What is the theory of non-linear forms (as contrasted to the theory of differential forms)?












17












$begingroup$


It is often said that differential forms (sections of an exterior power of the cotangent bundle) are the things that you can integrate. But unless I'm being thoroughly dense differential forms are not the only things that you can integrate, c.f. the arclength form (on a 2d manifold) $ds=sqrt{dx^2+dy^2}$, the unsigned 1-d forms $|f(x,y)dx+g(x,y)dy|$, or the unsigned area forms $|h(x,y)dxwedge dy|$.



My question is:




Where do the arclength form $ds=sqrt{dx^2+dy^2}$, the unsigned 1-d forms |f(x,y)dx+g(x,y)dy|, and the unsigned area forms $|h(x,y)dxwedge dy|$ live relative to the differentials $dx$ and $dy$, which I understand to live in the cotangent bundle of some 2-dimensional manifold?







differential-geometry







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    That's funny; I thought measurable functions were the things you can integrate...
    $endgroup$
    – Qiaochu Yuan
    Aug 22 '10 at 21:36






  • 5




    $begingroup$
    @Qiaochu: evidently, there's more than one kind of thing you can integrate.
    $endgroup$
    – Pete L. Clark
    Aug 22 '10 at 22:11






  • 1




    $begingroup$
    The notation used in the right hand side of «$ds=sqrt{dx^2+dy^2}$» is just a notation; in particular, it is not something that is built out of $dx$ and $dy$...
    $endgroup$
    – Mariano Suárez-Álvarez
    Aug 23 '10 at 2:00










  • $begingroup$
    @Mariano, I understand ds as a continuous function on the tangent space at a point. My limited understanding tells me that it is a non-linear form because c ds(v)=ds(c v) for positive constants c. I suspect that if you apply 'positively' homogeneous function of degree 1 in n variables to (dx_1, dx_2,..., dx_n), you would get a form.
    $endgroup$
    – Vladimir Sotirov
    Aug 24 '10 at 18:22










  • $begingroup$
    I guess these non-linear forms are taken from David Bachmann's book "A Geometric Approach to Differential Forms", aren't they?
    $endgroup$
    – shuhalo
    Feb 22 '12 at 9:57
















17












$begingroup$


It is often said that differential forms (sections of an exterior power of the cotangent bundle) are the things that you can integrate. But unless I'm being thoroughly dense differential forms are not the only things that you can integrate, c.f. the arclength form (on a 2d manifold) $ds=sqrt{dx^2+dy^2}$, the unsigned 1-d forms $|f(x,y)dx+g(x,y)dy|$, or the unsigned area forms $|h(x,y)dxwedge dy|$.



My question is:




Where do the arclength form $ds=sqrt{dx^2+dy^2}$, the unsigned 1-d forms |f(x,y)dx+g(x,y)dy|, and the unsigned area forms $|h(x,y)dxwedge dy|$ live relative to the differentials $dx$ and $dy$, which I understand to live in the cotangent bundle of some 2-dimensional manifold?







differential-geometry







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    That's funny; I thought measurable functions were the things you can integrate...
    $endgroup$
    – Qiaochu Yuan
    Aug 22 '10 at 21:36






  • 5




    $begingroup$
    @Qiaochu: evidently, there's more than one kind of thing you can integrate.
    $endgroup$
    – Pete L. Clark
    Aug 22 '10 at 22:11






  • 1




    $begingroup$
    The notation used in the right hand side of «$ds=sqrt{dx^2+dy^2}$» is just a notation; in particular, it is not something that is built out of $dx$ and $dy$...
    $endgroup$
    – Mariano Suárez-Álvarez
    Aug 23 '10 at 2:00










  • $begingroup$
    @Mariano, I understand ds as a continuous function on the tangent space at a point. My limited understanding tells me that it is a non-linear form because c ds(v)=ds(c v) for positive constants c. I suspect that if you apply 'positively' homogeneous function of degree 1 in n variables to (dx_1, dx_2,..., dx_n), you would get a form.
    $endgroup$
    – Vladimir Sotirov
    Aug 24 '10 at 18:22










  • $begingroup$
    I guess these non-linear forms are taken from David Bachmann's book "A Geometric Approach to Differential Forms", aren't they?
    $endgroup$
    – shuhalo
    Feb 22 '12 at 9:57














17












17








17


8



$begingroup$


It is often said that differential forms (sections of an exterior power of the cotangent bundle) are the things that you can integrate. But unless I'm being thoroughly dense differential forms are not the only things that you can integrate, c.f. the arclength form (on a 2d manifold) $ds=sqrt{dx^2+dy^2}$, the unsigned 1-d forms $|f(x,y)dx+g(x,y)dy|$, or the unsigned area forms $|h(x,y)dxwedge dy|$.



