Mixing
Relationship between Stokes's theorem and the Gauss-Bonnet theorem
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Stokes's theorem and the Gauss-Bonnet theorem are clearly very spiritually similar: they both relate the integral of a quantity $A$ over a region to the integral of some quantity $B$ over the boundary of the region, where $A$ can in some sense be thought of as a "curvature at one higher derivative" of $B$ or a closely related quantity. Is either of these theorems a special case of the other? If not, is there a more general theorem of which they are both special cases (which isn't too many levels higher up in abstraction)?
Edit: the answers to this follow-up question provide derivations of the Gauss-Bonnet theorem from Stokes's theorem in this paper, on pg. 105 of this textbook, and in Chapter 6 Section 1 of this textbook. Unfortunately, the derivations are too advanced for me to understand, as I haven't formally studied graduate-level differential geometry. I would appreciate any answer that summarizes the basic idea of the derivation.
differential-geometry algebraic-topology stokes-theorem
asked Sep 4 '16 at 7:09
tparkertparker
1,941834
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add a comment |
$begingroup$
Stokes's theorem and the Gauss-Bonnet theorem are clearly very spiritually similar: they both relate the integral of a quantity $A$ over a region to the integral of some quantity $B$ over the boundary of the region, where $A$ can in some sense be thought of as a "curvature at one higher derivative" of $B$ or a closely related quantity. Is either of these theorems a special case of the other? If not, is there a more general theorem of which they are both special cases (which isn't too many levels higher up in abstraction)?
Edit: the answers to this follow-up question provide derivations of the Gauss-Bonnet theorem from Stokes's theorem in this paper, on pg. 105 of this textbook, and in Chapter 6 Section 1 of this textbook. Unfortunately, the derivations are too advanced for me to understand, as I haven't formally studied graduate-level differential geometry. I would appreciate any answer that summarizes the basic idea of the derivation.
differential-geometry algebraic-topology stokes-theorem
asked Sep 4 '16 at 7:09
tparkertparker
1,941834
$endgroup$
$begingroup$
this is a good question!
$endgroup$
– GiantTortoise1729
Sep 4 '16 at 7:16
7
$begingroup$
Gauss Bonnet can be proved using Stokes theorem, but I cannot recall a way to prove Stokes using Gauss Bonnet. Indeed they are quite different: Stokes theorem consider only object in the smooth category while in Gauss Bonnet you consider object related to metric, curvatures...
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– user99914
Sep 4 '16 at 8:07
$begingroup$
@JohnMa Thank you, that is a very insightful distinction. Could you very briefly outline the proof of the GB theorem from Stokes' theorem (like in one or two sentences with no equations)?
$endgroup$
– tparker
Sep 4 '16 at 17:08
$begingroup$
Good question, I'd like to know the answer. This might be related: mathoverflow.net/questions/50051/…
$endgroup$
– David Herrero Martí
Oct 30 '16 at 1:08
add a comment |
$begingroup$
Stokes's theorem and the Gauss-Bonnet theorem are clearly very spiritually similar: they both relate the integral of a quantity $A$ over a region to the integral of some quantity $B$ over the boundary of the region, where $A$ can in some sense be thought of as a "curvature at one higher derivative" of $B$ or a closely related quantity. Is either of these theorems a special case of the other? If not, is there a more general theorem of which they are both special cases (which isn't too many levels higher up in abstraction)?
Edit: the answers to this follow-up question provide derivations of the Gauss-Bonnet theorem from Stokes's theorem in this paper, on pg. 105 of this textbook, and in Chapter 6 Section 1 of this textbook. Unfortunately, the derivations are too advanced for me to understand, as I haven't formally studied graduate-level differential geometry. I would appreciate any answer that summarizes the basic idea of the derivation.
differential-geometry algebraic-topology stokes-theorem
asked Sep 4 '16 at 7:09
tparkertparker
1,941834
$endgroup$
Stokes's theorem and the Gauss-Bonnet theorem are clearly very spiritually similar: they both relate the integral of a quantity $A$ over a region to the integral of some quantity $B$ over the boundary of the region, where $A$ can in some sense be thought of as a "curvature at one higher derivative" of $B$ or a closely related quantity. Is either of these theorems a special case of the other? If not, is there a more general theorem of which they are both special cases (which isn't too many levels higher up in abstraction)?
Edit: the answers to this follow-up question provide derivations of the Gauss-Bonnet theorem from Stokes's theorem in this paper, on pg. 105 of this textbook, and in Chapter 6 Section 1 of this textbook. Unfortunately, the derivations are too advanced for me to understand, as I haven't formally studied graduate-level differential geometry. I would appreciate any answer that summarizes the basic idea of the derivation.
differential-geometry algebraic-topology stokes-theorem
differential-geometry algebraic-topology stokes-theorem
asked Sep 4 '16 at 7:09
tparkertparker
1,941834
asked Sep 4 '16 at 7:09
tparkertparker
1,941834
edited Feb 2 at 15:17
tparker
asked Sep 4 '16 at 7:09
tparkertparker
1,941834
asked Sep 4 '16 at 7:09
tparkertparker
1,941834
asked Sep 4 '16 at 7:09
tparkertparker
1,941834
1,941834
$begingroup$
this is a good question!
