Reference request: proof of Stokes theorem using chains and boundaries
I am pretty inexperienced in differential geometry and wanted to go through the proof of Stokes theorem. The usual way I see it formulated is for an oriented manifold with boundary $M$. There was a particular way I saw it presented before, and was wondering if there was any reference which did things in this way.
I know what I am going to say is just a translation of the usual formulation of Stokes theorem, so if there is no such reference, I will just read a more standard treatment and translate the proof into a way I like.
So $M$ is an $n$-dimensional smooth manifold, and $Omega^k(M)$ is the space of differential $k$-forms on $M$. We have a cochain complex
$$cdots rightarrow Omega^0(M) rightarrow Omega^1(M) rightarrow Omega^2(M) rightarrow cdots $$
where $d: Omega^i(M) rightarrow Omega^{i+1}(M)$ is the exterior derivative. For each $k$, we also consider all smooth functions $[0,1]^k rightarrow M$ (meaning smooth on the interior $(0,1)^k$, with some nice boundary conditions), and let $C_k$ be the free vector space with all those smooth functions as a basis. There are natural (?) boundary maps $partial: C_i rightarrow C_{i-1}$ which give us a chain complex
$$cdots rightarrow C_2 rightarrow C_1 rightarrow C_0 rightarrow cdots$$
If $f: [0,1]^k rightarrow M$ is a basis element of $C_k$, and $omega in Omega^k(M)$, then the pullback $f^{ast} omega$ is a differential $k$-form on the manifold $(0,1)^k$, and one defines a pairing $C_k times Omega^k(M) rightarrow mathbb R$ by
$$langle f, omega rangle = intlimits_f omega = intlimits_{(0,1)^k} f^{ast} omega$$
where the right hand side is the usual integration of a top form in $mathbb R^k$. One extends $langle f, omega rangle$ by linearity to all $f in C_k$.
The version of Stokes theorem I heard is the following:
Theorem: Let $omega in Omega^{k-1}$, and $f in C_k$. Then
$$langle f, d omega rangle = langle partial f, omega rangle$$
or in other words
$$intlimits_f d omega = intlimits_{partial f} omega$$
Is there a nice reference which explains Stokes theorem in roughly the way I'm describing? If not, how is what I am saying related to the usual formulation? For one thing, I don't understand why Stokes theorem is stated for an oriented manifold. The formulation I have described says nothing about an orientation on $M$.
differential-geometry reference-request
asked Nov 18 at 22:35
D_S
13.3k51551
add a comment |
I am pretty inexperienced in differential geometry and wanted to go through the proof of Stokes theorem. The usual way I see it formulated is for an oriented manifold with boundary $M$. There was a particular way I saw it presented before, and was wondering if there was any reference which did things in this way.
I know what I am going to say is just a translation of the usual formulation of Stokes theorem, so if there is no such reference, I will just read a more standard treatment and translate the proof into a way I like.
So $M$ is an $n$-dimensional smooth manifold, and $Omega^k(M)$ is the space of differential $k$-forms on $M$. We have a cochain complex
$$cdots rightarrow Omega^0(M) rightarrow Omega^1(M) rightarrow Omega^2(M) rightarrow cdots $$
where $d: Omega^i(M) rightarrow Omega^{i+1}(M)$ is the exterior derivative. For each $k$, we also consider all smooth functions $[0,1]^k rightarrow M$ (meaning smooth on the interior $(0,1)^k$, with some nice boundary conditions), and let $C_k$ be the free vector space with all those smooth functions as a basis. There are natural (?) boundary maps $partial: C_i rightarrow C_{i-1}$ which give us a chain complex
$$cdots rightarrow C_2 rightarrow C_1 rightarrow C_0 rightarrow cdots$$
If $f: [0,1]^k rightarrow M$ is a basis element of $C_k$, and $omega in Omega^k(M)$, then the pullback $f^{ast} omega$ is a differential $k$-form on the manifold $(0,1)^k$, and one defines a pairing $C_k times Omega^k(M) rightarrow mathbb R$ by
$$langle f, omega rangle = intlimits_f omega = intlimits_{(0,1)^k} f^{ast} omega$$
where the right hand side is the usual integration of a top form in $mathbb R^k$. One extends $langle f, omega rangle$ by linearity to all $f in C_k$.
The version of Stokes theorem I heard is the following:
Theorem: Let $omega in Omega^{k-1}$, and $f in C_k$. Then
$$langle f, d omega rangle = langle partial f, omega rangle$$
or in other words
$$intlimits_f d omega = intlimits_{partial f} omega$$
Is there a nice reference which explains Stokes theorem in roughly the way I'm describing? If not, how is what I am saying related to the usual formulation? For one thing, I don't understand why Stokes theorem is stated for an oriented manifold. The formulation I have described says nothing about an orientation on $M$.
differential-geometry reference-request
asked Nov 18 at 22:35
D_S
13.3k51551
In H. Heuser Analysis II there is a proof involving chains. But this is a very special case. There is a reason, why manifolds are usually in the statement.
