Compute this integral of a form
$begingroup$
I need to compute the integral:
$$
int_{mathbb{S}^3}alpha,
$$
where $alphain Omega^3(mathbb{R}^4)$ is given by $alpha=-4t;dxwedge dywedge dz+4z;dxwedge dywedge dt-4y;dxwedge dzwedge dt+4x;dywedge dzwedge dt$.
My problem is that I am getting two different results. If I apply the definition:
$$
begin{array}{rccl}
varphi^+:& mathbb{S}^3cap{t geq 0}:=mathbb{S}^{3+}&longrightarrow&mathbb{D}^3\
&(x,y,z,t)&longmapsto&(x,y,z)
end{array}
$$
$$
begin{array}{rccl}
varphi^-:& mathbb{S}^{3-}cap{t leq 0}:=mathbb{S}^{3-}&longrightarrow&mathbb{D}^3\
&(x,y,z,t)&longmapsto&(x,y,z)
end{array}
$$
The inverses:
$$
begin{array}{rccl}
(varphi^+)^{-1}:& mathbb{D}^3&longrightarrow&mathbb{S}^{3+}\
&(u,v,w)&longmapsto&(u,v,w,sqrt{1-u^2-v^2-w^2})
end{array}
$$
$$
begin{array}{rccl}
(varphi^-)^{-1}:& mathbb{D}^3&longrightarrow&mathbb{S}^{3-}\
&(u,v,w)&longmapsto&(u,v,w,-sqrt{1-u^2-v^2-w^2})
end{array}
$$
If $beta=t;dxwedge dywedge dz$:
$$
int_{mathbb{S}^3}beta=int_{mathbb{S}^{3-}}beta+int_{mathbb{S}^{3+}}beta=int_{mathbb{D}^3}left[(varphi^-)^{-1}right]^*(beta)+int_{mathbb{D}^3}left[(varphi^+)^{-1}right]^*(beta)
$$
$$=-int_{mathbb{D}^3}sqrt{1-u^2-v^2-w^2};du;dv;dw+int_{mathbb{D}^3}sqrt{1-u^2-v^2-w^2};du;dv;dw=0
$$
And also $0$ for the other terms.
But using Stokes theorem I have:
$$
int_{mathbb{S}^3}alpha=int_{partial(mathbb{D}^4)}alpha=int_{mathbb{D}^4}partialalpha=int_{mathbb{D}^4}16;dxwedge dywedge dzwedge dt=16text{Vol}(mathbb{D}^4)not=0
$$
Where is the mistake? Which one is correct?
integration differential-geometry differential-forms
$endgroup$
add a comment |
$begingroup$
I need to compute the integral:
$$
int_{mathbb{S}^3}alpha,
$$
where $alphain Omega^3(mathbb{R}^4)$ is given by $alpha=-4t;dxwedge dywedge dz+4z;dxwedge dywedge dt-4y;dxwedge dzwedge dt+4x;dywedge dzwedge dt$.
My problem is that I am getting two different results. If I apply the definition:
$$
begin{array}{rccl}
varphi^+:& mathbb{S}^3cap{t geq 0}:=mathbb{S}^{3+}&longrightarrow&mathbb{D}^3\
&(x,y,z,t)&longmapsto&(x,y,z)
end{array}
$$
$$
begin{array}{rccl}
varphi^-:& mathbb{S}^{3-}cap{t leq 0}:=mathbb{S}^{3-}&longrightarrow&mathbb{D}^3\
&(x,y,z,t)&longmapsto&(x,y,z)
end{array}
$$
The inverses:
$$
begin{array}{rccl}
(varphi^+)^{-1}:& mathbb{D}^3&longrightarrow&mathbb{S}^{3+}\
&(u,v,w)&longmapsto&(u,v,w,sqrt{1-u^2-v^2-w^2})
end{array}
$$
$$
begin{array}{rccl}
(varphi^-)^{-1}:& mathbb{D}^3&longrightarrow&mathbb{S}^{3-}\
&(u,v,w)&longmapsto&(u,v,w,-sqrt{1-u^2-v^2-w^2})
end{array}
$$
If $beta=t;dxwedge dywedge dz$:
$$
int_{mathbb{S}^3}beta=int_{mathbb{S}^{3-}}beta+int_{mathbb{S}^{3+}}beta=int_{mathbb{D}^3}left[(varphi^-)^{-1}right]^*(beta)+int_{mathbb{D}^3}left[(varphi^+)^{-1}right]^*(beta)
$$
$$=-int_{mathbb{D}^3}sqrt{1-u^2-v^2-w^2};du;dv;dw+int_{mathbb{D}^3}sqrt{1-u^2-v^2-w^2};du;dv;dw=0
$$
And also $0$ for the other terms.