$endgroup$
– GiantTortoise1729
Sep 4 '16 at 7:16
7
$begingroup$
Gauss Bonnet can be proved using Stokes theorem, but I cannot recall a way to prove Stokes using Gauss Bonnet. Indeed they are quite different: Stokes theorem consider only object in the smooth category while in Gauss Bonnet you consider object related to metric, curvatures...
$endgroup$
– user99914
Sep 4 '16 at 8:07
$begingroup$
@JohnMa Thank you, that is a very insightful distinction. Could you very briefly outline the proof of the GB theorem from Stokes' theorem (like in one or two sentences with no equations)?
$endgroup$
– tparker
Sep 4 '16 at 17:08
$begingroup$
Good question, I'd like to know the answer. This might be related: mathoverflow.net/questions/50051/…
$endgroup$
– David Herrero Martí
Oct 30 '16 at 1:08
add a comment |
$begingroup$
this is a good question!
$endgroup$
– GiantTortoise1729
Sep 4 '16 at 7:16
7
$begingroup$
Gauss Bonnet can be proved using Stokes theorem, but I cannot recall a way to prove Stokes using Gauss Bonnet. Indeed they are quite different: Stokes theorem consider only object in the smooth category while in Gauss Bonnet you consider object related to metric, curvatures...
$endgroup$
– user99914
Sep 4 '16 at 8:07
$begingroup$
@JohnMa Thank you, that is a very insightful distinction. Could you very briefly outline the proof of the GB theorem from Stokes' theorem (like in one or two sentences with no equations)?
$endgroup$
– tparker
Sep 4 '16 at 17:08
$begingroup$
Good question, I'd like to know the answer. This might be related: mathoverflow.net/questions/50051/…
$endgroup$
– David Herrero Martí
Oct 30 '16 at 1:08
$begingroup$
this is a good question!
$endgroup$
– GiantTortoise1729
Sep 4 '16 at 7:16
$begingroup$
this is a good question!
$endgroup$
– GiantTortoise1729
Sep 4 '16 at 7:16
7
7
$begingroup$
Gauss Bonnet can be proved using Stokes theorem, but I cannot recall a way to prove Stokes using Gauss Bonnet. Indeed they are quite different: Stokes theorem consider only object in the smooth category while in Gauss Bonnet you consider object related to metric, curvatures...
$endgroup$
– user99914
Sep 4 '16 at 8:07
$begingroup$
Gauss Bonnet can be proved using Stokes theorem, but I cannot recall a way to prove Stokes using Gauss Bonnet. Indeed they are quite different: Stokes theorem consider only object in the smooth category while in Gauss Bonnet you consider object related to metric, curvatures...
$endgroup$
– user99914
Sep 4 '16 at 8:07
$begingroup$
@JohnMa Thank you, that is a very insightful distinction. Could you very briefly outline the proof of the GB theorem from Stokes' theorem (like in one or two sentences with no equations)?
$endgroup$
– tparker
Sep 4 '16 at 17:08
$begingroup$
@JohnMa Thank you, that is a very insightful distinction. Could you very briefly outline the proof of the GB theorem from Stokes' theorem (like in one or two sentences with no equations)?
$endgroup$
– tparker
Sep 4 '16 at 17:08
$begingroup$
Good question, I'd like to know the answer. This might be related: mathoverflow.net/questions/50051/…
$endgroup$
– David Herrero Martí
Oct 30 '16 at 1:08
$begingroup$
Good question, I'd like to know the answer. This might be related: mathoverflow.net/questions/50051/…
$endgroup$
– David Herrero Martí
Oct 30 '16 at 1:08
add a comment |
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$begingroup$
this is a good question!
$endgroup$
– GiantTortoise1729
Sep 4 '16 at 7:16
7
$begingroup$
Gauss Bonnet can be proved using Stokes theorem, but I cannot recall a way to prove Stokes using Gauss Bonnet. Indeed they are quite different: Stokes theorem consider only object in the smooth category while in Gauss Bonnet you consider object related to metric, curvatures...
$endgroup$
– user99914
Sep 4 '16 at 8:07
$begingroup$
@JohnMa Thank you, that is a very insightful distinction. Could you very briefly outline the proof of the GB theorem from Stokes' theorem (like in one or two sentences with no equations)?
$endgroup$
– tparker
Sep 4 '16 at 17:08
$begingroup$
Good question, I'd like to know the answer. This might be related: mathoverflow.net/questions/50051/…
$endgroup$
– David Herrero Martí
Oct 30 '16 at 1:08