Decomposition of $C^{(k)}$ function
0
I am stuck at the following exercise in Zorich.
Let $f,gin C^{(k)}(D;mathbb{R})$, and suppose that $f(x)=0Rightarrow g(x)=0$ in the domain $D$. Show that if grad $f neq 0$, them there is a decomposition $g=hcdot f$ in $D$, where $hin C^{(k-1)}(D;mathbb{R})$.
Could you give me a hint? Thanks in advance!
analysis
asked Dec 5 '18 at 3:07
JiuJiu
466111
|
show 1 more comment
0
I am stuck at the following exercise in Zorich.
Let $f,gin C^{(k)}(D;mathbb{R})$, and suppose that $f(x)=0Rightarrow g(x)=0$ in the domain $D$. Show that if grad $f neq 0$, them there is a decomposition $g=hcdot f$ in $D$, where $hin C^{(k-1)}(D;mathbb{R})$.
Could you give me a hint? Thanks in advance!
analysis
asked Dec 5 '18 at 3:07
JiuJiu
466111
Dear @Jiu, is $D$ a domain in $mathbb{R}^n$? I'm confused about $h$, because the image of $f$ lies in $mathbb{R}$, not in $D$. You should require $h(0)=0$ and perhaps some additional constraints on $h$.
– user90189
Dec 5 '18 at 4:36
@user90189 thanks for your comment! Now that you make me realize this problem, I just checked and see that in the English version, the $circ$ is replaced by $cdot$... So the $circ$ should be a typo in the German version!
– Jiu
Dec 5 '18 at 4:53
So, I think you can try to check that $g/f$ is well-defined :)
– user90189
Dec 5 '18 at 4:56
@user90189 yes now I know how to solve it :D
– Jiu
Dec 5 '18 at 7:55
@user90189 In fact when I try to write down the solution I realize that it is not correct :(
– Jiu
Dec 5 '18 at 8:57
|
show 1 more comment
0
0
0
I am stuck at the following exercise in Zorich.
Let $f,gin C^{(k)}(D;mathbb{R})$, and suppose that $f(x)=0Rightarrow g(x)=0$ in the domain $D$. Show that if grad $f neq 0$, them there is a decomposition $g=hcdot f$ in $D$, where $hin C^{(k-1)}(D;mathbb{R})$.
Could you give me a hint? Thanks in advance!
analysis
asked Dec 5 '18 at 3:07
JiuJiu
466111
I am stuck at the following exercise in Zorich.
Let $f,gin C^{(k)}(D;mathbb{R})$, and suppose that $f(x)=0Rightarrow g(x)=0$ in the domain $D$. Show that if grad $f neq 0$, them there is a decomposition $g=hcdot f$ in $D$, where $hin C^{(k-1)}(D;mathbb{R})$.
Could you give me a hint? Thanks in advance!
analysis
analysis
asked Dec 5 '18 at 3:07
JiuJiu
466111
asked Dec 5 '18 at 3:07
JiuJiu
466111
edited Dec 5 '18 at 4:54
Jiu
asked Dec 5 '18 at 3:07
JiuJiu
466111
asked Dec 5 '18 at 3:07
JiuJiu
466111
asked Dec 5 '18 at 3:07
JiuJiu
466111
466111
Dear @Jiu, is $D$ a domain in $mathbb{R}^n$? I'm confused about $h$, because the image of $f$ lies in $mathbb{R}$, not in $D$. You should require $h(0)=0$ and perhaps some additional constraints on $h$.
– user90189
Dec 5 '18 at 4:36
@user90189 thanks for your comment! Now that you make me realize this problem, I just checked and see that in the English version, the $circ$ is replaced by $cdot$... So the $circ$ should be a typo in the German version!
