Mixing
Problem with Chain Rule and Product Rule in Tensor Notation
$begingroup$
From the paper "Flow By Mean Curvature Of Convex Surfaces into Spheres," I encounter some problem in the derivation of Lemma $5.2$. Here is my steps,
$$frac{partial}{partial t} left( frac{|A|^2}{H^{alpha}} - frac{1}{n} H^{2 - alpha} right)$$
By using chain rule and quotient rule,
$$= frac{H^{alpha}partial_t |A|^2-alpha |A|^2H^{alpha-1} partial _t H
}{H^{alpha+2}}- frac{1}{n} (2 - alpha)H^{1-alpha}partial_t H$$
Since we know that $partial_t |A|^2 = Delta |A|^2 - 2 |nabla A|^2 + 2 |A|^4$ and we also know that $partial_t H = Delta H + |A|^2 H,$ I try to plug in and I got the term with $H$ to the power of $-3$ and $-2.$ But in the paper, the term has power $-alpha$ or $-alpha-1.$ My guess might be that the way I use chain rule and quotient rule was wrong.
Can someone spot my issue?
calculus
asked Jan 23 at 13:53
Zack NiZack Ni
3,480729
$endgroup$
add a comment |
$begingroup$
From the paper "Flow By Mean Curvature Of Convex Surfaces into Spheres," I encounter some problem in the derivation of Lemma $5.2$. Here is my steps,
$$frac{partial}{partial t} left( frac{|A|^2}{H^{alpha}} - frac{1}{n} H^{2 - alpha} right)$$
By using chain rule and quotient rule,
$$= frac{H^{alpha}partial_t |A|^2-alpha |A|^2H^{alpha-1} partial _t H
}{H^{alpha+2}}- frac{1}{n} (2 - alpha)H^{1-alpha}partial_t H$$
Since we know that $partial_t |A|^2 = Delta |A|^2 - 2 |nabla A|^2 + 2 |A|^4$ and we also know that $partial_t H = Delta H + |A|^2 H,$ I try to plug in and I got the term with $H$ to the power of $-3$ and $-2.$ But in the paper, the term has power $-alpha$ or $-alpha-1.$ My guess might be that the way I use chain rule and quotient rule was wrong.
Can someone spot my issue?
calculus
asked Jan 23 at 13:53
Zack NiZack Ni
3,480729
$endgroup$
add a comment |
$begingroup$
From the paper "Flow By Mean Curvature Of Convex Surfaces into Spheres," I encounter some problem in the derivation of Lemma $5.2$. Here is my steps,
$$frac{partial}{partial t} left( frac{|A|^2}{H^{alpha}} - frac{1}{n} H^{2 - alpha} right)$$
By using chain rule and quotient rule,
$$= frac{H^{alpha}partial_t |A|^2-alpha |A|^2H^{alpha-1} partial _t H
}{H^{alpha+2}}- frac{1}{n} (2 - alpha)H^{1-alpha}partial_t H$$
Since we know that $partial_t |A|^2 = Delta |A|^2 - 2 |nabla A|^2 + 2 |A|^4$ and we also know that $partial_t H = Delta H + |A|^2 H,$ I try to plug in and I got the term with $H$ to the power of $-3$ and $-2.$ But in the paper, the term has power $-alpha$ or $-alpha-1.$ My guess might be that the way I use chain rule and quotient rule was wrong.
Can someone spot my issue?
calculus
asked Jan 23 at 13:53
Zack NiZack Ni
3,480729
$endgroup$
From the paper "Flow By Mean Curvature Of Convex Surfaces into Spheres," I encounter some problem in the derivation of Lemma $5.2$. Here is my steps,
$$frac{partial}{partial t} left( frac{|A|^2}{H^{alpha}} - frac{1}{n} H^{2 - alpha} right)$$
By using chain rule and quotient rule,
$$= frac{H^{alpha}partial_t |A|^2-alpha |A|^2H^{alpha-1} partial _t H
}{H^{alpha+2}}- frac{1}{n} (2 - alpha)H^{1-alpha}partial_t H$$
Since we know that $partial_t |A|^2 = Delta |A|^2 - 2 |nabla A|^2 + 2 |A|^4$ and we also know that $partial_t H = Delta H + |A|^2 H,$ I try to plug in and I got the term with $H$ to the power of $-3$ and $-2.$ But in the paper, the term has power $-alpha$ or $-alpha-1.$ My guess might be that the way I use chain rule and quotient rule was wrong.
Can someone spot my issue?
calculus
calculus
asked Jan 23 at 13:53
Zack NiZack Ni
3,480729
asked Jan 23 at 13:53
Zack NiZack Ni
3,480729
asked Jan 23 at 13:53
Zack NiZack Ni
3,480729
asked Jan 23 at 13:53
Zack NiZack Ni
3,480729
asked Jan 23 at 13:53
Zack NiZack Ni
3,480729
3,480729
add a comment |
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%2f3084494%2fproblem-with-chain-rule-and-product-rule-in-tensor-notation%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%2f3084494%2fproblem-with-chain-rule-and-product-rule-in-tensor-notation%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
