What is the theory of non-linear forms (as contrasted to the theory of differential forms)?
$begingroup$
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
asked Aug 22 '10 at 20:51
Vladimir SotirovVladimir Sotirov
8,69111949
$endgroup$
add a comment |
$begingroup$
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
asked Aug 22 '10 at 20:51
Vladimir SotirovVladimir Sotirov
8,69111949
$endgroup$
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
add a comment |
$begingroup$
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
asked Aug 22 '10 at 20:51
Vladimir SotirovVladimir Sotirov
8,69111949
$endgroup$
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
asked Aug 22 '10 at 20:51
Vladimir SotirovVladimir Sotirov
8,69111949
asked Aug 22 '10 at 20:51
Vladimir SotirovVladimir Sotirov
8,69111949
asked Aug 22 '10 at 20:51
Vladimir SotirovVladimir Sotirov
8,69111949
asked Aug 22 '10 at 20:51
Vladimir SotirovVladimir Sotirov
8,69111949
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
add a comment |
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
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
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.
answered Aug 22 '10 at 22:45
Jack LeeJack Lee
27.5k54868
$endgroup$
add a comment |
$begingroup$
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.
answered Jan 4 at 3:19
goblingoblin
37.1k1159193
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3059%2fwhat-is-the-theory-of-non-linear-forms-as-contrasted-to-the-theory-of-different%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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.
answered Aug 22 '10 at 22:45
Jack LeeJack Lee
27.5k54868
$endgroup$
add a comment |
$begingroup$
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.
answered Aug 22 '10 at 22:45
Jack LeeJack Lee
27.5k54868
$endgroup$
add a comment |
$begingroup$
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.
answered Aug 22 '10 at 22:45
Jack LeeJack Lee
27.5k54868
$endgroup$
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.
answered Aug 22 '10 at 22:45
Jack LeeJack Lee
27.5k54868
answered Aug 22 '10 at 22:45
Jack LeeJack Lee
27.5k54868
answered Aug 22 '10 at 22:45
Jack LeeJack Lee
27.5k54868
answered Aug 22 '10 at 22:45
Jack LeeJack Lee
27.5k54868
27.5k54868
add a comment |
add a comment |
$begingroup$
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.
answered Jan 4 at 3:19
goblingoblin
37.1k1159193
$endgroup$
add a comment |
$begingroup$
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.
answered Jan 4 at 3:19
goblingoblin
37.1k1159193
$endgroup$
add a comment |
$begingroup$
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.
answered Jan 4 at 3:19
goblingoblin
37.1k1159193
$endgroup$
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.
answered Jan 4 at 3:19
goblingoblin
37.1k1159193
edited Jan 5 at 12:16
answered Jan 4 at 3:19
goblingoblin
37.1k1159193
answered Jan 4 at 3:19
goblingoblin
37.1k1159193
answered Jan 4 at 3:19
goblingoblin
37.1k1159193
37.1k1159193
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3059%2fwhat-is-the-theory-of-non-linear-forms-as-contrasted-to-the-theory-of-different%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
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