– Jiu
Dec 5 '18 at 4:53
So, I think you can try to check that $g/f$ is well-defined :)
– user90189
Dec 5 '18 at 4:56
@user90189 yes now I know how to solve it :D
– Jiu
Dec 5 '18 at 7:55
@user90189 In fact when I try to write down the solution I realize that it is not correct :(
– Jiu
Dec 5 '18 at 8:57
|
show 1 more comment
Dear @Jiu, is $D$ a domain in $mathbb{R}^n$? I'm confused about $h$, because the image of $f$ lies in $mathbb{R}$, not in $D$. You should require $h(0)=0$ and perhaps some additional constraints on $h$.
– user90189
Dec 5 '18 at 4:36
@user90189 thanks for your comment! Now that you make me realize this problem, I just checked and see that in the English version, the $circ$ is replaced by $cdot$... So the $circ$ should be a typo in the German version!
– Jiu
Dec 5 '18 at 4:53
So, I think you can try to check that $g/f$ is well-defined :)
– user90189
Dec 5 '18 at 4:56
@user90189 yes now I know how to solve it :D
– Jiu
Dec 5 '18 at 7:55
@user90189 In fact when I try to write down the solution I realize that it is not correct :(
– Jiu
Dec 5 '18 at 8:57
Dear @Jiu, is $D$ a domain in $mathbb{R}^n$? I'm confused about $h$, because the image of $f$ lies in $mathbb{R}$, not in $D$. You should require $h(0)=0$ and perhaps some additional constraints on $h$.
– user90189
Dec 5 '18 at 4:36
Dear @Jiu, is $D$ a domain in $mathbb{R}^n$? I'm confused about $h$, because the image of $f$ lies in $mathbb{R}$, not in $D$. You should require $h(0)=0$ and perhaps some additional constraints on $h$.
– user90189
Dec 5 '18 at 4:36
@user90189 thanks for your comment! Now that you make me realize this problem, I just checked and see that in the English version, the $circ$ is replaced by $cdot$... So the $circ$ should be a typo in the German version!
– Jiu
Dec 5 '18 at 4:53
@user90189 thanks for your comment! Now that you make me realize this problem, I just checked and see that in the English version, the $circ$ is replaced by $cdot$... So the $circ$ should be a typo in the German version!
– Jiu
Dec 5 '18 at 4:53
So, I think you can try to check that $g/f$ is well-defined :)
– user90189
Dec 5 '18 at 4:56
So, I think you can try to check that $g/f$ is well-defined :)
– user90189
Dec 5 '18 at 4:56
@user90189 yes now I know how to solve it :D
– Jiu
Dec 5 '18 at 7:55
@user90189 yes now I know how to solve it :D
– Jiu
Dec 5 '18 at 7:55
@user90189 In fact when I try to write down the solution I realize that it is not correct :(
– Jiu
Dec 5 '18 at 8:57
@user90189 In fact when I try to write down the solution I realize that it is not correct :(
– Jiu
Dec 5 '18 at 8:57
|
show 1 more 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
});
}
});
draft saved
draft discarded
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%2f3026548%2fdecomposition-of-ck-function%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
draft saved
draft discarded
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
To learn more, see our tips on writing great answers.
draft saved
draft discarded
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%2f3026548%2fdecomposition-of-ck-function%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
Dear @Jiu, is $D$ a domain in $mathbb{R}^n$? I'm confused about $h$, because the image of $f$ lies in $mathbb{R}$, not in $D$. You should require $h(0)=0$ and perhaps some additional constraints on $h$.
– user90189
Dec 5 '18 at 4:36
@user90189 thanks for your comment! Now that you make me realize this problem, I just checked and see that in the English version, the $circ$ is replaced by $cdot$... So the $circ$ should be a typo in the German version!
– Jiu
Dec 5 '18 at 4:53
So, I think you can try to check that $g/f$ is well-defined :)
– user90189
Dec 5 '18 at 4:56
@user90189 yes now I know how to solve it :D
– Jiu
Dec 5 '18 at 7:55
@user90189 In fact when I try to write down the solution I realize that it is not correct :(
– Jiu
Dec 5 '18 at 8:57