Understanding k- jets












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K-jets can be realized as space of (up to equivalence classes) of Taylor polynomials of order k. There is a statement in V. Arnold's lectures in PDE stating that 1-jets of functions in the plane can be viewed a 5-dimensional space. Why is that?
1) Can someone clarify the concept?
2)How are jets defined for Riemannian manifolds?






differential-geometry pde







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    $begingroup$


    K-jets can be realized as space of (up to equivalence classes) of Taylor polynomials of order k. There is a statement in V. Arnold's lectures in PDE stating that 1-jets of functions in the plane can be viewed a 5-dimensional space. Why is that?
    1) Can someone clarify the concept?
    2)How are jets defined for Riemannian manifolds?






    differential-geometry pde







    share|cite|improve this question









    $endgroup$















      0












      0








      0


      1



      $begingroup$


      K-jets can be realized as space of (up to equivalence classes) of Taylor polynomials of order k. There is a statement in V. Arnold's lectures in PDE stating that 1-jets of functions in the plane can be viewed a 5-dimensional space. Why is that?
      1) Can someone clarify the concept?
      2)How are jets defined for Riemannian manifolds?






      differential-geometry pde







      share|cite|improve this question









      $endgroup$




      K-jets can be realized as space of (up to equivalence classes) of Taylor polynomials of order k. There is a statement in V. Arnold's lectures in PDE stating that 1-jets of functions in the plane can be viewed a 5-dimensional space. Why is that?
      1) Can someone clarify the concept?
      2)How are jets defined for Riemannian manifolds?






      differential-geometry pde




      differential-geometry pde






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      asked Jan 25 at 3:42









      BigMBigM

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          At each point in the plane, a $1$-jet is determined by three numbers, since
          a Taylor series at point $(x_0,y_0)$ up to the linear terms has the form $a+b(x-x_0)+c(y-y_0)$. As a point then, the $1$-jets form a three-dimensional vector space. Then the $1$-jets in the plane are a rank three vector bundle over the
          plane, so the total space has dimension $3+2=5$.



          All this works just as well over a smooth manifold. Taking a coordinate
          patch one can define jets as Taylor expansions in the local coordinates there.
          Then one checks that these behave well when we transition between coordinate
          patches. It's the same sort of argument that's used to show that the tangent
          bundle of a manifold makes sense.






          share|cite|improve this answer









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            $begingroup$

            At each point in the plane, a $1$-jet is determined by three numbers, since
            a Taylor series at point $(x_0,y_0)$ up to the linear terms has the form $a+b(x-x_0)+c(y-y_0)$. As a point then, the $1$-jets form a three-dimensional vector space. Then the $1$-jets in the plane are a rank three vector bundle over the
            plane, so the total space has dimension $3+2=5$.



            All this works just as well over a smooth manifold. Taking a coordinate
            patch one can define jets as Taylor expansions in the local coordinates there.
            Then one checks that these behave well when we transition between coordinate
            patches. It's the same sort of argument that's used to show that the tangent
            bundle of a manifold makes sense.






            share|cite|improve this answer









            $endgroup$


















              2












              $begingroup$

              At each point in the plane, a $1$-jet is determined by three numbers, since
              a Taylor series at point $(x_0,y_0)$ up to the linear terms has the form $a+b(x-x_0)+c(y-y_0)$. As a point then, the $1$-jets form a three-dimensional vector space. Then the $1$-jets in the plane are a rank three vector bundle over the
              plane, so the total space has dimension $3+2=5$.



              All this works just as well over a smooth manifold. Taking a coordinate
              patch one can define jets as Taylor expansions in the local coordinates there.
              Then one checks that these behave well when we transition between coordinate
              patches. It's the same sort of argument that's used to show that the tangent
              bundle of a manifold makes sense.






              share|cite|improve this answer









              $endgroup$
















                2












                2








                2





                $begingroup$

                At each point in the plane, a $1$-jet is determined by three numbers, since
                a Taylor series at point $(x_0,y_0)$ up to the linear terms has the form $a+b(x-x_0)+c(y-y_0)$. As a point then, the $1$-jets form a three-dimensional vector space. Then the $1$-jets in the plane are a rank three vector bundle over the
                plane, so the total space has dimension $3+2=5$.



                All this works just as well over a smooth manifold. Taking a coordinate
                patch one can define jets as Taylor expansions in the local coordinates there.
                Then one checks that these behave well when we transition between coordinate
                patches. It's the same sort of argument that's used to show that the tangent
                bundle of a manifold makes sense.






                share|cite|improve this answer









                $endgroup$



                At each point in the plane, a $1$-jet is determined by three numbers, since
                a Taylor series at point $(x_0,y_0)$ up to the linear terms has the form $a+b(x-x_0)+c(y-y_0)$. As a point then, the $1$-jets form a three-dimensional vector space. Then the $1$-jets in the plane are a rank three vector bundle over the
                plane, so the total space has dimension $3+2=5$.



                All this works just as well over a smooth manifold. Taking a coordinate
                patch one can define jets as Taylor expansions in the local coordinates there.
                Then one checks that these behave well when we transition between coordinate
                patches. It's the same sort of argument that's used to show that the tangent
                bundle of a manifold makes sense.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Jan 25 at 3:58









                Lord Shark the UnknownLord Shark the Unknown

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