise notion of a tensor. Also, by the technology of the construction of the tensor product we can identify (i.e. stablish a isomorphism between vector spaces) the vector space $Votimes W$ and $mathfrakLin^2(V^* times W^*; mathbbK)$:




          $$Votimes W cong mathfrakLin^2(V^* times W^*; mathbbK) tag5$$



          where $mathfrakLin^2(V^* times W^*; mathbbK)$ is the dual vector space of all bilinear functionals.




          So a tensor have the form:




          $$mathbfT = sum^n_i=1sum^n_j=1 T^ij (mathbfe_iotimesmathbfe_j) tag6$$
          And $mathbfT in Votimes W$.




          Well, given the transformation rule $(3)$ the space, $(5)$, and form, $(6)$, you can talk precisely about what tensor really is. It's clear that the "object tensor" isn't just a transformation of coordinates. Also, in $(6)$ the tensor basis $(mathbfe_iotimesmathbfe_j)$ spans $Votimes W$.



          By virtue of the general construction of tensor product and the identification given by $(5)$, you'll also encounter the definition of a tensor as a multilinear object which spits scalars:




          $$beginarrayrl
          mathbfT :V^*times W^* &to mathbbK \
          (mathbfv,mathbfw)&mapsto mathbfT(mathbfv,mathbfw)=: v^icdot_mathbbKw^j
          endarray$$



          Where the operation $cdot_mathbbK$ is the product defined in the field.




          With this picture we say that a tensor like $(6)$ is a tensor of rank 2. And a vector a tensor of rank 1. Furthermore a scalar a tensor of rank 0.



          II) The tensor object and physics



          The well stablish physics, in general, deals with spacetime (like Newtonian physics), and the theory of spacetime is geometry. So, in order to really apply the tensor theory in physics first we have to give the geometry of physics.
          The geometry is basically classical Manifold Theory (which, again, is beyond the scope). And by Manifold Theory we can apply tensors on Manifolds introducing the concept of a tangent vector space. In parallel, we can construct another algebraic structure called Fibre Bundle of tangent spaces and then create the precise notion of Vector Field and Tensor Field.



          Tensor Fields are the real objects defined in physics books as tensors and we use the word of a tensor and tensor field as synonyms (IN FACT THEY ARE NOT THE SAME CONCEPT!). A tensor field is a section of the tensor bundle and a vector field, a section of vector bundle. But the intuitive definition (by far, general to physics) of a tensor field is then:




          $$[mathbfT(x^k)] = sum^n_i=1sum^n_j=1 [T^ij(x^k)] ([mathbfe_i(x^k)]otimes[mathbfe_j(x^k)]) tag5$$
          A tensor field is the object which attaches a tensor to every point p of the Manifold.




          With the manifold theory, the transformation rule becomes:




          $$[T'^ij(x^m)] = sum^n_k=1sum^n_l=1 fracpartial x'^ipartial x^kfracpartial x'^jpartial x^l [T^kl(x^m)] equiv T'^ij = sum^n_k=1sum^n_l=1 fracpartial x'^ipartial x^kfracpartial x'^jpartial x^l T^kl tag7$$




          Notice that the partials are simply the transformation matrices $M$. The matrices $M$ are called the Jacobians transformation matrices and the matrices $M$ became these jacobians by virtue of Manifold theory.



          In a restric way, these Jacobians are rotations,lorentz transformations,galilean transformation, and so on.



          III) What is the difference between zero-rank tensor x (scalar) and 1D vector [x]?



          So, in order to talk about lengths we have to realize that we are talking about a scalar field, or a tensor of rank 0. Then the difference between a scalar and a 1D vector (which is a tensor of rank 1) is that one is a scalar field and the other is a vector field. From a "Pure" mathematical point of view, (section I) if this answer) one is a member of the field $mathbbK$ and the other is a member of a vector space.
          Also, you're quite right, a scalar (or a scalar field) is a rank 0 tensor or "a object which do not have "matrices of change"; an object which do not suffer a change under a transformation of coordinates (we say that a scalar quantity is a invariant quantity).




          $$phi'= phi$$




          $$* * *$$



          [*] ROMAN.S. Advanced Linear Algebra. Springer. chapter 14. 1 ed. 1992.






          share|cite|improve this answer











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            2 Answers
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            2 Answers
            2






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            active

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            active

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            12












            $begingroup$

            “Scalar”, “vector”, and “tensor” have no meaning without specifying the group of transformations. In physics we focus on groups such as rotations, Galilean transformations, Lorentz transformations, Poincaire transformations, and gauge transformations because these are symmetries of various physical theories, built in to reflect symmetries of the natural world.



            The length of a writing pen is a scalar under rotations and Galilean transformations. This is a significant physical fact about our world.



            But the fact that you can measure its length in various units is not significant, because units are inventions of humans, not of Nature. Physicists never say that the length of a writing pen “transforms” because you can choose to measure it in different length units. Different units such as inches and centimeters for a particular physical quantity like length do not have any physical significance at all.



            Going back to your original question, the difference between a scalar and a vector under rotations should now be obvious: a scalar is a single number that stays the same under a rotation, while a vector is a directed quantity that requires three numbers to describe it, and under rotations these numbers transform into linear combinations of each other, as specified by the relevant rotation matrix.



