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Is vector field a field (in the algebraic sense) also?


Representing a vector field locallyWhen is a gradient vector field also an algebraic field?Extension of Vector field along a curve always exists?Reconstruct a smooth vector field from direction field (Arnold's ODE)Vector fields as rank 1 contravariant tensor fieldsOutward-pointing vector field on Projective spaceSmooth extension of a coordinate vector fieldHow can a vector field eat a smooth function?Intuition of a smooth vector fieldFinding an explicit formula for a Hamiltonian vector field













3












$begingroup$


This is something about basic confusion.



Let $M$ be a smooth manifold of dimension $n$.



We denote $X$ to mean the vector field.




Is the vector field $X$, a field (in the sense of abstract algebra) also?











share|cite|improve this question











$endgroup$
















    3












    $begingroup$


    This is something about basic confusion.



    Let $M$ be a smooth manifold of dimension $n$.



    We denote $X$ to mean the vector field.




    Is the vector field $X$, a field (in the sense of abstract algebra) also?











    share|cite|improve this question











    $endgroup$














      3












      3








      3


      1



      $begingroup$


      This is something about basic confusion.



      Let $M$ be a smooth manifold of dimension $n$.



      We denote $X$ to mean the vector field.




      Is the vector field $X$, a field (in the sense of abstract algebra) also?











      share|cite|improve this question











      $endgroup$




      This is something about basic confusion.



      Let $M$ be a smooth manifold of dimension $n$.



      We denote $X$ to mean the vector field.




      Is the vector field $X$, a field (in the sense of abstract algebra) also?








      vector-fields






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Mar 14 at 11:19









      Asaf Karagila

      306k33438769




      306k33438769










      asked Mar 14 at 8:31









      M. A. SARKARM. A. SARKAR

      2,3991819




      2,3991819




















          1 Answer
          1






          active

          oldest

          votes


















          5












          $begingroup$

          A single vector field is simply a choice of vector for each point in the plane. It doesn't have "elements" in any sensible way, and especially not elements one can add or multiply.



          The collection $mathscr X$ of all vector fields on $M$ (or all continuous vector fields, or all smooth vector fields) certainly has algebraic structure, as they can be added, and they can be multiplied by (real or complex) scalars. We call such a structure a module (in this case it isn't wrong to call $mathscr X$ a vector space either). But vector fields cannot in any sensible way be multiplied with one another, so such a set cannot be a ring, and therefore cannot be a field.






          share|cite|improve this answer









          $endgroup$












          • $begingroup$
            A module over a field is a vector space. Why call it "module" then?
            $endgroup$
            – enedil
            Mar 14 at 8:57











          • $begingroup$
            @enedil Because once we get to this stage, the word "vector" is already in heavy use. Having one more thing called a "vector space" and an entirely new set of "vectors" to keep track of as we're working will not help. Also, it's convention. And as you say, it is exactly the same thing in this case, so we don't lose anything.
            $endgroup$
            – Arthur
            Mar 14 at 9:01











          Your Answer





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          1 Answer
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          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          5












          $begingroup$

          A single vector field is simply a choice of vector for each point in the plane. It doesn't have "elements" in any sensible way, and especially not elements one can add or multiply.



          The collection $mathscr X$ of all vector fields on $M$ (or all continuous vector fields, or all smooth vector fields) certainly has algebraic structure, as they can be added, and they can be multiplied by (real or complex) scalars. We call such a structure a module (in this case it isn't wrong to call $mathscr X$ a vector space either). But vector fields cannot in any sensible way be multiplied with one another, so such a set cannot be a ring, and therefore cannot be a field.






          share|cite|improve this answer









          $endgroup$












          • $begingroup$
            A module over a field is a vector space. Why call it "module" then?
            $endgroup$
            – enedil
            Mar 14 at 8:57











          • $begingroup$
            @enedil Because once we get to this stage, the word "vector" is already in heavy use. Having one more thing called a "vector space" and an entirely new set of "vectors" to keep track of as we're working will not help. Also, it's convention. And as you say, it is exactly the same thing in this case, so we don't lose anything.
            $endgroup$
            – Arthur
            Mar 14 at 9:01
















          5












          $begingroup$

          A single vector field is simply a choice of vector for each point in the plane. It doesn't have "elements" in any sensible way, and especially not elements one can add or multiply.



