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




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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














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0
down vote

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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




share|cite|improve this question





















  • 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












up vote
0
down vote

favorite
1









up vote
0
down vote

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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




share|cite|improve this question














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





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Aug 7 at 4:31









Vinícius Novelli

1,905716




1,905716









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
















  • 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















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