Example 3 - Section 5.5.2 - Partial Differential Equations by Evans (2)
$begingroup$
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 '18 at 2:16
StudentStudent
11
$endgroup$
add a comment |
$begingroup$
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 '18 at 2:16
StudentStudent
11
$endgroup$
$begingroup$
If I am remembering correctly, he usually means $$|x| = |x|_2=left(sum_{i=1}^{n}x_i^2right)^{1/2}$$
$endgroup$
– whpowell96
Dec 5 '18 at 2:21
add a comment |
$begingroup$
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 '18 at 2:16
StudentStudent
11
$endgroup$
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 '18 at 2:16
StudentStudent
11
asked Dec 5 '18 at 2:16
StudentStudent
11
asked Dec 5 '18 at 2:16
StudentStudent
11
asked Dec 5 '18 at 2:16
StudentStudent
11
asked Dec 5 '18 at 2:16
StudentStudent
11
11
$begingroup$
If I am remembering correctly, he usually means $$|x| = |x|_2=left(sum_{i=1}^{n}x_i^2right)^{1/2}$$
$endgroup$
– whpowell96
Dec 5 '18 at 2:21
add a comment |
$begingroup$
If I am remembering correctly, he usually means $$|x| = |x|_2=left(sum_{i=1}^{n}x_i^2right)^{1/2}$$
$endgroup$
– whpowell96
Dec 5 '18 at 2:21
$begingroup$
If I am remembering correctly, he usually means $$|x| = |x|_2=left(sum_{i=1}^{n}x_i^2right)^{1/2}$$
$endgroup$
– whpowell96
Dec 5 '18 at 2:21
$begingroup$
If I am remembering correctly, he usually means $$|x| = |x|_2=left(sum_{i=1}^{n}x_i^2right)^{1/2}$$
$endgroup$
– whpowell96
Dec 5 '18 at 2:21
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3026507%2fexample-3-section-5-5-2-partial-differential-equations-by-evans-2%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3026507%2fexample-3-section-5-5-2-partial-differential-equations-by-evans-2%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
If I am remembering correctly, he usually means $$|x| = |x|_2=left(sum_{i=1}^{n}x_i^2right)^{1/2}$$
$endgroup$
– whpowell96
Dec 5 '18 at 2:21