            Under any other transformation group, the distinction between scalars and vectors is similar.






            share|cite|improve this answer











            $endgroup$












            • $begingroup$
              I'm sorry if it might sound dumb, but ... Is 1D vector invariant under rotation? I mean is there rotation in 1D space? If so how it's different from scalar?
              $endgroup$
              – coobit
              8 hours ago











            • $begingroup$
              @coobit Consider the group of reflections along that one dimension. A vector changes sign, but a scalar doesn't.
              $endgroup$
              – Chiral Anomaly
              8 hours ago










            • $begingroup$
              Whoops, I completely overlooked the fact that you were asking about 1D. (Since you had referred to scalars as rank 0, I was thinking "rank 1" , not "1D", when you said "vector".) There are no proper rotations in 1D. As @ChiralAnomaly explains, you can consider 1D reflections, and scalar and vectors transform differently under these, even though both are only a single number.
              $endgroup$
              – G. Smith
              8 hours ago











            • $begingroup$
              @coobit there's a difference in what a mathematician and a physicist mean when they say "vector". A mathematician would just use "vector" to mean "an element of a vector space", where the vector space in question could be anything depending on context. In physics, "vector" almost always means "an element of the tangent space of some manifold", where the manifold in question depends on context (often Euclidean or Minkowski space). See also this question on math stack exchange
              $endgroup$
              – Carmeister
              4 hours ago















            12












            $begingroup$

            “Scalar”, “vector”, and “tensor” have no meaning without specifying the group of transformations. In physics we focus on groups such as rotations, Galilean transformations, Lorentz transformations, Poincaire transformations, and gauge transformations because these are symmetries of various physical theories, built in to reflect symmetries of the natural world.



            The length of a writing pen is a scalar under rotations and Galilean transformations. This is a significant physical fact about our world.



            But the fact that you can measure its length in various units is not significant, because units are inventions of humans, not of Nature. Physicists never say that the length of a writing pen “transforms” because you can choose to measure it in different length units. Different units such as inches and centimeters for a particular physical quantity like length do not have any physical significance at all.



            Going back to your original question, the difference between a scalar and a vector under rotations should now be obvious: a scalar is a single number that stays the same under a rotation, while a vector is a directed quantity that requires three numbers to describe it, and under rotations these numbers transform into linear combinations of each other, as specified by the relevant rotation matrix.



            Under any other transformation group, the distinction between scalars and vectors is similar.






            share|cite|improve this answer











            $endgroup$












            • $begingroup$
              I'm sorry if it might sound dumb, but ... Is 1D vector invariant under rotation? I mean is there rotation in 1D space? If so how it's different from scalar?
              $endgroup$
              – coobit
              8 hours ago











            • $begingroup$
              @coobit Consider the group of reflections along that one dimension. A vector changes sign, but a scalar doesn't.
              $endgroup$
              – Chiral Anomaly
              8 hours ago










            • $begingroup$
              Whoops, I completely overlooked the fact that you were asking about 1D. (Since you had referred to scalars as rank 0, I was thinking "rank 1" , not "1D", when you said "vector".) There are no proper rotations in 1D. As @ChiralAnomaly explains, you can consider 1D reflections, and scalar and vectors transform differently under these, even though both are only a single number.
              $endgroup$
              – G. Smith
              8 hours ago











            • $begingroup$
              @coobit there's a difference in what a mathematician and a physicist mean when they say "vector". A mathematician would just use "vector" to mean "an element of a vector space", where the vector space in question could be anything depending on context. In physics, "vector" almost always means "an element of the tangent space of some manifold", where the manifold in question depends on context (often Euclidean or Minkowski space). See also this question on math stack exchange
              $endgroup$
              – Carmeister
              4 hours ago













            12












            12








            12





            $begingroup$

            “Scalar”, “vector”, and “tensor” have no meaning without specifying the group of transformations. In physics we focus on groups such as rotations, Galilean transformations, Lorentz transformations, Poincaire transformations, and gauge transformations because these are symmetries of various physical theories, built in to reflect symmetries of the natural world.



            The length of a writing pen is a scalar under rotations and Galilean transformations. This is a significant physical fact about our world.



            But the fact that you can measure its length in various units is not significant, because units are inventions of humans, not of Nature. Physicists never say that the length of a writing pen “transforms” because you can choose to measure it in different length units. Different units such as inches and centimeters for a particular physical quantity like length do not have any physical significance at all.



            Going back to your original question, the difference between a scalar and a vector under rotations should now be obvious: a scalar is a single number that stays the same under a rotation, while a vector is a directed quantity that requires three numbers to describe it, and under rotations these numbers transform into linear combinations of each other, as specified by the relevant rotation matrix.



            Under any other transformation group, the distinction between scalars and vectors is similar.






            share|cite|improve this answer











            $endgroup$



            “Scalar”, “vector”, and “tensor” have no meaning without specifying the group of transformations. In physics we focus on groups such as rotations, Galilean transformations, Lorentz transformations, Poincaire transformations, and gauge transformations because these are symmetries of various physical theories, built in to reflect symmetries of the natural world.



            The length of a writing pen is a scalar under rotations and Galilean transformations. This is a significant physical fact about our world.



            But the fact that you can measure its length in various units is not significant, because units are inventions of humans, not of Nature. Physicists never say that the length of a writing pen “transforms” because you can choose to measure it in different length units. Different units such as inches and centimeters for a particular physical quantity like length do not have any physical significance at all.



            Going back to your original question, the difference between a scalar and a vector under rotations should now be obvious: a scalar is a single number that stays the same under a rotation, while a vector is a directed quantity that requires three numbers to describe it, and under rotations these numbers transform into linear combinations of each other, as specified by the relevant rotation matrix.



            Under any other transformation group, the distinction between scalars and vectors is similar.







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited 9 hours ago

























            answered 9 hours ago









            G. SmithG. Smith

            10.2k11429




            10.2k11429











            • $begingroup$
              I'm sorry if it might sound dumb, but ... Is 1D vector invariant under rotation? I mean is there rotation in 1D space? If so how it's different from scalar?
              $endgroup$
              – coobit
              8 hours ago











            • $begingroup$
              @coobit Consider the group of reflections along that one dimension. A vector changes sign, but a scalar doesn't.
              $endgroup$
              – Chiral Anomaly
              8 hours ago










            • $begingroup$
              Whoops, I completely overlooked the fact that you were asking about 1D. (Since you had referred to scalars as rank 0, I was thinking "rank 1" , not "1D", when you said "vector".) There are no proper rotations in 1D. As @ChiralAnomaly explains, you can consider 1D reflections, and scalar and vectors transform differently under these, even though both are only a single number.
              $endgroup$
              – G. Smith
              8 hours ago