But using Stokes theorem I have:
$$
int_{mathbb{S}^3}alpha=int_{partial(mathbb{D}^4)}alpha=int_{mathbb{D}^4}partialalpha=int_{mathbb{D}^4}16;dxwedge dywedge dzwedge dt=16text{Vol}(mathbb{D}^4)not=0
$$
Where is the mistake? Which one is correct?
integration differential-geometry differential-forms
$endgroup$
add a comment |
$begingroup$
I need to compute the integral:
$$
int_{mathbb{S}^3}alpha,
$$
where $alphain Omega^3(mathbb{R}^4)$ is given by $alpha=-4t;dxwedge dywedge dz+4z;dxwedge dywedge dt-4y;dxwedge dzwedge dt+4x;dywedge dzwedge dt$.
My problem is that I am getting two different results. If I apply the definition:
$$
begin{array}{rccl}
varphi^+:& mathbb{S}^3cap{t geq 0}:=mathbb{S}^{3+}&longrightarrow&mathbb{D}^3\
&(x,y,z,t)&longmapsto&(x,y,z)
end{array}
$$
$$
begin{array}{rccl}
varphi^-:& mathbb{S}^{3-}cap{t leq 0}:=mathbb{S}^{3-}&longrightarrow&mathbb{D}^3\
&(x,y,z,t)&longmapsto&(x,y,z)
end{array}
$$
The inverses:
$$
begin{array}{rccl}
(varphi^+)^{-1}:& mathbb{D}^3&longrightarrow&mathbb{S}^{3+}\
&(u,v,w)&longmapsto&(u,v,w,sqrt{1-u^2-v^2-w^2})
end{array}
$$
$$
begin{array}{rccl}
(varphi^-)^{-1}:& mathbb{D}^3&longrightarrow&mathbb{S}^{3-}\
&(u,v,w)&longmapsto&(u,v,w,-sqrt{1-u^2-v^2-w^2})
end{array}
$$
If $beta=t;dxwedge dywedge dz$:
$$
int_{mathbb{S}^3}beta=int_{mathbb{S}^{3-}}beta+int_{mathbb{S}^{3+}}beta=int_{mathbb{D}^3}left[(varphi^-)^{-1}right]^*(beta)+int_{mathbb{D}^3}left[(varphi^+)^{-1}right]^*(beta)
$$
$$=-int_{mathbb{D}^3}sqrt{1-u^2-v^2-w^2};du;dv;dw+int_{mathbb{D}^3}sqrt{1-u^2-v^2-w^2};du;dv;dw=0
$$
And also $0$ for the other terms.
But using Stokes theorem I have:
$$
int_{mathbb{S}^3}alpha=int_{partial(mathbb{D}^4)}alpha=int_{mathbb{D}^4}partialalpha=int_{mathbb{D}^4}16;dxwedge dywedge dzwedge dt=16text{Vol}(mathbb{D}^4)not=0
$$
Where is the mistake? Which one is correct?
integration differential-geometry differential-forms
$endgroup$
I need to compute the integral:
$$
int_{mathbb{S}^3}alpha,
$$
where $alphain Omega^3(mathbb{R}^4)$ is given by $alpha=-4t;dxwedge dywedge dz+4z;dxwedge dywedge dt-4y;dxwedge dzwedge dt+4x;dywedge dzwedge dt$.
