Example 3 - Section 5.5.2 - Partial Differential Equations by Evans (2)
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Could somebody help me: there is example from book:Example
And I don't understand why:
$
u_{x_i}=-frac{alpha x_i}{|x|^{alpha+2}}
$.
I thought that $|x|=|x_1+x_2+...+x_n|$. And here is my calculations:
$
frac{partial u} {partial x_i}=-alpha |x|^{-alpha-1} frac{partial x} {partial x_i}=frac{-alpha}{|x|^{alpha+2}}frac{1}{x} frac{partial x} {partial x_i} =frac{-alpha}{|x|^{alpha+2}}frac{1}{x}frac{partial (x_1,...,x_i,...,x_n)} {partial x_i}
$.
So, if I was right, why does it equal:
$
frac{1}{x}frac{partial (x_1,...,x_i,...,x_n)} {partial x_i}=frac{x_i}{|x|}
$?
Thank you!
pde sobolev-spaces
asked Dec 5 at 2:16
Student
11
add a comment |
up vote
0
down vote
favorite
Could somebody help me: there is example from book:Example
And I don't understand why:
$
u_{x_i}=-frac{alpha x_i}{|x|^{alpha+2}}
$.
I thought that $|x|=|x_1+x_2+...+x_n|$. And here is my calculations:
$
frac{partial u} {partial x_i}=-alpha |x|^{-alpha-1} frac{partial x} {partial x_i}=frac{-alpha}{|x|^{alpha+2}}frac{1}{x} frac{partial x} {partial x_i} =frac{-alpha}{|x|^{alpha+2}}frac{1}{x}frac{partial (x_1,...,x_i,...,x_n)} {partial x_i}
$.
So, if I was right, why does it equal:
$
frac{1}{x}frac{partial (x_1,...,x_i,...,x_n)} {partial x_i}=frac{x_i}{|x|}
$?
Thank you!
pde sobolev-spaces
asked Dec 5 at 2:16
Student
11
If I am remembering correctly, he usually means $$|x| = |x|_2=left(sum_{i=1}^{n}x_i^2right)^{1/2}$$
– whpowell96
Dec 5 at 2:21
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Could somebody help me: there is example from book:Example
And I don't understand why:
$
u_{x_i}=-frac{alpha x_i}{|x|^{alpha+2}}
$.
I thought that $|x|=|x_1+x_2+...+x_n|$. And here is my calculations:
$
frac{partial u} {partial x_i}=-alpha |x|^{-alpha-1} frac{partial x} {partial x_i}=frac{-alpha}{|x|^{alpha+2}}frac{1}{x} frac{partial x} {partial x_i} =frac{-alpha}{|x|^{alpha+2}}frac{1}{x}frac{partial (x_1,...,x_i,...,x_n)} {partial x_i}
$.
So, if I was right, why does it equal:
$
frac{1}{x}frac{partial (x_1,...,x_i,...,x_n)} {partial x_i}=frac{x_i}{|x|}
$?
Thank you!
pde sobolev-spaces
asked Dec 5 at 2:16
Student
11
Could somebody help me: there is example from book:Example
And I don't understand why:
$
u_{x_i}=-frac{alpha x_i}{|x|^{alpha+2}}
$.
I thought that $|x|=|x_1+x_2+...+x_n|$. And here is my calculations:
$
frac{partial u} {partial x_i}=-alpha |x|^{-alpha-1} frac{partial x} {partial x_i}=frac{-alpha}{|x|^{alpha+2}}frac{1}{x} frac{partial x} {partial x_i} =frac{-alpha}{|x|^{alpha+2}}frac{1}{x}frac{partial (x_1,...,x_i,...,x_n)} {partial x_i}
$.
So, if I was right, why does it equal:
$
frac{1}{x}frac{partial (x_1,...,x_i,...,x_n)} {partial x_i}=frac{x_i}{|x|}
$?
Thank you!
pde sobolev-spaces
pde sobolev-spaces
asked Dec 5 at 2:16
Student
11
asked Dec 5 at 2:16
Student
11
asked Dec 5 at 2:16
Student
11
asked Dec 5 at 2:16
Student
11
asked Dec 5 at 2:16
Student
11
11
If I am remembering correctly, he usually means $$|x| = |x|_2=left(sum_{i=1}^{n}x_i^2right)^{1/2}$$
– whpowell96
Dec 5 at 2:21
add a comment |
If I am remembering correctly, he usually means $$|x| = |x|_2=left(sum_{i=1}^{n}x_i^2right)^{1/2}$$
– whpowell96
Dec 5 at 2:21
If I am remembering correctly, he usually means $$|x| = |x|_2=left(sum_{i=1}^{n}x_i^2right)^{1/2}$$
– whpowell96
Dec 5 at 2:21
If I am remembering correctly, he usually means $$|x| = |x|_2=left(sum_{i=1}^{n}x_i^2right)^{1/2}$$
– whpowell96
Dec 5 at 2:21
add a comment |
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If I am remembering correctly, he usually means $$|x| = |x|_2=left(sum_{i=1}^{n}x_i^2right)^{1/2}$$
– whpowell96
Dec 5 at 2:21