            • $begingroup$
              @coobit there's a difference in what a mathematician and a physicist mean when they say "vector". A mathematician would just use "vector" to mean "an element of a vector space", where the vector space in question could be anything depending on context. In physics, "vector" almost always means "an element of the tangent space of some manifold", where the manifold in question depends on context (often Euclidean or Minkowski space). See also this question on math stack exchange
              $endgroup$
              – Carmeister
              4 hours ago
















            • $begingroup$
              I'm sorry if it might sound dumb, but ... Is 1D vector invariant under rotation? I mean is there rotation in 1D space? If so how it's different from scalar?
              $endgroup$
              – coobit
              8 hours ago











            • $begingroup$
              @coobit Consider the group of reflections along that one dimension. A vector changes sign, but a scalar doesn't.
              $endgroup$
              – Chiral Anomaly
              8 hours ago










            • $begingroup$
              Whoops, I completely overlooked the fact that you were asking about 1D. (Since you had referred to scalars as rank 0, I was thinking "rank 1" , not "1D", when you said "vector".) There are no proper rotations in 1D. As @ChiralAnomaly explains, you can consider 1D reflections, and scalar and vectors transform differently under these, even though both are only a single number.
              $endgroup$
              – G. Smith
              8 hours ago











            • $begingroup$
              @coobit there's a difference in what a mathematician and a physicist mean when they say "vector". A mathematician would just use "vector" to mean "an element of a vector space", where the vector space in question could be anything depending on context. In physics, "vector" almost always means "an element of the tangent space of some manifold", where the manifold in question depends on context (often Euclidean or Minkowski space). See also this question on math stack exchange
              $endgroup$
              – Carmeister
              4 hours ago















            $begingroup$
            I'm sorry if it might sound dumb, but ... Is 1D vector invariant under rotation? I mean is there rotation in 1D space? If so how it's different from scalar?
            $endgroup$
            – coobit
            8 hours ago





            $begingroup$
            I'm sorry if it might sound dumb, but ... Is 1D vector invariant under rotation? I mean is there rotation in 1D space? If so how it's different from scalar?
            $endgroup$
            – coobit
            8 hours ago













            $begingroup$
            @coobit Consider the group of reflections along that one dimension. A vector changes sign, but a scalar doesn't.
            $endgroup$
            – Chiral Anomaly
            8 hours ago




            $begingroup$
            @coobit Consider the group of reflections along that one dimension. A vector changes sign, but a scalar doesn't.
            $endgroup$
            – Chiral Anomaly
            8 hours ago












            $begingroup$
            Whoops, I completely overlooked the fact that you were asking about 1D. (Since you had referred to scalars as rank 0, I was thinking "rank 1" , not "1D", when you said "vector".) There are no proper rotations in 1D. As @ChiralAnomaly explains, you can consider 1D reflections, and scalar and vectors transform differently under these, even though both are only a single number.
            $endgroup$
            – G. Smith
            8 hours ago





            $begingroup$
            Whoops, I completely overlooked the fact that you were asking about 1D. (Since you had referred to scalars as rank 0, I was thinking "rank 1" , not "1D", when you said "vector".) There are no proper rotations in 1D. As @ChiralAnomaly explains, you can consider 1D reflections, and scalar and vectors transform differently under these, even though both are only a single number.
            $endgroup$
            – G. Smith
            8 hours ago













            $begingroup$
            @coobit there's a difference in what a mathematician and a physicist mean when they say "vector". A mathematician would just use "vector" to mean "an element of a vector space", where the vector space in question could be anything depending on context. In physics, "vector" almost always means "an element of the tangent space of some manifold", where the manifold in question depends on context (often Euclidean or Minkowski space). See also this question on math stack exchange
            $endgroup$
            – Carmeister
            4 hours ago




            $begingroup$
            @coobit there's a difference in what a mathematician and a physicist mean when they say "vector". A mathematician would just use "vector" to mean "an element of a vector space", where the vector space in question could be anything depending on context. In physics, "vector" almost always means "an element of the tangent space of some manifold", where the manifold in question depends on context (often Euclidean or Minkowski space). See also this question on math stack exchange
            $endgroup$
            – Carmeister
            4 hours ago











            0












            $begingroup$

            First of all, I'll constrain the discussion assuming:



            1) Finite-dimensional vector spaces



            2) Real Vector spaces



            3) Talking just about contravariant tensors



            4) Physics which use the standard notion of Spacetime



            $$* * *$$



            To answer your question I need to talk a little bit about Tensors.



            I) The tensor object and pure mathematics:



            The precise answer to the question "What is a tensor?" is, by far:




            A tensor is a object of a vector space called Tensor Product.




            In order to this general statement become something that have some value to you, I would like you to think a little bit about vectors and their algebra : the linear algebra.



            I.1) What truly is a Vector?



            First of all, if you look on linear algebra texts, you'll rapidly realize that the answer to the question "What are vectors after all? Matrices? Arrows? Functions?" is:




            A vector is a element of a algebraic structure called vector space.




            So after the study of the definition of a vector space you can talk with all rigour in the world that a vector isn't a arrow or a matrix, but a element of a vector space.



            I.1.1) Some facts about vectors



            Consider then a vector formed by a linear combination of basis vectors:



            $$mathbfv = sum_j = 1^n v^jmathbfe_j tag1$$



            This is well-known fact about vectors. So, there's another key point about basis vectors: the vector space is spanned by these basis vectors. You can create a "constrain machinery" to verify if a set of vectors spans a entire vector space (i.e. forms a basis):




            A set $mathcalS$ is a basis for a vector space $mathfrakV$ if:



            1) the vectors of the set $mathcalS$ are linear independent



            2) the vectors of the set $mathcalS$ spanned the vector space $mathfrakV$, i.e. $mathfrakV equiv span(mathcalS)$




            So another point of view to "form" an entire vector space is from basis vectors. The intuitive idea is that, more or less, like if the basis vectors "constructs" (they span) and entire vector space.



            Another fact is that you can change the basis $mathbfe_j$ to another set basis of basis $mathbfe'_j$. Well, when you do this the vector components suffer a change too. And then the components transforms like:



            $$v'^k = sum^n_j=1M^k_jv^jtag2$$



            but, of course, the vector object, remains the same:



            $$mathbfv = sum_j = 1^n v^jmathbfe_j = sum_j = 1^n v'^jmathbfe'_j$$



            So, a vector, truly, is a object of a vector space, which have the form of $(1)$ and their components transforms like $(2)$.