My problem is that I am getting two different results. If I apply the definition:
$$
begin{array}{rccl}
varphi^+:& mathbb{S}^3cap{t geq 0}:=mathbb{S}^{3+}&longrightarrow&mathbb{D}^3\
&(x,y,z,t)&longmapsto&(x,y,z)
end{array}
$$
$$
begin{array}{rccl}
varphi^-:& mathbb{S}^{3-}cap{t leq 0}:=mathbb{S}^{3-}&longrightarrow&mathbb{D}^3\
&(x,y,z,t)&longmapsto&(x,y,z)
end{array}
$$
The inverses:
$$
begin{array}{rccl}
(varphi^+)^{-1}:& mathbb{D}^3&longrightarrow&mathbb{S}^{3+}\
&(u,v,w)&longmapsto&(u,v,w,sqrt{1-u^2-v^2-w^2})
end{array}
$$
$$
begin{array}{rccl}
(varphi^-)^{-1}:& mathbb{D}^3&longrightarrow&mathbb{S}^{3-}\
&(u,v,w)&longmapsto&(u,v,w,-sqrt{1-u^2-v^2-w^2})
end{array}
$$
If $beta=t;dxwedge dywedge dz$:
$$
int_{mathbb{S}^3}beta=int_{mathbb{S}^{3-}}beta+int_{mathbb{S}^{3+}}beta=int_{mathbb{D}^3}left[(varphi^-)^{-1}right]^*(beta)+int_{mathbb{D}^3}left[(varphi^+)^{-1}right]^*(beta)
$$
$$=-int_{mathbb{D}^3}sqrt{1-u^2-v^2-w^2};du;dv;dw+int_{mathbb{D}^3}sqrt{1-u^2-v^2-w^2};du;dv;dw=0
$$
And also $0$ for the other terms.
But using Stokes theorem I have:
$$
int_{mathbb{S}^3}alpha=int_{partial(mathbb{D}^4)}alpha=int_{mathbb{D}^4}partialalpha=int_{mathbb{D}^4}16;dxwedge dywedge dzwedge dt=16text{Vol}(mathbb{D}^4)not=0
$$
Where is the mistake? Which one is correct?
integration differential-geometry differential-forms
integration differential-geometry differential-forms
asked Dec 7 '18 at 16:00
davidivadfuldavidivadful
1189
1189
add a comment |
add a comment |
1 Answer
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$begingroup$
The Stokes's Theorem application is correct. The mistake comes in half of the first part: With the boundary orientation on $Bbb S^3$, the chart $varphi^+$ is orientation-reversing. So you should get two positive terms for your integrals. (In your computation, in the $varphi^-$ integral, $-4tge 0$, and in the $varphi^+$ integral, you must change sign to compensate for the orientation reversal, and so $-(-4t)ge 0$ once again.)
$endgroup$
add a comment |
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1 Answer
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1 Answer
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active
oldest
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votes
$begingroup$
The Stokes's Theorem application is correct. The mistake comes in half of the first part: With the boundary orientation on $Bbb S^3$, the chart $varphi^+$ is orientation-reversing. So you should get two positive terms for your integrals. (In your computation, in the $varphi^-$ integral, $-4tge 0$, and in the $varphi^+$ integral, you must change sign to compensate for the orientation reversal, and so $-(-4t)ge 0$ once again.)
$endgroup$
add a comment |
$begingroup$
The Stokes's Theorem application is correct. The mistake comes in half of the first part: With the boundary orientation on $Bbb S^3$, the chart $varphi^+$ is orientation-reversing. So you should get two positive terms for your integrals. (In your computation, in the $varphi^-$ integral, $-4tge 0$, and in the $varphi^+$ integral, you must change sign to compensate for the orientation reversal, and so $-(-4t)ge 0$ once again.)
$endgroup$
add a comment |
$begingroup$
The Stokes's Theorem application is correct. The mistake comes in half of the first part: With the boundary orientation on $Bbb S^3$, the chart $varphi^+$ is orientation-reversing. So you should get two positive terms for your integrals. (In your computation, in the $varphi^-$ integral, $-4tge 0$, and in the $varphi^+$ integral, you must change sign to compensate for the orientation reversal, and so $-(-4t)ge 0$ once again.)
$endgroup$
The Stokes's Theorem application is correct. The mistake comes in half of the first part: With the boundary orientation on $Bbb S^3$, the chart $varphi^+$ is orientation-reversing. So you should get two positive terms for your integrals. (In your computation, in the $varphi^-$ integral, $-4tge 0$, and in the $varphi^+$ integral, you must change sign to compensate for the orientation reversal, and so $-(-4t)ge 0$ once again.)
answered Dec 7 '18 at 18:21
Ted ShifrinTed Shifrin
63.1k44489
63.1k44489
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