My question is:




Where do the arclength form $ds=sqrt{dx^2+dy^2}$, the unsigned 1-d forms |f(x,y)dx+g(x,y)dy|, and the unsigned area forms $|h(x,y)dxwedge dy|$ live relative to the differentials $dx$ and $dy$, which I understand to live in the cotangent bundle of some 2-dimensional manifold?







differential-geometry







share|cite|improve this question









$endgroup$




It is often said that differential forms (sections of an exterior power of the cotangent bundle) are the things that you can integrate. But unless I'm being thoroughly dense differential forms are not the only things that you can integrate, c.f. the arclength form (on a 2d manifold) $ds=sqrt{dx^2+dy^2}$, the unsigned 1-d forms $|f(x,y)dx+g(x,y)dy|$, or the unsigned area forms $|h(x,y)dxwedge dy|$.



My question is:




Where do the arclength form $ds=sqrt{dx^2+dy^2}$, the unsigned 1-d forms |f(x,y)dx+g(x,y)dy|, and the unsigned area forms $|h(x,y)dxwedge dy|$ live relative to the differentials $dx$ and $dy$, which I understand to live in the cotangent bundle of some 2-dimensional manifold?







differential-geometry




differential-geometry






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Aug 22 '10 at 20:51









Vladimir SotirovVladimir Sotirov

8,69111949




8,69111949








  • 1




    $begingroup$
    That's funny; I thought measurable functions were the things you can integrate...
    $endgroup$
    – Qiaochu Yuan
    Aug 22 '10 at 21:36






  • 5




    $begingroup$
    @Qiaochu: evidently, there's more than one kind of thing you can integrate.
    $endgroup$
    – Pete L. Clark
    Aug 22 '10 at 22:11






  • 1




    $begingroup$
    The notation used in the right hand side of «$ds=sqrt{dx^2+dy^2}$» is just a notation; in particular, it is not something that is built out of $dx$ and $dy$...
    $endgroup$
    – Mariano Suárez-Álvarez
    Aug 23 '10 at 2:00










  • $begingroup$
    @Mariano, I understand ds as a continuous function on the tangent space at a point. My limited understanding tells me that it is a non-linear form because c ds(v)=ds(c v) for positive constants c. I suspect that if you apply 'positively' homogeneous function of degree 1 in n variables to (dx_1, dx_2,..., dx_n), you would get a form.
    $endgroup$
    – Vladimir Sotirov
    Aug 24 '10 at 18:22










  • $begingroup$
    I guess these non-linear forms are taken from David Bachmann's book "A Geometric Approach to Differential Forms", aren't they?
    $endgroup$
    – shuhalo
    Feb 22 '12 at 9:57














  • 1




    $begingroup$
    That's funny; I thought measurable functions were the things you can integrate...
    $endgroup$
    – Qiaochu Yuan
    Aug 22 '10 at 21:36






  • 5




    $begingroup$
    @Qiaochu: evidently, there's more than one kind of thing you can integrate.
    $endgroup$
    – Pete L. Clark
    Aug 22 '10 at 22:11






  • 1




    $begingroup$
    The notation used in the right hand side of «$ds=sqrt{dx^2+dy^2}$» is just a notation; in particular, it is not something that is built out of $dx$ and $dy$...
    $endgroup$
    – Mariano Suárez-Álvarez
    Aug 23 '10 at 2:00










  • $begingroup$
    @Mariano, I understand ds as a continuous function on the tangent space at a point. My limited understanding tells me that it is a non-linear form because c ds(v)=ds(c v) for positive constants c. I suspect that if you apply 'positively' homogeneous function of degree 1 in n variables to (dx_1, dx_2,..., dx_n), you would get a form.
    $endgroup$
    – Vladimir Sotirov
    Aug 24 '10 at 18:22










  • $begingroup$
    I guess these non-linear forms are taken from David Bachmann's book "A Geometric Approach to Differential Forms", aren't they?
    $endgroup$
    – shuhalo
    Feb 22 '12 at 9:57








1




1




$begingroup$
That's funny; I thought measurable functions were the things you can integrate...
$endgroup$
– Qiaochu Yuan
Aug 22 '10 at 21:36




$begingroup$
That's funny; I thought measurable functions were the things you can integrate...
$endgroup$
– Qiaochu Yuan
Aug 22 '10 at 21:36