            I.1.2) The "physicist way" of definition of a Tensor



            When you're searching about tensors on physics/engeneering texts you certainly will encounter the following definition of a tensor:




            A Tensor is defined as the kind of object which transforms, under a coordinate transformation, like:



            $$T'^ij = sum^n_k=1sum^n_l=1 M^i_kM^j_l T^kl tag3$$




            This definition serves to encode the notion that a valid physical law must be independent of coordinate systems (or all that G.Smith said).
            Well, there's some interesting happening here. A vector, is a object which have a precise formulation in terms of a algebraic structure, have a precise form (that of $(1)$, which the basis vectors spans the entire $mathfrakV$) and their components have "transformation behaviour" like $(2)$. If you compare what I exposed about vector and $(3)$, you may reach the conclusion that, concerning about tensors, some information about their nature is missing.
            The fact is, the definition $(3)$ isn't a tensor, but the "transformation behaviour" of the components of a tensor $mathbfT$.



            I.2) What truly is a Tensor?



            Well, you have the transformation of components of a tensor, i.e. $(3)$, well defined. But what about their "space" and "form" (something like $(1)$)?



            So, the space is called tensor product of two vector spaces:




            $$Votimes W tag4$$




            The construction of tensor product is something beyond the scope of this answer [*]. But the mathematical considerations about tensor products are that they generalize the concept of products of vectors (remember that, in linear algebra and analytic geometry you're able to "multiply" vector just using the inner product and vector product), they construct a concept of products of vector spaces (remember that a Direct sum of vector space gives you a notion of Sum of vector spaces), and they construct the precise notion of a tensor. Also, by the technology of the construction of the tensor product we can identify (i.e. stablish a isomorphism between vector spaces) the vector space $Votimes W$ and $mathfrakLin^2(V^* times W^*; mathbbK)$:




            $$Votimes W cong mathfrakLin^2(V^* times W^*; mathbbK) tag5$$



            where $mathfrakLin^2(V^* times W^*; mathbbK)$ is the dual vector space of all bilinear functionals.




            So a tensor have the form:




            $$mathbfT = sum^n_i=1sum^n_j=1 T^ij (mathbfe_iotimesmathbfe_j) tag6$$
            And $mathbfT in Votimes W$.




            Well, given the transformation rule $(3)$ the space, $(5)$, and form, $(6)$, you can talk precisely about what tensor really is. It's clear that the "object tensor" isn't just a transformation of coordinates. Also, in $(6)$ the tensor basis $(mathbfe_iotimesmathbfe_j)$ spans $Votimes W$.



            By virtue of the general construction of tensor product and the identification given by $(5)$, you'll also encounter the definition of a tensor as a multilinear object which spits scalars:




            $$beginarrayrl
            mathbfT :V^*times W^* &to mathbbK \
            (mathbfv,mathbfw)&mapsto mathbfT(mathbfv,mathbfw)=: v^icdot_mathbbKw^j
            endarray$$



            Where the operation $cdot_mathbbK$ is the product defined in the field.




            With this picture we say that a tensor like $(6)$ is a tensor of rank 2. And a vector a tensor of rank 1. Furthermore a scalar a tensor of rank 0.



            II) The tensor object and physics



            The well stablish physics, in general, deals with spacetime (like Newtonian physics), and the theory of spacetime is geometry. So, in order to really apply the tensor theory in physics first we have to give the geometry of physics.
            The geometry is basically classical Manifold Theory (which, again, is beyond the scope). And by Manifold Theory we can apply tensors on Manifolds introducing the concept of a tangent vector space. In parallel, we can construct another algebraic structure called Fibre Bundle of tangent spaces and then create the precise notion of Vector Field and Tensor Field.



            Tensor Fields are the real objects defined in physics books as tensors and we use the word of a tensor and tensor field as synonyms (IN FACT THEY ARE NOT THE SAME CONCEPT!). A tensor field is a section of the tensor bundle and a vector field, a section of vector bundle. But the intuitive definition (by far, general to physics) of a tensor field is then:




            $$[mathbfT(x^k)] = sum^n_i=1sum^n_j=1 [T^ij(x^k)] ([mathbfe_i(x^k)]otimes[mathbfe_j(x^k)]) tag5$$
            A tensor field is the object which attaches a tensor to every point p of the Manifold.




            With the manifold theory, the transformation rule becomes:




            $$[T'^ij(x^m)] = sum^n_k=1sum^n_l=1 fracpartial x'^ipartial x^kfracpartial x'^jpartial x^l [T^kl(x^m)] equiv T'^ij = sum^n_k=1sum^n_l=1 fracpartial x'^ipartial x^kfracpartial x'^jpartial x^l T^kl tag7$$




            Notice that the partials are simply the transformation matrices $M$. The matrices $M$ are called the Jacobians transformation matrices and the matrices $M$ became these jacobians by virtue of Manifold theory.



            In a restric way, these Jacobians are rotations,lorentz transformations,galilean transformation, and so on.



            III) What is the difference between zero-rank tensor x (scalar) and 1D vector [x]?



            So, in order to talk about lengths we have to realize that we are talking about a scalar field, or a tensor of rank 0. Then the difference between a scalar and a 1D vector (which is a tensor of rank 1) is that one is a scalar field and the other is a vector field. From a "Pure" mathematical point of view, (section I) if this answer) one is a member of the field $mathbbK$ and the other is a member of a vector space.
            Also, you're quite right, a scalar (or a scalar field) is a rank 0 tensor or "a object which do not have "matrices of change"; an object which do not suffer a change under a transformation of coordinates (we say that a scalar quantity is a invariant quantity).




            $$phi'= phi$$




            $$* * *$$



            [*] ROMAN.S. Advanced Linear Algebra. Springer. chapter 14. 1 ed. 1992.






            share|cite|improve this answer











            $endgroup$

















              0












              $begingroup$

              First of all, I'll constrain the discussion assuming:



              1) Finite-dimensional vector spaces



              2) Real Vector spaces



              3) Talking just about contravariant tensors



              4) Physics which use the standard notion of Spacetime



              $$* * *$$



              To answer your question I need to talk a little bit about Tensors.