          The collection $mathscr X$ of all vector fields on $M$ (or all continuous vector fields, or all smooth vector fields) certainly has algebraic structure, as they can be added, and they can be multiplied by (real or complex) scalars. We call such a structure a module (in this case it isn't wrong to call $mathscr X$ a vector space either). But vector fields cannot in any sensible way be multiplied with one another, so such a set cannot be a ring, and therefore cannot be a field.






          share|cite|improve this answer









          $endgroup$












          • $begingroup$
            A module over a field is a vector space. Why call it "module" then?
            $endgroup$
            – enedil
            Mar 14 at 8:57











          • $begingroup$
            @enedil Because once we get to this stage, the word "vector" is already in heavy use. Having one more thing called a "vector space" and an entirely new set of "vectors" to keep track of as we're working will not help. Also, it's convention. And as you say, it is exactly the same thing in this case, so we don't lose anything.
            $endgroup$
            – Arthur
            Mar 14 at 9:01














          5












          5








          5





          $begingroup$

          A single vector field is simply a choice of vector for each point in the plane. It doesn't have "elements" in any sensible way, and especially not elements one can add or multiply.



          The collection $mathscr X$ of all vector fields on $M$ (or all continuous vector fields, or all smooth vector fields) certainly has algebraic structure, as they can be added, and they can be multiplied by (real or complex) scalars. We call such a structure a module (in this case it isn't wrong to call $mathscr X$ a vector space either). But vector fields cannot in any sensible way be multiplied with one another, so such a set cannot be a ring, and therefore cannot be a field.






          share|cite|improve this answer









          $endgroup$



          A single vector field is simply a choice of vector for each point in the plane. It doesn't have "elements" in any sensible way, and especially not elements one can add or multiply.



          The collection $mathscr X$ of all vector fields on $M$ (or all continuous vector fields, or all smooth vector fields) certainly has algebraic structure, as they can be added, and they can be multiplied by (real or complex) scalars. We call such a structure a module (in this case it isn't wrong to call $mathscr X$ a vector space either). But vector fields cannot in any sensible way be multiplied with one another, so such a set cannot be a ring, and therefore cannot be a field.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Mar 14 at 8:34









          ArthurArthur

          119k7118202




          119k7118202











          • $begingroup$
            A module over a field is a vector space. Why call it "module" then?
            $endgroup$
            – enedil
            Mar 14 at 8:57











          • $begingroup$
            @enedil Because once we get to this stage, the word "vector" is already in heavy use. Having one more thing called a "vector space" and an entirely new set of "vectors" to keep track of as we're working will not help. Also, it's convention. And as you say, it is exactly the same thing in this case, so we don't lose anything.
            $endgroup$
            – Arthur
            Mar 14 at 9:01

















          • $begingroup$
            A module over a field is a vector space. Why call it "module" then?
            $endgroup$
            – enedil
            Mar 14 at 8:57











          • $begingroup$
            @enedil Because once we get to this stage, the word "vector" is already in heavy use. Having one more thing called a "vector space" and an entirely new set of "vectors" to keep track of as we're working will not help. Also, it's convention. And as you say, it is exactly the same thing in this case, so we don't lose anything.
            $endgroup$
            – Arthur
            Mar 14 at 9:01
















          $begingroup$
          A module over a field is a vector space. Why call it "module" then?
          $endgroup$
          – enedil
          Mar 14 at 8:57





          $begingroup$
          A module over a field is a vector space. Why call it "module" then?
          $endgroup$
          – enedil
          Mar 14 at 8:57













          $begingroup$
          @enedil Because once we get to this stage, the word "vector" is already in heavy use. Having one more thing called a "vector space" and an entirely new set of "vectors" to keep track of as we're working will not help. Also, it's convention. And as you say, it is exactly the same thing in this case, so we don't lose anything.
          $endgroup$
          – Arthur
          Mar 14 at 9:01





          $begingroup$
          @enedil Because once we get to this stage, the word "vector" is already in heavy use. Having one more thing called a "vector space" and an entirely new set of "vectors" to keep track of as we're working will not help. Also, it's convention. And as you say, it is exactly the same thing in this case, so we don't lose anything.
          $endgroup$
          – Arthur
          Mar 14 at 9:01


















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