Integral Transformation from circle to unit sphere
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I want to show that
$displaystyle frac{1}{2pi r}int_{partial B(x,r)}u(y),mathrm{d}s(y)=frac{1}{|S^1|}int_{S^1}u(x+rtheta),mathrm{d}s(theta)$
This is essentially a shift and dilation from (or) to the unit sphere.
I defined a diffeomorphism
$Phicolon S^1rightarrowpartial B(x,r)subsetmathbb{R}^2\
thetamapsto x+rtheta$
where $theta$ is a point on $S^1$.
It follows that $|det(DPhi(S^1))|= r$
and therefore
$displaystyle frac{1}{2pi r}int_{partial B(x,r)}u(y),mathrm{d}s(y)=frac{1}{2pi r}int_{S^1}u(Phi(S^1))r,mathrm{d}s(theta)=frac{1}{2pi(=|S^1|)}int_{S^1}u(x+rtheta),mathrm{d}s(theta)$
Is this correct? If yes, I appreciate to have a look at solutions using a different approach.
integration lebesgue-integral integral-transforms
asked Nov 16 at 13:21
EpsilonDelta
5921515
add a comment |
up vote
0
down vote
favorite
I want to show that
$displaystyle frac{1}{2pi r}int_{partial B(x,r)}u(y),mathrm{d}s(y)=frac{1}{|S^1|}int_{S^1}u(x+rtheta),mathrm{d}s(theta)$
This is essentially a shift and dilation from (or) to the unit sphere.
I defined a diffeomorphism
$Phicolon S^1rightarrowpartial B(x,r)subsetmathbb{R}^2\
thetamapsto x+rtheta$
where $theta$ is a point on $S^1$.
It follows that $|det(DPhi(S^1))|= r$
and therefore
$displaystyle frac{1}{2pi r}int_{partial B(x,r)}u(y),mathrm{d}s(y)=frac{1}{2pi r}int_{S^1}u(Phi(S^1))r,mathrm{d}s(theta)=frac{1}{2pi(=|S^1|)}int_{S^1}u(x+rtheta),mathrm{d}s(theta)$
Is this correct? If yes, I appreciate to have a look at solutions using a different approach.
integration lebesgue-integral integral-transforms
asked Nov 16 at 13:21
EpsilonDelta
5921515
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I want to show that
$displaystyle frac{1}{2pi r}int_{partial B(x,r)}u(y),mathrm{d}s(y)=frac{1}{|S^1|}int_{S^1}u(x+rtheta),mathrm{d}s(theta)$
This is essentially a shift and dilation from (or) to the unit sphere.
I defined a diffeomorphism
$Phicolon S^1rightarrowpartial B(x,r)subsetmathbb{R}^2\
thetamapsto x+rtheta$
where $theta$ is a point on $S^1$.
It follows that $|det(DPhi(S^1))|= r$
and therefore
$displaystyle frac{1}{2pi r}int_{partial B(x,r)}u(y),mathrm{d}s(y)=frac{1}{2pi r}int_{S^1}u(Phi(S^1))r,mathrm{d}s(theta)=frac{1}{2pi(=|S^1|)}int_{S^1}u(x+rtheta),mathrm{d}s(theta)$
Is this correct? If yes, I appreciate to have a look at solutions using a different approach.
integration lebesgue-integral integral-transforms
asked Nov 16 at 13:21
EpsilonDelta
5921515
I want to show that
$displaystyle frac{1}{2pi r}int_{partial B(x,r)}u(y),mathrm{d}s(y)=frac{1}{|S^1|}int_{S^1}u(x+rtheta),mathrm{d}s(theta)$
This is essentially a shift and dilation from (or) to the unit sphere.
I defined a diffeomorphism
$Phicolon S^1rightarrowpartial B(x,r)subsetmathbb{R}^2\
thetamapsto x+rtheta$
where $theta$ is a point on $S^1$.
It follows that $|det(DPhi(S^1))|= r$
and therefore
$displaystyle frac{1}{2pi r}int_{partial B(x,r)}u(y),mathrm{d}s(y)=frac{1}{2pi r}int_{S^1}u(Phi(S^1))r,mathrm{d}s(theta)=frac{1}{2pi(=|S^1|)}int_{S^1}u(x+rtheta),mathrm{d}s(theta)$
Is this correct? If yes, I appreciate to have a look at solutions using a different approach.
integration lebesgue-integral integral-transforms
integration lebesgue-integral integral-transforms
asked Nov 16 at 13:21
EpsilonDelta
5921515
asked Nov 16 at 13:21
EpsilonDelta
5921515
edited Nov 16 at 14:24
asked Nov 16 at 13:21
EpsilonDelta
5921515
asked Nov 16 at 13:21
EpsilonDelta
5921515
asked Nov 16 at 13:21
EpsilonDelta
5921515
5921515
add a comment |
add a comment |
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