              I) The tensor object and pure mathematics:



              The precise answer to the question "What is a tensor?" is, by far:




              A tensor is a object of a vector space called Tensor Product.




              In order to this general statement become something that have some value to you, I would like you to think a little bit about vectors and their algebra : the linear algebra.



              I.1) What truly is a Vector?



              First of all, if you look on linear algebra texts, you'll rapidly realize that the answer to the question "What are vectors after all? Matrices? Arrows? Functions?" is:




              A vector is a element of a algebraic structure called vector space.




              So after the study of the definition of a vector space you can talk with all rigour in the world that a vector isn't a arrow or a matrix, but a element of a vector space.



              I.1.1) Some facts about vectors



              Consider then a vector formed by a linear combination of basis vectors:



              $$mathbfv = sum_j = 1^n v^jmathbfe_j tag1$$



              This is well-known fact about vectors. So, there's another key point about basis vectors: the vector space is spanned by these basis vectors. You can create a "constrain machinery" to verify if a set of vectors spans a entire vector space (i.e. forms a basis):




              A set $mathcalS$ is a basis for a vector space $mathfrakV$ if:



              1) the vectors of the set $mathcalS$ are linear independent



              2) the vectors of the set $mathcalS$ spanned the vector space $mathfrakV$, i.e. $mathfrakV equiv span(mathcalS)$




              So another point of view to "form" an entire vector space is from basis vectors. The intuitive idea is that, more or less, like if the basis vectors "constructs" (they span) and entire vector space.



              Another fact is that you can change the basis $mathbfe_j$ to another set basis of basis $mathbfe'_j$. Well, when you do this the vector components suffer a change too. And then the components transforms like:



              $$v'^k = sum^n_j=1M^k_jv^jtag2$$



              but, of course, the vector object, remains the same:



              $$mathbfv = sum_j = 1^n v^jmathbfe_j = sum_j = 1^n v'^jmathbfe'_j$$



              So, a vector, truly, is a object of a vector space, which have the form of $(1)$ and their components transforms like $(2)$.



              I.1.2) The "physicist way" of definition of a Tensor



              When you're searching about tensors on physics/engeneering texts you certainly will encounter the following definition of a tensor:




              A Tensor is defined as the kind of object which transforms, under a coordinate transformation, like:



              $$T'^ij = sum^n_k=1sum^n_l=1 M^i_kM^j_l T^kl tag3$$




              This definition serves to encode the notion that a valid physical law must be independent of coordinate systems (or all that G.Smith said).
              Well, there's some interesting happening here. A vector, is a object which have a precise formulation in terms of a algebraic structure, have a precise form (that of $(1)$, which the basis vectors spans the entire $mathfrakV$) and their components have "transformation behaviour" like $(2)$. If you compare what I exposed about vector and $(3)$, you may reach the conclusion that, concerning about tensors, some information about their nature is missing.
              The fact is, the definition $(3)$ isn't a tensor, but the "transformation behaviour" of the components of a tensor $mathbfT$.



              I.2) What truly is a Tensor?



              Well, you have the transformation of components of a tensor, i.e. $(3)$, well defined. But what about their "space" and "form" (something like $(1)$)?



              So, the space is called tensor product of two vector spaces:




              $$Votimes W tag4$$




              The construction of tensor product is something beyond the scope of this answer [*]. But the mathematical considerations about tensor products are that they generalize the concept of products of vectors (remember that, in linear algebra and analytic geometry you're able to "multiply" vector just using the inner product and vector product), they construct a concept of products of vector spaces (remember that a Direct sum of vector space gives you a notion of Sum of vector spaces), and they construct the precise notion of a tensor. Also, by the technology of the construction of the tensor product we can identify (i.e. stablish a isomorphism between vector spaces) the vector space $Votimes W$ and $mathfrakLin^2(V^* times W^*; mathbbK)$:




              $$Votimes W cong mathfrakLin^2(V^* times W^*; mathbbK) tag5$$



              where $mathfrakLin^2(V^* times W^*; mathbbK)$ is the dual vector space of all bilinear functionals.




              So a tensor have the form:




              $$mathbfT = sum^n_i=1sum^n_j=1 T^ij (mathbfe_iotimesmathbfe_j) tag6$$
              And $mathbfT in Votimes W$.




              Well, given the transformation rule $(3)$ the space, $(5)$, and form, $(6)$, you can talk precisely about what tensor really is. It's clear that the "object tensor" isn't just a transformation of coordinates. Also, in $(6)$ the tensor basis $(mathbfe_iotimesmathbfe_j)$ spans $Votimes W$.



              By virtue of the general construction of tensor product and the identification given by $(5)$, you'll also encounter the definition of a tensor as a multilinear object which spits scalars:




              $$beginarrayrl
              mathbfT :V^*times W^* &to mathbbK \
              (mathbfv,mathbfw)&mapsto mathbfT(mathbfv,mathbfw)=: v^icdot_mathbbKw^j
              endarray$$



              Where the operation $cdot_mathbbK$ is the product defined in the field.




              With this picture we say that a tensor like $(6)$ is a tensor of rank 2. And a vector a tensor of rank 1. Furthermore a scalar a tensor of rank 0.



              II) The tensor object and physics



              The well stablish physics, in general, deals with spacetime (like Newtonian physics), and the theory of spacetime is geometry. So, in order to really apply the tensor theory in physics first we have to give the geometry of physics.
              The geometry is basically classical Manifold Theory (which, again, is beyond the scope). And by Manifold Theory we can apply tensors on Manifolds introducing the concept of a tangent vector space. In parallel, we can construct another algebraic structure called Fibre Bundle of tangent spaces and then create the precise notion of Vector Field and Tensor Field.