5




5




$begingroup$
@Qiaochu: evidently, there's more than one kind of thing you can integrate.
$endgroup$
– Pete L. Clark
Aug 22 '10 at 22:11




$begingroup$
@Qiaochu: evidently, there's more than one kind of thing you can integrate.
$endgroup$
– Pete L. Clark
Aug 22 '10 at 22:11




1




1




$begingroup$
The notation used in the right hand side of «$ds=sqrt{dx^2+dy^2}$» is just a notation; in particular, it is not something that is built out of $dx$ and $dy$...
$endgroup$
– Mariano Suárez-Álvarez
Aug 23 '10 at 2:00




$begingroup$
The notation used in the right hand side of «$ds=sqrt{dx^2+dy^2}$» is just a notation; in particular, it is not something that is built out of $dx$ and $dy$...
$endgroup$
– Mariano Suárez-Álvarez
Aug 23 '10 at 2:00












$begingroup$
@Mariano, I understand ds as a continuous function on the tangent space at a point. My limited understanding tells me that it is a non-linear form because c ds(v)=ds(c v) for positive constants c. I suspect that if you apply 'positively' homogeneous function of degree 1 in n variables to (dx_1, dx_2,..., dx_n), you would get a form.
$endgroup$
– Vladimir Sotirov
Aug 24 '10 at 18:22




$begingroup$
@Mariano, I understand ds as a continuous function on the tangent space at a point. My limited understanding tells me that it is a non-linear form because c ds(v)=ds(c v) for positive constants c. I suspect that if you apply 'positively' homogeneous function of degree 1 in n variables to (dx_1, dx_2,..., dx_n), you would get a form.
$endgroup$
– Vladimir Sotirov
Aug 24 '10 at 18:22












$begingroup$
I guess these non-linear forms are taken from David Bachmann's book "A Geometric Approach to Differential Forms", aren't they?
$endgroup$
– shuhalo
Feb 22 '12 at 9:57




$begingroup$
I guess these non-linear forms are taken from David Bachmann's book "A Geometric Approach to Differential Forms", aren't they?
$endgroup$
– shuhalo
Feb 22 '12 at 9:57










2 Answers
2






active

oldest

votes


















26












$begingroup$

The answer to "what kinds of things can you integrate" depends on the context.





  • Measurable functions are things you can integrate over measure spaces, which includes in particular measurable subsets of R^n.


  • Differential forms are things you can integrate over oriented smooth manifolds -- the key thing about them is that their integrals are invariant under smooth, orientation-preserving changes of coordinates.


  • Densities are things that can be integrated in a coordinate-independent way on any smooth manifold, regardless of whether it has an orientation or not.

  • Coming full circle, every Riemannian manifold (i.e., smooth manifold endowed with a Riemannian metric) has a naturally-defined density dV, so in that context you can integrate measurable functions again: the integral of the function f is defined to be the integral of the density f dV.


All three of the expressions you asked about are examples of densities. For details, see my book Introduction to Smooth Manifolds, pp. 375-382.






share|cite|improve this answer









$endgroup$





















    0












    $begingroup$

    In my opinion, you're looking for the notion of a cogerm.



    If I understand correctly, the fact that such things act on paths (and not just vectors) allows for "higher order" forms like $d^2 x$, and the fact that such things aren't assumed linear allows for "non-linear" forms like $ds := sqrt{dx^2+dy^2}$. And yes, there is indeed a notion of integration for such forms; see the link.






    share|cite|improve this answer











    $endgroup$













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      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      26












      $begingroup$

      The answer to "what kinds of things can you integrate" depends on the context.





      • Measurable functions are things you can integrate over measure spaces, which includes in particular measurable subsets of R^n.


      • Differential forms are things you can integrate over oriented smooth manifolds -- the key thing about them is that their integrals are invariant under smooth, orientation-preserving changes of coordinates.


      • Densities are things that can be integrated in a coordinate-independent way on any smooth manifold, regardless of whether it has an orientation or not.

      • Coming full circle, every Riemannian manifold (i.e., smooth manifold endowed with a Riemannian metric) has a naturally-defined density dV, so in that context you can integrate measurable functions again: the integral of the function f is defined to be the integral of the density f dV.


      All three of the expressions you asked about are examples of densities. For details, see my book Introduction to Smooth Manifolds, pp. 375-382.






      share|cite|improve this answer









      $endgroup$


















        26












        $begingroup$

        The answer to "what kinds of things can you integrate" depends on the context.





