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Prove: $lim_varepsilonto 0int_S(x_0,varepsilon)u(x)operatornamegrad(varphi(x))cdot N,ds=u(x_0)$
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For $x_0inmathbbR^n,ngeq 3$, let $varphi(x)=frac1(n-2)omega_ncdotfrac1$, where $omega_n$ is the surface area of the sphere $S_n-1==1$.
Prove that, if $uin C^1(mathbbR^n)$,
then:
$$lim_varepsilonto 0int_S(x_0,varepsilon)u(x) operatornamegrad(varphi(x))cdot N,ds = u(x_0),$$
where $S(x_0,varepsilon)=xin mathbbR^n:$ and $N$ is internal normal vector.
Where can I find a proof to this? or in which theorem should I look?
calculus multivariable-calculus vector-analysis
edited Aug 7 at 4:31
VinÃcius Novelli
1,905716
asked Aug 6 at 19:14
gbox
5,30851841
add a comment |Â
up vote
0
down vote
favorite
For $x_0inmathbbR^n,ngeq 3$, let $varphi(x)=frac1(n-2)omega_ncdotfrac1$, where $omega_n$ is the surface area of the sphere $S_n-1==1$.
Prove that, if $uin C^1(mathbbR^n)$,
then:
$$lim_varepsilonto 0int_S(x_0,varepsilon)u(x) operatornamegrad(varphi(x))cdot N,ds = u(x_0),$$
where $S(x_0,varepsilon)=xin mathbbR^n:$ and $N$ is internal normal vector.
Where can I find a proof to this? or in which theorem should I look?
calculus multivariable-calculus vector-analysis
edited Aug 7 at 4:31
VinÃcius Novelli
1,905716
asked Aug 6 at 19:14
gbox
5,30851841
The two big theorems relating surface integrals to other integrals are the Divergence Theorem and Stokes's Theorem. Do either of those look fruitful?
– Matthew Leingang
Aug 6 at 19:22
I just realized you're not assuming $n=3$. So one of those is not as applicable as the other.
– Matthew Leingang
Aug 6 at 19:25
@MatthewLeingang So it is related to Stokes?
– gbox
Aug 6 at 19:26
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
For $x_0inmathbbR^n,ngeq 3$, let $varphi(x)=frac1(n-2)omega_ncdotfrac1$, where $omega_n$ is the surface area of the sphere $S_n-1==1$.
Prove that, if $uin C^1(mathbbR^n)$,
then:
$$lim_varepsilonto 0int_S(x_0,varepsilon)u(x) operatornamegrad(varphi(x))cdot N,ds = u(x_0),$$
where $S(x_0,varepsilon)=xin mathbbR^n:$ and $N$ is internal normal vector.
Where can I find a proof to this? or in which theorem should I look?
calculus multivariable-calculus vector-analysis
edited Aug 7 at 4:31
VinÃcius Novelli
1,905716
asked Aug 6 at 19:14
gbox
5,30851841
For $x_0inmathbbR^n,ngeq 3$, let $varphi(x)=frac1(n-2)omega_ncdotfrac1$, where $omega_n$ is the surface area of the sphere $S_n-1==1$.
Prove that, if $uin C^1(mathbbR^n)$,
then:
$$lim_varepsilonto 0int_S(x_0,varepsilon)u(x) operatornamegrad(varphi(x))cdot N,ds = u(x_0),$$
where $S(x_0,varepsilon)=xin mathbbR^n:$ and $N$ is internal normal vector.
Where can I find a proof to this? or in which theorem should I look?
calculus multivariable-calculus vector-analysis
edited Aug 7 at 4:31
VinÃcius Novelli
1,905716
asked Aug 6 at 19:14
gbox
5,30851841
edited Aug 7 at 4:31
VinÃcius Novelli
1,905716
edited Aug 7 at 4:31
VinÃcius Novelli
1,905716
edited Aug 7 at 4:31
VinÃcius Novelli
1,905716
1,905716
asked Aug 6 at 19:14
gbox
5,30851841
asked Aug 6 at 19:14
gbox
5,30851841
asked Aug 6 at 19:14
gbox
5,30851841
5,30851841
The two big theorems relating surface integrals to other integrals are the Divergence Theorem and Stokes's Theorem. Do either of those look fruitful?
– Matthew Leingang
Aug 6 at 19:22
I just realized you're not assuming $n=3$. So one of those is not as applicable as the other.
– Matthew Leingang
Aug 6 at 19:25
@MatthewLeingang So it is related to Stokes?
– gbox
Aug 6 at 19:26
add a comment |Â
The two big theorems relating surface integrals to other integrals are the Divergence Theorem and Stokes's Theorem. Do either of those look fruitful?
– Matthew Leingang
Aug 6 at 19:22
I just realized you're not assuming $n=3$. So one of those is not as applicable as the other.
– Matthew Leingang
Aug 6 at 19:25
@MatthewLeingang So it is related to Stokes?
– gbox
Aug 6 at 19:26
The two big theorems relating surface integrals to other integrals are the Divergence Theorem and Stokes's Theorem. Do either of those look fruitful?
– Matthew Leingang
Aug 6 at 19:22
The two big theorems relating surface integrals to other integrals are the Divergence Theorem and Stokes's Theorem. Do either of those look fruitful?
– Matthew Leingang
Aug 6 at 19:22
I just realized you're not assuming $n=3$. So one of those is not as applicable as the other.
– Matthew Leingang
Aug 6 at 19:25
I just realized you're not assuming $n=3$. So one of those is not as applicable as the other.
– Matthew Leingang
Aug 6 at 19:25
@MatthewLeingang So it is related to Stokes?
– gbox
Aug 6 at 19:26
@MatthewLeingang So it is related to Stokes?
– gbox
Aug 6 at 19:26
add a comment |Â
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The two big theorems relating surface integrals to other integrals are the Divergence Theorem and Stokes's Theorem. Do either of those look fruitful?
– Matthew Leingang
Aug 6 at 19:22
I just realized you're not assuming $n=3$. So one of those is not as applicable as the other.
– Matthew Leingang
Aug 6 at 19:25
@MatthewLeingang So it is related to Stokes?
– gbox
Aug 6 at 19:26