              Tensor Fields are the real objects defined in physics books as tensors and we use the word of a tensor and tensor field as synonyms (IN FACT THEY ARE NOT THE SAME CONCEPT!). A tensor field is a section of the tensor bundle and a vector field, a section of vector bundle. But the intuitive definition (by far, general to physics) of a tensor field is then:




              $$[mathbfT(x^k)] = sum^n_i=1sum^n_j=1 [T^ij(x^k)] ([mathbfe_i(x^k)]otimes[mathbfe_j(x^k)]) tag5$$
              A tensor field is the object which attaches a tensor to every point p of the Manifold.




              With the manifold theory, the transformation rule becomes:




              $$[T'^ij(x^m)] = sum^n_k=1sum^n_l=1 fracpartial x'^ipartial x^kfracpartial x'^jpartial x^l [T^kl(x^m)] equiv T'^ij = sum^n_k=1sum^n_l=1 fracpartial x'^ipartial x^kfracpartial x'^jpartial x^l T^kl tag7$$




              Notice that the partials are simply the transformation matrices $M$. The matrices $M$ are called the Jacobians transformation matrices and the matrices $M$ became these jacobians by virtue of Manifold theory.



              In a restric way, these Jacobians are rotations,lorentz transformations,galilean transformation, and so on.



              III) What is the difference between zero-rank tensor x (scalar) and 1D vector [x]?



              So, in order to talk about lengths we have to realize that we are talking about a scalar field, or a tensor of rank 0. Then the difference between a scalar and a 1D vector (which is a tensor of rank 1) is that one is a scalar field and the other is a vector field. From a "Pure" mathematical point of view, (section I) if this answer) one is a member of the field $mathbbK$ and the other is a member of a vector space.
              Also, you're quite right, a scalar (or a scalar field) is a rank 0 tensor or "a object which do not have "matrices of change"; an object which do not suffer a change under a transformation of coordinates (we say that a scalar quantity is a invariant quantity).




              $$phi'= phi$$




              $$* * *$$



              [*] ROMAN.S. Advanced Linear Algebra. Springer. chapter 14. 1 ed. 1992.






              share|cite|improve this answer











              $endgroup$















                0












                0








                0





                $begingroup$

                First of all, I'll constrain the discussion assuming:



                1) Finite-dimensional vector spaces



                2) Real Vector spaces



                3) Talking just about contravariant tensors



                4) Physics which use the standard notion of Spacetime



                $$* * *$$



                To answer your question I need to talk a little bit about Tensors.



                I) The tensor object and pure mathematics:



                The precise answer to the question "What is a tensor?" is, by far:




                A tensor is a object of a vector space called Tensor Product.




                In order to this general statement become something that have some value to you, I would like you to think a little bit about vectors and their algebra : the linear algebra.



                I.1) What truly is a Vector?



                First of all, if you look on linear algebra texts, you'll rapidly realize that the answer to the question "What are vectors after all? Matrices? Arrows? Functions?" is:




                A vector is a element of a algebraic structure called vector space.




                So after the study of the definition of a vector space you can talk with all rigour in the world that a vector isn't a arrow or a matrix, but a element of a vector space.



                I.1.1) Some facts about vectors



                Consider then a vector formed by a linear combination of basis vectors:



                $$mathbfv = sum_j = 1^n v^jmathbfe_j tag1$$



                This is well-known fact about vectors. So, there's another key point about basis vectors: the vector space is spanned by these basis vectors. You can create a "constrain machinery" to verify if a set of vectors spans a entire vector space (i.e. forms a basis):




                A set $mathcalS$ is a basis for a vector space $mathfrakV$ if:



                1) the vectors of the set $mathcalS$ are linear independent



                2) the vectors of the set $mathcalS$ spanned the vector space $mathfrakV$, i.e. $mathfrakV equiv span(mathcalS)$




                So another point of view to "form" an entire vector space is from basis vectors. The intuitive idea is that, more or less, like if the basis vectors "constructs" (they span) and entire vector space.



                Another fact is that you can change the basis $mathbfe_j$ to another set basis of basis $mathbfe'_j$. Well, when you do this the vector components suffer a change too. And then the components transforms like:



                $$v'^k = sum^n_j=1M^k_jv^jtag2$$



                but, of course, the vector object, remains the same:



                $$mathbfv = sum_j = 1^n v^jmathbfe_j = sum_j = 1^n v'^jmathbfe'_j$$



                So, a vector, truly, is a object of a vector space, which have the form of $(1)$ and their components transforms like $(2)$.



                I.1.2) The "physicist way" of definition of a Tensor



                When you're searching about tensors on physics/engeneering texts you certainly will encounter the following definition of a tensor:




                A Tensor is defined as the kind of object which transforms, under a coordinate transformation, like:



                $$T'^ij = sum^n_k=1sum^n_l=1 M^i_kM^j_l T^kl tag3$$




                This definition serves to encode the notion that a valid physical law must be independent of coordinate systems (or all that G.Smith said).
                Well, there's some interesting happening here. A vector, is a object which have a precise formulation in terms of a algebraic structure, have a precise form (that of $(1)$, which the basis vectors spans the entire $mathfrakV$) and their components have "transformation behaviour" like $(2)$. If you compare what I exposed about vector and $(3)$, you may reach the conclusion that, concerning about tensors, some information about their nature is missing.
                The fact is, the definition $(3)$ isn't a tensor, but the "transformation behaviour" of the components of a tensor $mathbfT$.



                I.2) What truly is a Tensor?



                Well, you have the transformation of components of a tensor, i.e. $(3)$, well defined. But what about their "space" and "form" (something like $(1)$)?



                So, the space is called tensor product of two vector spaces:




                $$Votimes W tag4$$




                The construction of tensor product is something beyond the scope of this answer [*]. But the mathematical considerations about tensor products are that they generalize the concept of products of vectors (remember that, in linear algebra and analytic geometry you're able to "multiply" vector just using the inner product and vector product), they construct a concept of products of vector spaces (remember that a Direct sum of vector space gives you a notion of Sum of vector spaces), and they construct the precise notion of a tensor. Also, by the technology of the construction of the tensor product we can identify (i.e. stablish a isomorphism between vector spaces) the vector space $Votimes W$ and $mathfrakLin^2(V^* times W^*; mathbbK)$:




                $$Votimes W cong mathfrakLin^2(V^* times W^*; mathbbK) tag5$$



                where $mathfrakLin^2(V^* times W^*; mathbbK)$ is the dual vector space of all bilinear functionals.