        • Measurable functions are things you can integrate over measure spaces, which includes in particular measurable subsets of R^n.


        • Differential forms are things you can integrate over oriented smooth manifolds -- the key thing about them is that their integrals are invariant under smooth, orientation-preserving changes of coordinates.


        • Densities are things that can be integrated in a coordinate-independent way on any smooth manifold, regardless of whether it has an orientation or not.

        • Coming full circle, every Riemannian manifold (i.e., smooth manifold endowed with a Riemannian metric) has a naturally-defined density dV, so in that context you can integrate measurable functions again: the integral of the function f is defined to be the integral of the density f dV.


        All three of the expressions you asked about are examples of densities. For details, see my book Introduction to Smooth Manifolds, pp. 375-382.






        share|cite|improve this answer









        $endgroup$
















          26












          26








          26





          $begingroup$

          The answer to "what kinds of things can you integrate" depends on the context.





          • Measurable functions are things you can integrate over measure spaces, which includes in particular measurable subsets of R^n.


          • Differential forms are things you can integrate over oriented smooth manifolds -- the key thing about them is that their integrals are invariant under smooth, orientation-preserving changes of coordinates.


          • Densities are things that can be integrated in a coordinate-independent way on any smooth manifold, regardless of whether it has an orientation or not.

          • Coming full circle, every Riemannian manifold (i.e., smooth manifold endowed with a Riemannian metric) has a naturally-defined density dV, so in that context you can integrate measurable functions again: the integral of the function f is defined to be the integral of the density f dV.


          All three of the expressions you asked about are examples of densities. For details, see my book Introduction to Smooth Manifolds, pp. 375-382.






          share|cite|improve this answer









          $endgroup$



          The answer to "what kinds of things can you integrate" depends on the context.





          • Measurable functions are things you can integrate over measure spaces, which includes in particular measurable subsets of R^n.


          • Differential forms are things you can integrate over oriented smooth manifolds -- the key thing about them is that their integrals are invariant under smooth, orientation-preserving changes of coordinates.


          • Densities are things that can be integrated in a coordinate-independent way on any smooth manifold, regardless of whether it has an orientation or not.

          • Coming full circle, every Riemannian manifold (i.e., smooth manifold endowed with a Riemannian metric) has a naturally-defined density dV, so in that context you can integrate measurable functions again: the integral of the function f is defined to be the integral of the density f dV.


          All three of the expressions you asked about are examples of densities. For details, see my book Introduction to Smooth Manifolds, pp. 375-382.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Aug 22 '10 at 22:45









          Jack LeeJack Lee

          27.5k54868




          27.5k54868























              0












              $begingroup$

              In my opinion, you're looking for the notion of a cogerm.



              If I understand correctly, the fact that such things act on paths (and not just vectors) allows for "higher order" forms like $d^2 x$, and the fact that such things aren't assumed linear allows for "non-linear" forms like $ds := sqrt{dx^2+dy^2}$. And yes, there is indeed a notion of integration for such forms; see the link.






              share|cite|improve this answer











              $endgroup$


















                0












                $begingroup$

                In my opinion, you're looking for the notion of a cogerm.



                If I understand correctly, the fact that such things act on paths (and not just vectors) allows for "higher order" forms like $d^2 x$, and the fact that such things aren't assumed linear allows for "non-linear" forms like $ds := sqrt{dx^2+dy^2}$. And yes, there is indeed a notion of integration for such forms; see the link.






                share|cite|improve this answer











                $endgroup$
















                  0












                  0








                  0





                  $begingroup$

                  In my opinion, you're looking for the notion of a cogerm.



                  If I understand correctly, the fact that such things act on paths (and not just vectors) allows for "higher order" forms like $d^2 x$, and the fact that such things aren't assumed linear allows for "non-linear" forms like $ds := sqrt{dx^2+dy^2}$. And yes, there is indeed a notion of integration for such forms; see the link.






                  share|cite|improve this answer











                  $endgroup$



                  In my opinion, you're looking for the notion of a cogerm.



                  If I understand correctly, the fact that such things act on paths (and not just vectors) allows for "higher order" forms like $d^2 x$, and the fact that such things aren't assumed linear allows for "non-linear" forms like $ds := sqrt{dx^2+dy^2}$. And yes, there is indeed a notion of integration for such forms; see the link.







                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited Jan 5 at 12:16

























                  answered Jan 4 at 3:19









                  goblingoblin

                  37.1k1159193




                  37.1k1159193






























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