– Martin Erhardt
Nov 18 at 22:47
There is no simplification in the proof gained by working with simplices as opposed to general manifolds with corners. The proof is always to use partitions of unity to reduce to Stokes' theorem for compactly supported $n$-forms in $[0,infty)^k times Bbb R^{n-k}$. Now, simplices are canonically oriented, which is where the orientation is secretly hiding here. But it is true that $Omega^*(M)$ pairs against $C_*(M)$ for any manifold $M$, not necessarily oriented.
– Mike Miller
Nov 18 at 23:21
add a comment |
I am pretty inexperienced in differential geometry and wanted to go through the proof of Stokes theorem. The usual way I see it formulated is for an oriented manifold with boundary $M$. There was a particular way I saw it presented before, and was wondering if there was any reference which did things in this way.
I know what I am going to say is just a translation of the usual formulation of Stokes theorem, so if there is no such reference, I will just read a more standard treatment and translate the proof into a way I like.
So $M$ is an $n$-dimensional smooth manifold, and $Omega^k(M)$ is the space of differential $k$-forms on $M$. We have a cochain complex
$$cdots rightarrow Omega^0(M) rightarrow Omega^1(M) rightarrow Omega^2(M) rightarrow cdots $$
where $d: Omega^i(M) rightarrow Omega^{i+1}(M)$ is the exterior derivative. For each $k$, we also consider all smooth functions $[0,1]^k rightarrow M$ (meaning smooth on the interior $(0,1)^k$, with some nice boundary conditions), and let $C_k$ be the free vector space with all those smooth functions as a basis. There are natural (?) boundary maps $partial: C_i rightarrow C_{i-1}$ which give us a chain complex
$$cdots rightarrow C_2 rightarrow C_1 rightarrow C_0 rightarrow cdots$$
If $f: [0,1]^k rightarrow M$ is a basis element of $C_k$, and $omega in Omega^k(M)$, then the pullback $f^{ast} omega$ is a differential $k$-form on the manifold $(0,1)^k$, and one defines a pairing $C_k times Omega^k(M) rightarrow mathbb R$ by
$$langle f, omega rangle = intlimits_f omega = intlimits_{(0,1)^k} f^{ast} omega$$
where the right hand side is the usual integration of a top form in $mathbb R^k$. One extends $langle f, omega rangle$ by linearity to all $f in C_k$.
The version of Stokes theorem I heard is the following:
Theorem: Let $omega in Omega^{k-1}$, and $f in C_k$. Then
$$langle f, d omega rangle = langle partial f, omega rangle$$
or in other words
$$intlimits_f d omega = intlimits_{partial f} omega$$
Is there a nice reference which explains Stokes theorem in roughly the way I'm describing? If not, how is what I am saying related to the usual formulation? For one thing, I don't understand why Stokes theorem is stated for an oriented manifold. The formulation I have described says nothing about an orientation on $M$.
differential-geometry reference-request
asked Nov 18 at 22:35
D_S
13.3k51551
I am pretty inexperienced in differential geometry and wanted to go through the proof of Stokes theorem. The usual way I see it formulated is for an oriented manifold with boundary $M$. There was a particular way I saw it presented before, and was wondering if there was any reference which did things in this way.
I know what I am going to say is just a translation of the usual formulation of Stokes theorem, so if there is no such reference, I will just read a more standard treatment and translate the proof into a way I like.
So $M$ is an $n$-dimensional smooth manifold, and $Omega^k(M)$ is the space of differential $k$-forms on $M$. We have a cochain complex
$$cdots rightarrow Omega^0(M) rightarrow Omega^1(M) rightarrow Omega^2(M) rightarrow cdots $$
where $d: Omega^i(M) rightarrow Omega^{i+1}(M)$ is the exterior derivative. For each $k$, we also consider all smooth functions $[0,1]^k rightarrow M$ (meaning smooth on the interior $(0,1)^k$, with some nice boundary conditions), and let $C_k$ be the free vector space with all those smooth functions as a basis. There are natural (?) boundary maps $partial: C_i rightarrow C_{i-1}$ which give us a chain complex
$$cdots rightarrow C_2 rightarrow C_1 rightarrow C_0 rightarrow cdots$$
If $f: [0,1]^k rightarrow M$ is a basis element of $C_k$, and $omega in Omega^k(M)$, then the pullback $f^{ast} omega$ is a differential $k$-form on the manifold $(0,1)^k$, and one defines a pairing $C_k times Omega^k(M) rightarrow mathbb R$ by
$$langle f, omega rangle = intlimits_f omega = intlimits_{(0,1)^k} f^{ast} omega$$
where the right hand side is the usual integration of a top form in $mathbb R^k$. One extends $langle f, omega rangle$ by linearity to all $f in C_k$.