                So a tensor have the form:




                $$mathbfT = sum^n_i=1sum^n_j=1 T^ij (mathbfe_iotimesmathbfe_j) tag6$$
                And $mathbfT in Votimes W$.




                Well, given the transformation rule $(3)$ the space, $(5)$, and form, $(6)$, you can talk precisely about what tensor really is. It's clear that the "object tensor" isn't just a transformation of coordinates. Also, in $(6)$ the tensor basis $(mathbfe_iotimesmathbfe_j)$ spans $Votimes W$.



                By virtue of the general construction of tensor product and the identification given by $(5)$, you'll also encounter the definition of a tensor as a multilinear object which spits scalars:




                $$beginarrayrl
                mathbfT :V^*times W^* &to mathbbK \
                (mathbfv,mathbfw)&mapsto mathbfT(mathbfv,mathbfw)=: v^icdot_mathbbKw^j
                endarray$$



                Where the operation $cdot_mathbbK$ is the product defined in the field.




                With this picture we say that a tensor like $(6)$ is a tensor of rank 2. And a vector a tensor of rank 1. Furthermore a scalar a tensor of rank 0.



                II) The tensor object and physics



                The well stablish physics, in general, deals with spacetime (like Newtonian physics), and the theory of spacetime is geometry. So, in order to really apply the tensor theory in physics first we have to give the geometry of physics.
                The geometry is basically classical Manifold Theory (which, again, is beyond the scope). And by Manifold Theory we can apply tensors on Manifolds introducing the concept of a tangent vector space. In parallel, we can construct another algebraic structure called Fibre Bundle of tangent spaces and then create the precise notion of Vector Field and Tensor Field.



                Tensor Fields are the real objects defined in physics books as tensors and we use the word of a tensor and tensor field as synonyms (IN FACT THEY ARE NOT THE SAME CONCEPT!). A tensor field is a section of the tensor bundle and a vector field, a section of vector bundle. But the intuitive definition (by far, general to physics) of a tensor field is then:




                $$[mathbfT(x^k)] = sum^n_i=1sum^n_j=1 [T^ij(x^k)] ([mathbfe_i(x^k)]otimes[mathbfe_j(x^k)]) tag5$$
                A tensor field is the object which attaches a tensor to every point p of the Manifold.




                With the manifold theory, the transformation rule becomes:




                $$[T'^ij(x^m)] = sum^n_k=1sum^n_l=1 fracpartial x'^ipartial x^kfracpartial x'^jpartial x^l [T^kl(x^m)] equiv T'^ij = sum^n_k=1sum^n_l=1 fracpartial x'^ipartial x^kfracpartial x'^jpartial x^l T^kl tag7$$




                Notice that the partials are simply the transformation matrices $M$. The matrices $M$ are called the Jacobians transformation matrices and the matrices $M$ became these jacobians by virtue of Manifold theory.



                In a restric way, these Jacobians are rotations,lorentz transformations,galilean transformation, and so on.



                III) What is the difference between zero-rank tensor x (scalar) and 1D vector [x]?



                So, in order to talk about lengths we have to realize that we are talking about a scalar field, or a tensor of rank 0. Then the difference between a scalar and a 1D vector (which is a tensor of rank 1) is that one is a scalar field and the other is a vector field. From a "Pure" mathematical point of view, (section I) if this answer) one is a member of the field $mathbbK$ and the other is a member of a vector space.
                Also, you're quite right, a scalar (or a scalar field) is a rank 0 tensor or "a object which do not have "matrices of change"; an object which do not suffer a change under a transformation of coordinates (we say that a scalar quantity is a invariant quantity).




                $$phi'= phi$$




                $$* * *$$



                [*] ROMAN.S. Advanced Linear Algebra. Springer. chapter 14. 1 ed. 1992.






                share|cite|improve this answer











                $endgroup$



                First of all, I'll constrain the discussion assuming:



                1) Finite-dimensional vector spaces



                2) Real Vector spaces



                3) Talking just about contravariant tensors



                4) Physics which use the standard notion of Spacetime



                $$* * *$$



                To answer your question I need to talk a little bit about Tensors.



                I) The tensor object and pure mathematics:



                The precise answer to the question "What is a tensor?" is, by far:




                A tensor is a object of a vector space called Tensor Product.




                In order to this general statement become something that have some value to you, I would like you to think a little bit about vectors and their algebra : the linear algebra.



                I.1) What truly is a Vector?



                First of all, if you look on linear algebra texts, you'll rapidly realize that the answer to the question "What are vectors after all? Matrices? Arrows? Functions?" is:




                A vector is a element of a algebraic structure called vector space.




                So after the study of the definition of a vector space you can talk with all rigour in the world that a vector isn't a arrow or a matrix, but a element of a vector space.



                I.1.1) Some facts about vectors



                Consider then a vector formed by a linear combination of basis vectors:



                $$mathbfv = sum_j = 1^n v^jmathbfe_j tag1$$



                This is well-known fact about vectors. So, there's another key point about basis vectors: the vector space is spanned by these basis vectors. You can create a "constrain machinery" to verify if a set of vectors spans a entire vector space (i.e. forms a basis):




                A set $mathcalS$ is a basis for a vector space $mathfrakV$ if:



                1) the vectors of the set $mathcalS$ are linear independent



                2) the vectors of the set $mathcalS$ spanned the vector space $mathfrakV$, i.e. $mathfrakV equiv span(mathcalS)$




                So another point of view to "form" an entire vector space is from basis vectors. The intuitive idea is that, more or less, like if the basis vectors "constructs" (they span) and entire vector space.



