Derivation in Evans not reproducible
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 Nov 30 at 14:44
EpsilonDelta
6211615
add a comment |
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 Nov 30 at 14:44
EpsilonDelta
6211615
The question seems to be "How do we differentiate the first expression with respect to $r$ to get the second expression?"
– marty cohen
Nov 30 at 15:09
@ZacharySelk Made an edit.
– EpsilonDelta
Nov 30 at 18:08
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.
Nov 30 at 20:10
@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
Nov 30 at 23:38
But there was an $r$ in the numerator to begin with. Your edit agrees with the formula in the comment
– Umberto P.
Nov 30 at 23:43
add a comment |
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 Nov 30 at 14:44
EpsilonDelta
6211615
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 Nov 30 at 14:44
EpsilonDelta
6211615
asked Nov 30 at 14:44
EpsilonDelta
6211615
edited Nov 30 at 23:37
asked Nov 30 at 14:44
EpsilonDelta
6211615
asked Nov 30 at 14:44
EpsilonDelta
6211615
asked Nov 30 at 14:44
EpsilonDelta
6211615
6211615
The question seems to be "How do we differentiate the first expression with respect to $r$ to get the second expression?"
– marty cohen
Nov 30 at 15:09
@ZacharySelk Made an edit.
– EpsilonDelta
Nov 30 at 18:08
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.
Nov 30 at 20:10
@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
Nov 30 at 23:38
But there was an $r$ in the numerator to begin with. Your edit agrees with the formula in the comment
– Umberto P.
Nov 30 at 23:43
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
Nov 30 at 15:09
@ZacharySelk Made an edit.
– EpsilonDelta
Nov 30 at 18:08
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.
Nov 30 at 20:10
@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
Nov 30 at 23:38
But there was an $r$ in the numerator to begin with. Your edit agrees with the formula in the comment
– Umberto P.
Nov 30 at 23:43
The question seems to be "How do we differentiate the first expression with respect to $r$ to get the second expression?"
– marty cohen
Nov 30 at 15:09
The question seems to be "How do we differentiate the first expression with respect to $r$ to get the second expression?"
– marty cohen
Nov 30 at 15:09
@ZacharySelk Made an edit.
– EpsilonDelta
Nov 30 at 18:08
@ZacharySelk Made an edit.
– EpsilonDelta
Nov 30 at 18:08
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.
Nov 30 at 20:10
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.
Nov 30 at 20:10
@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
Nov 30 at 23:38
@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
Nov 30 at 23:38
But there was an $r$ in the numerator to begin with. Your edit agrees with the formula in the comment
– Umberto P.
Nov 30 at 23:43
But there was an $r$ in the numerator to begin with. Your edit agrees with the formula in the comment
– Umberto P.
Nov 30 at 23:43
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
Nov 30 at 15:09
@ZacharySelk Made an edit.
– EpsilonDelta
Nov 30 at 18:08
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.
Nov 30 at 20:10
@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
Nov 30 at 23:38
But there was an $r$ in the numerator to begin with. Your edit agrees with the formula in the comment
– Umberto P.
Nov 30 at 23:43