The version of Stokes theorem I heard is the following:
Theorem: Let $omega in Omega^{k-1}$, and $f in C_k$. Then
$$langle f, d omega rangle = langle partial f, omega rangle$$
or in other words
$$intlimits_f d omega = intlimits_{partial f} omega$$
Is there a nice reference which explains Stokes theorem in roughly the way I'm describing? If not, how is what I am saying related to the usual formulation? For one thing, I don't understand why Stokes theorem is stated for an oriented manifold. The formulation I have described says nothing about an orientation on $M$.
differential-geometry reference-request
differential-geometry reference-request
asked Nov 18 at 22:35
D_S
13.3k51551
asked Nov 18 at 22:35
D_S
13.3k51551
edited Nov 18 at 22:41
asked Nov 18 at 22:35
D_S
13.3k51551
asked Nov 18 at 22:35
D_S
13.3k51551
asked Nov 18 at 22:35
D_S
13.3k51551
13.3k51551
In H. Heuser Analysis II there is a proof involving chains. But this is a very special case. There is a reason, why manifolds are usually in the statement.
– Martin Erhardt
Nov 18 at 22:47
There is no simplification in the proof gained by working with simplices as opposed to general manifolds with corners. The proof is always to use partitions of unity to reduce to Stokes' theorem for compactly supported $n$-forms in $[0,infty)^k times Bbb R^{n-k}$. Now, simplices are canonically oriented, which is where the orientation is secretly hiding here. But it is true that $Omega^*(M)$ pairs against $C_*(M)$ for any manifold $M$, not necessarily oriented.
– Mike Miller
Nov 18 at 23:21
add a comment |
In H. Heuser Analysis II there is a proof involving chains. But this is a very special case. There is a reason, why manifolds are usually in the statement.
– Martin Erhardt
Nov 18 at 22:47
There is no simplification in the proof gained by working with simplices as opposed to general manifolds with corners. The proof is always to use partitions of unity to reduce to Stokes' theorem for compactly supported $n$-forms in $[0,infty)^k times Bbb R^{n-k}$. Now, simplices are canonically oriented, which is where the orientation is secretly hiding here. But it is true that $Omega^*(M)$ pairs against $C_*(M)$ for any manifold $M$, not necessarily oriented.
– Mike Miller
Nov 18 at 23:21
In H. Heuser Analysis II there is a proof involving chains. But this is a very special case. There is a reason, why manifolds are usually in the statement.
– Martin Erhardt
Nov 18 at 22:47
In H. Heuser Analysis II there is a proof involving chains. But this is a very special case. There is a reason, why manifolds are usually in the statement.
– Martin Erhardt
Nov 18 at 22:47
There is no simplification in the proof gained by working with simplices as opposed to general manifolds with corners. The proof is always to use partitions of unity to reduce to Stokes' theorem for compactly supported $n$-forms in $[0,infty)^k times Bbb R^{n-k}$. Now, simplices are canonically oriented, which is where the orientation is secretly hiding here. But it is true that $Omega^*(M)$ pairs against $C_*(M)$ for any manifold $M$, not necessarily oriented.
– Mike Miller
Nov 18 at 23:21
There is no simplification in the proof gained by working with simplices as opposed to general manifolds with corners. The proof is always to use partitions of unity to reduce to Stokes' theorem for compactly supported $n$-forms in $[0,infty)^k times Bbb R^{n-k}$. Now, simplices are canonically oriented, which is where the orientation is secretly hiding here. But it is true that $Omega^*(M)$ pairs against $C_*(M)$ for any manifold $M$, not necessarily oriented.
– Mike Miller
Nov 18 at 23:21
add a comment |
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In H. Heuser Analysis II there is a proof involving chains. But this is a very special case. There is a reason, why manifolds are usually in the statement.
– Martin Erhardt
Nov 18 at 22:47
There is no simplification in the proof gained by working with simplices as opposed to general manifolds with corners. The proof is always to use partitions of unity to reduce to Stokes' theorem for compactly supported $n$-forms in $[0,infty)^k times Bbb R^{n-k}$. Now, simplices are canonically oriented, which is where the orientation is secretly hiding here. But it is true that $Omega^*(M)$ pairs against $C_*(M)$ for any manifold $M$, not necessarily oriented.
– Mike Miller
Nov 18 at 23:21