                Another fact is that you can change the basis $mathbfe_j$ to another set basis of basis $mathbfe'_j$. Well, when you do this the vector components suffer a change too. And then the components transforms like:



                $$v'^k = sum^n_j=1M^k_jv^jtag2$$



                but, of course, the vector object, remains the same:



                $$mathbfv = sum_j = 1^n v^jmathbfe_j = sum_j = 1^n v'^jmathbfe'_j$$



                So, a vector, truly, is a object of a vector space, which have the form of $(1)$ and their components transforms like $(2)$.



                I.1.2) The "physicist way" of definition of a Tensor



                When you're searching about tensors on physics/engeneering texts you certainly will encounter the following definition of a tensor:




                A Tensor is defined as the kind of object which transforms, under a coordinate transformation, like:



                $$T'^ij = sum^n_k=1sum^n_l=1 M^i_kM^j_l T^kl tag3$$




                This definition serves to encode the notion that a valid physical law must be independent of coordinate systems (or all that G.Smith said).
                Well, there's some interesting happening here. A vector, is a object which have a precise formulation in terms of a algebraic structure, have a precise form (that of $(1)$, which the basis vectors spans the entire $mathfrakV$) and their components have "transformation behaviour" like $(2)$. If you compare what I exposed about vector and $(3)$, you may reach the conclusion that, concerning about tensors, some information about their nature is missing.
                The fact is, the definition $(3)$ isn't a tensor, but the "transformation behaviour" of the components of a tensor $mathbfT$.



                I.2) What truly is a Tensor?



                Well, you have the transformation of components of a tensor, i.e. $(3)$, well defined. But what about their "space" and "form" (something like $(1)$)?



                So, the space is called tensor product of two vector spaces:




                $$Votimes W tag4$$




                The construction of tensor product is something beyond the scope of this answer [*]. But the mathematical considerations about tensor products are that they generalize the concept of products of vectors (remember that, in linear algebra and analytic geometry you're able to "multiply" vector just using the inner product and vector product), they construct a concept of products of vector spaces (remember that a Direct sum of vector space gives you a notion of Sum of vector spaces), and they construct the precise notion of a tensor. Also, by the technology of the construction of the tensor product we can identify (i.e. stablish a isomorphism between vector spaces) the vector space $Votimes W$ and $mathfrakLin^2(V^* times W^*; mathbbK)$:




                $$Votimes W cong mathfrakLin^2(V^* times W^*; mathbbK) tag5$$



                where $mathfrakLin^2(V^* times W^*; mathbbK)$ is the dual vector space of all bilinear functionals.




                So a tensor have the form:




                $$mathbfT = sum^n_i=1sum^n_j=1 T^ij (mathbfe_iotimesmathbfe_j) tag6$$
                And $mathbfT in Votimes W$.




                Well, given the transformation rule $(3)$ the space, $(5)$, and form, $(6)$, you can talk precisely about what tensor really is. It's clear that the "object tensor" isn't just a transformation of coordinates. Also, in $(6)$ the tensor basis $(mathbfe_iotimesmathbfe_j)$ spans $Votimes W$.



                By virtue of the general construction of tensor product and the identification given by $(5)$, you'll also encounter the definition of a tensor as a multilinear object which spits scalars:




                $$beginarrayrl
                mathbfT :V^*times W^* &to mathbbK \
                (mathbfv,mathbfw)&mapsto mathbfT(mathbfv,mathbfw)=: v^icdot_mathbbKw^j
                endarray$$



                Where the operation $cdot_mathbbK$ is the product defined in the field.




                With this picture we say that a tensor like $(6)$ is a tensor of rank 2. And a vector a tensor of rank 1. Furthermore a scalar a tensor of rank 0.



                II) The tensor object and physics



                The well stablish physics, in general, deals with spacetime (like Newtonian physics), and the theory of spacetime is geometry. So, in order to really apply the tensor theory in physics first we have to give the geometry of physics.
                The geometry is basically classical Manifold Theory (which, again, is beyond the scope). And by Manifold Theory we can apply tensors on Manifolds introducing the concept of a tangent vector space. In parallel, we can construct another algebraic structure called Fibre Bundle of tangent spaces and then create the precise notion of Vector Field and Tensor Field.



                Tensor Fields are the real objects defined in physics books as tensors and we use the word of a tensor and tensor field as synonyms (IN FACT THEY ARE NOT THE SAME CONCEPT!). A tensor field is a section of the tensor bundle and a vector field, a section of vector bundle. But the intuitive definition (by far, general to physics) of a tensor field is then:




                $$[mathbfT(x^k)] = sum^n_i=1sum^n_j=1 [T^ij(x^k)] ([mathbfe_i(x^k)]otimes[mathbfe_j(x^k)]) tag5$$
                A tensor field is the object which attaches a tensor to every point p of the Manifold.




                With the manifold theory, the transformation rule becomes:




                $$[T'^ij(x^m)] = sum^n_k=1sum^n_l=1 fracpartial x'^ipartial x^kfracpartial x'^jpartial x^l [T^kl(x^m)] equiv T'^ij = sum^n_k=1sum^n_l=1 fracpartial x'^ipartial x^kfracpartial x'^jpartial x^l T^kl tag7$$




                Notice that the partials are simply the transformation matrices $M$. The matrices $M$ are called the Jacobians transformation matrices and the matrices $M$ became these jacobians by virtue of Manifold theory.



                In a restric way, these Jacobians are rotations,lorentz transformations,galilean transformation, and so on.



                III) What is the difference between zero-rank tensor x (scalar) and 1D vector [x]?



                So, in order to talk about lengths we have to realize that we are talking about a scalar field, or a tensor of rank 0. Then the difference between a scalar and a 1D vector (which is a tensor of rank 1) is that one is a scalar field and the other is a vector field. From a "Pure" mathematical point of view, (section I) if this answer) one is a member of the field $mathbbK$ and the other is a member of a vector space.
                Also, you're quite right, a scalar (or a scalar field) is a rank 0 tensor or "a object which do not have "matrices of change"; an object which do not suffer a change under a transformation of coordinates (we say that a scalar quantity is a invariant quantity).




                $$phi'= phi$$




                $$* * *$$



                [*] ROMAN.S. Advanced Linear Algebra. Springer. chapter 14. 1 ed. 1992.







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                M.N.RaiaM.N.Raia

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