Derivation in Evans not reproducible
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I am currently reading in Evans where he discusses the Euler-Poisson equation on page 81.
There we have that $displaystyle partial_rU(x;r,t)=frac{r}{nalpha(n)r^n}int_{B(x,r)}Delta u(y,t)mathrm{d}y$
Now he makes another derivative which is not reproducible for me
$displaystyle partial_{rr}U(x;r,t)=frac{1}{alpha(n)r^{n-1}n}int_{partial B(x,r)}Delta umathrm{d}S+left ( frac{1}{n}-1right ) frac{1}{alpha(n)r^n}int_{B(x,r)}Delta u mathrm{d}y$
where $alpha(n)$ is the volume of the n-th unit sphere. How is this derivative obtained?
integration pde
asked yesterday
EpsilonDelta
6071615
add a comment |
up vote
1
down vote
favorite
I am currently reading in Evans where he discusses the Euler-Poisson equation on page 81.
There we have that $displaystyle partial_rU(x;r,t)=frac{r}{nalpha(n)r^n}int_{B(x,r)}Delta u(y,t)mathrm{d}y$
Now he makes another derivative which is not reproducible for me
$displaystyle partial_{rr}U(x;r,t)=frac{1}{alpha(n)r^{n-1}n}int_{partial B(x,r)}Delta umathrm{d}S+left ( frac{1}{n}-1right ) frac{1}{alpha(n)r^n}int_{B(x,r)}Delta u mathrm{d}y$
where $alpha(n)$ is the volume of the n-th unit sphere. How is this derivative obtained?
integration pde
asked yesterday
EpsilonDelta
6071615
The question seems to be "How do we differentiate the first expression with respect to $r$ to get the second expression?"
– marty cohen
yesterday
@ZacharySelk Made an edit.
– EpsilonDelta
yesterday
You've written the initial integral incorrectly. It should be $$partial_r U(x;r,t) = frac{1}{r^{n-1}n alpha(n)} int_{B(x,r)} Delta u(y,t) , dy.$$
– Umberto P.
yesterday
@UmbertoP. I corrected this. But I think since it is the integral over the ball, it should be $r^n$ not $r^{n-1}$.
– EpsilonDelta
yesterday
But there was an $r$ in the numerator to begin with. Your edit agrees with the formula in the comment
– Umberto P.
yesterday
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I am currently reading in Evans where he discusses the Euler-Poisson equation on page 81.
There we have that $displaystyle partial_rU(x;r,t)=frac{r}{nalpha(n)r^n}int_{B(x,r)}Delta u(y,t)mathrm{d}y$
Now he makes another derivative which is not reproducible for me
$displaystyle partial_{rr}U(x;r,t)=frac{1}{alpha(n)r^{n-1}n}int_{partial B(x,r)}Delta umathrm{d}S+left ( frac{1}{n}-1right ) frac{1}{alpha(n)r^n}int_{B(x,r)}Delta u mathrm{d}y$
where $alpha(n)$ is the volume of the n-th unit sphere. How is this derivative obtained?
integration pde
asked yesterday
EpsilonDelta
6071615
I am currently reading in Evans where he discusses the Euler-Poisson equation on page 81.
There we have that $displaystyle partial_rU(x;r,t)=frac{r}{nalpha(n)r^n}int_{B(x,r)}Delta u(y,t)mathrm{d}y$
Now he makes another derivative which is not reproducible for me
$displaystyle partial_{rr}U(x;r,t)=frac{1}{alpha(n)r^{n-1}n}int_{partial B(x,r)}Delta umathrm{d}S+left ( frac{1}{n}-1right ) frac{1}{alpha(n)r^n}int_{B(x,r)}Delta u mathrm{d}y$
where $alpha(n)$ is the volume of the n-th unit sphere. How is this derivative obtained?
integration pde
integration pde
asked yesterday
EpsilonDelta
6071615
asked yesterday
EpsilonDelta
6071615
edited yesterday
asked yesterday
EpsilonDelta
6071615
asked yesterday
EpsilonDelta
6071615
asked yesterday
EpsilonDelta
6071615
6071615
The question seems to be "How do we differentiate the first expression with respect to $r$ to get the second expression?"
– marty cohen
yesterday
@ZacharySelk Made an edit.
– EpsilonDelta
yesterday
You've written the initial integral incorrectly. It should be $$partial_r U(x;r,t) = frac{1}{r^{n-1}n alpha(n)} int_{B(x,r)} Delta u(y,t) , dy.$$
– Umberto P.
yesterday
@UmbertoP. I corrected this. But I think since it is the integral over the ball, it should be $r^n$ not $r^{n-1}$.
– EpsilonDelta
yesterday
But there was an $r$ in the numerator to begin with. Your edit agrees with the formula in the comment
– Umberto P.
yesterday
add a comment |
The question seems to be "How do we differentiate the first expression with respect to $r$ to get the second expression?"
– marty cohen
yesterday
@ZacharySelk Made an edit.
– EpsilonDelta
yesterday
You've written the initial integral incorrectly. It should be $$partial_r U(x;r,t) = frac{1}{r^{n-1}n alpha(n)} int_{B(x,r)} Delta u(y,t) , dy.$$
– Umberto P.
yesterday
@UmbertoP. I corrected this. But I think since it is the integral over the ball, it should be $r^n$ not $r^{n-1}$.
– EpsilonDelta
yesterday
But there was an $r$ in the numerator to begin with. Your edit agrees with the formula in the comment
– Umberto P.
yesterday
The question seems to be "How do we differentiate the first expression with respect to $r$ to get the second expression?"
– marty cohen
yesterday
The question seems to be "How do we differentiate the first expression with respect to $r$ to get the second expression?"
– marty cohen
yesterday
@ZacharySelk Made an edit.
– EpsilonDelta
yesterday
@ZacharySelk Made an edit.
– EpsilonDelta
yesterday
You've written the initial integral incorrectly. It should be $$partial_r U(x;r,t) = frac{1}{r^{n-1}n alpha(n)} int_{B(x,r)} Delta u(y,t) , dy.$$
– Umberto P.
yesterday
You've written the initial integral incorrectly. It should be $$partial_r U(x;r,t) = frac{1}{r^{n-1}n alpha(n)} int_{B(x,r)} Delta u(y,t) , dy.$$
– Umberto P.
yesterday
@UmbertoP. I corrected this. But I think since it is the integral over the ball, it should be $r^n$ not $r^{n-1}$.
– EpsilonDelta
yesterday
@UmbertoP. I corrected this. But I think since it is the integral over the ball, it should be $r^n$ not $r^{n-1}$.
– EpsilonDelta
yesterday
But there was an $r$ in the numerator to begin with. Your edit agrees with the formula in the comment
– Umberto P.
yesterday
But there was an $r$ in the numerator to begin with. Your edit agrees with the formula in the comment
– Umberto P.
yesterday
add a comment |
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The question seems to be "How do we differentiate the first expression with respect to $r$ to get the second expression?"
– marty cohen
yesterday
@ZacharySelk Made an edit.
– EpsilonDelta
yesterday
You've written the initial integral incorrectly. It should be $$partial_r U(x;r,t) = frac{1}{r^{n-1}n alpha(n)} int_{B(x,r)} Delta u(y,t) , dy.$$
– Umberto P.
yesterday
@UmbertoP. I corrected this. But I think since it is the integral over the ball, it should be $r^n$ not $r^{n-1}$.
– EpsilonDelta
yesterday
But there was an $r$ in the numerator to begin with. Your edit agrees with the formula in the comment
– Umberto P.
yesterday