Derivation of curl in two dimension
$begingroup$
Let $f(x,y) = u(x,y) e_x + v(x,y) e_y$ be a vector valued function on $mathbb{R}^2$. I'm not sure how the formula is determined.
I take this definition of curl:
The curl at point $P$ is the (directed) anglular velocity a "infinitesemal" disc would rotate with, when it's center is aligned with $P$
From this, it's easy to see (by eg: treating it as torque) for a curve $gamma$ with enclosed area $A$, the torque experined by it is $frac{1}{|A|} oint_{gamma} f cdot dS $ where the integral is taken along the surface of the curve.
Now, we need to prove that $displaystyle lim_{|A| rightarrow 0} frac{1}{|A|} oint_{gamma} f cdot dS = frac{partial u}{partial x} - frac{partial v}{partial y}$ for any curve $gamma$, but how do you do that ?
I'm stuck even for the very simple case when $gamma$ is a circle, and I'm finding it very counterintuive why would the limit even be the same and not depending on the shape of $gamma$.
calculus multivariable-calculus curl
asked Jan 7 at 13:50
alxchenalxchen
644421
$endgroup$
add a comment |
$begingroup$
Let $f(x,y) = u(x,y) e_x + v(x,y) e_y$ be a vector valued function on $mathbb{R}^2$. I'm not sure how the formula is determined.
I take this definition of curl:
The curl at point $P$ is the (directed) anglular velocity a "infinitesemal" disc would rotate with, when it's center is aligned with $P$
From this, it's easy to see (by eg: treating it as torque) for a curve $gamma$ with enclosed area $A$, the torque experined by it is $frac{1}{|A|} oint_{gamma} f cdot dS $ where the integral is taken along the surface of the curve.
Now, we need to prove that $displaystyle lim_{|A| rightarrow 0} frac{1}{|A|} oint_{gamma} f cdot dS = frac{partial u}{partial x} - frac{partial v}{partial y}$ for any curve $gamma$, but how do you do that ?
I'm stuck even for the very simple case when $gamma$ is a circle, and I'm finding it very counterintuive why would the limit even be the same and not depending on the shape of $gamma$.
calculus multivariable-calculus curl
asked Jan 7 at 13:50
alxchenalxchen
644421
$endgroup$
add a comment |
$begingroup$
Let $f(x,y) = u(x,y) e_x + v(x,y) e_y$ be a vector valued function on $mathbb{R}^2$. I'm not sure how the formula is determined.
I take this definition of curl:
The curl at point $P$ is the (directed) anglular velocity a "infinitesemal" disc would rotate with, when it's center is aligned with $P$
From this, it's easy to see (by eg: treating it as torque) for a curve $gamma$ with enclosed area $A$, the torque experined by it is $frac{1}{|A|} oint_{gamma} f cdot dS $ where the integral is taken along the surface of the curve.
Now, we need to prove that $displaystyle lim_{|A| rightarrow 0} frac{1}{|A|} oint_{gamma} f cdot dS = frac{partial u}{partial x} - frac{partial v}{partial y}$ for any curve $gamma$, but how do you do that ?
I'm stuck even for the very simple case when $gamma$ is a circle, and I'm finding it very counterintuive why would the limit even be the same and not depending on the shape of $gamma$.
calculus multivariable-calculus curl
asked Jan 7 at 13:50
alxchenalxchen
644421
$endgroup$
Let $f(x,y) = u(x,y) e_x + v(x,y) e_y$ be a vector valued function on $mathbb{R}^2$. I'm not sure how the formula is determined.
I take this definition of curl:
The curl at point $P$ is the (directed) anglular velocity a "infinitesemal" disc would rotate with, when it's center is aligned with $P$
From this, it's easy to see (by eg: treating it as torque) for a curve $gamma$ with enclosed area $A$, the torque experined by it is $frac{1}{|A|} oint_{gamma} f cdot dS $ where the integral is taken along the surface of the curve.
Now, we need to prove that $displaystyle lim_{|A| rightarrow 0} frac{1}{|A|} oint_{gamma} f cdot dS = frac{partial u}{partial x} - frac{partial v}{partial y}$ for any curve $gamma$, but how do you do that ?
I'm stuck even for the very simple case when $gamma$ is a circle, and I'm finding it very counterintuive why would the limit even be the same and not depending on the shape of $gamma$.
calculus multivariable-calculus curl
calculus multivariable-calculus curl
asked Jan 7 at 13:50
alxchenalxchen
644421
asked Jan 7 at 13:50
alxchenalxchen
644421
asked Jan 7 at 13:50
alxchenalxchen
644421
asked Jan 7 at 13:50
alxchenalxchen
644421
asked Jan 7 at 13:50
alxchenalxchen
644421
644421
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%2f3065026%2fderivation-of-curl-in-two-dimension%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%2f3065026%2fderivation-of-curl-in-two-dimension%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
