Gauss's divergence theorem and change of coordination system
$begingroup$
Gauss's divergence theorem:
Let $V$ be a compact subset of $mathbb R^3$ with a piecewise smooth boundary $partial V in C^1_{pw}$.
Further, let $v$ be a continuous differentiable vector field on $V$. Then we have:
$$int_{partial V}vcdot n do=int_V div(v)dmu$$
wheras $n$ denotes the outwards pointing normal along $partial V$.
Question:
So I hope I can formulate my question correctly. When I first started calculating integrals using Gauss's convergence theorem, I did a lot of mistakes when changing the coordinate system. E.g. if I have to calculate the flow of a simple vector field over the unit sphere I'd use spherical coordinates and of course, since I just changed the coordinate system, I'd take the determinant of the jacobian matrix into account. But that's actually "wrong".
It is wrong because Gauss divergence theorem is tightly connected to the actually flow integral - we have an equation of integrals! So if we transform $int_V div(v)dmu$ to let's say spherical coordinates, we'd have to transform $int_{partial V}vcdot n do$ too.
Would we do that, we would see the determinant of the jacobian matrix of the coordination change show up on both sides: We can cancel it out.
Or simply: The flow does not depend on the chosen coordinate system.
Is that observation correct? Could I just cancel the determinant of the jacobian matrix of my coordination change?
vector-analysis
asked Dec 28 '18 at 15:47
xotixxotix
291411
$endgroup$
add a comment |
$begingroup$
Gauss's divergence theorem:
Let $V$ be a compact subset of $mathbb R^3$ with a piecewise smooth boundary $partial V in C^1_{pw}$.
Further, let $v$ be a continuous differentiable vector field on $V$. Then we have:
$$int_{partial V}vcdot n do=int_V div(v)dmu$$
wheras $n$ denotes the outwards pointing normal along $partial V$.
Question:
So I hope I can formulate my question correctly. When I first started calculating integrals using Gauss's convergence theorem, I did a lot of mistakes when changing the coordinate system. E.g. if I have to calculate the flow of a simple vector field over the unit sphere I'd use spherical coordinates and of course, since I just changed the coordinate system, I'd take the determinant of the jacobian matrix into account. But that's actually "wrong".
It is wrong because Gauss divergence theorem is tightly connected to the actually flow integral - we have an equation of integrals! So if we transform $int_V div(v)dmu$ to let's say spherical coordinates, we'd have to transform $int_{partial V}vcdot n do$ too.
Would we do that, we would see the determinant of the jacobian matrix of the coordination change show up on both sides: We can cancel it out.
Or simply: The flow does not depend on the chosen coordinate system.
Is that observation correct? Could I just cancel the determinant of the jacobian matrix of my coordination change?
vector-analysis
asked Dec 28 '18 at 15:47
xotixxotix
291411
$endgroup$
add a comment |
$begingroup$
Gauss's divergence theorem:
Let $V$ be a compact subset of $mathbb R^3$ with a piecewise smooth boundary $partial V in C^1_{pw}$.
Further, let $v$ be a continuous differentiable vector field on $V$. Then we have:
$$int_{partial V}vcdot n do=int_V div(v)dmu$$
wheras $n$ denotes the outwards pointing normal along $partial V$.
Question:
So I hope I can formulate my question correctly. When I first started calculating integrals using Gauss's convergence theorem, I did a lot of mistakes when changing the coordinate system. E.g. if I have to calculate the flow of a simple vector field over the unit sphere I'd use spherical coordinates and of course, since I just changed the coordinate system, I'd take the determinant of the jacobian matrix into account. But that's actually "wrong".
It is wrong because Gauss divergence theorem is tightly connected to the actually flow integral - we have an equation of integrals! So if we transform $int_V div(v)dmu$ to let's say spherical coordinates, we'd have to transform $int_{partial V}vcdot n do$ too.
Would we do that, we would see the determinant of the jacobian matrix of the coordination change show up on both sides: We can cancel it out.
Or simply: The flow does not depend on the chosen coordinate system.
Is that observation correct? Could I just cancel the determinant of the jacobian matrix of my coordination change?
vector-analysis
asked Dec 28 '18 at 15:47
xotixxotix
291411
$endgroup$
Gauss's divergence theorem:
Let $V$ be a compact subset of $mathbb R^3$ with a piecewise smooth boundary $partial V in C^1_{pw}$.
Further, let $v$ be a continuous differentiable vector field on $V$. Then we have:
$$int_{partial V}vcdot n do=int_V div(v)dmu$$
wheras $n$ denotes the outwards pointing normal along $partial V$.
Question:
So I hope I can formulate my question correctly. When I first started calculating integrals using Gauss's convergence theorem, I did a lot of mistakes when changing the coordinate system. E.g. if I have to calculate the flow of a simple vector field over the unit sphere I'd use spherical coordinates and of course, since I just changed the coordinate system, I'd take the determinant of the jacobian matrix into account. But that's actually "wrong".
It is wrong because Gauss divergence theorem is tightly connected to the actually flow integral - we have an equation of integrals! So if we transform $int_V div(v)dmu$ to let's say spherical coordinates, we'd have to transform $int_{partial V}vcdot n do$ too.
Would we do that, we would see the determinant of the jacobian matrix of the coordination change show up on both sides: We can cancel it out.
Or simply: The flow does not depend on the chosen coordinate system.
Is that observation correct? Could I just cancel the determinant of the jacobian matrix of my coordination change?
vector-analysis
vector-analysis
asked Dec 28 '18 at 15:47
xotixxotix
291411
asked Dec 28 '18 at 15:47
xotixxotix
291411
asked Dec 28 '18 at 15:47
xotixxotix
291411
asked Dec 28 '18 at 15:47
xotixxotix
291411
asked Dec 28 '18 at 15:47
xotixxotix
291411
291411
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%2f3055001%2fgausss-divergence-theorem-and-change-of-coordination-system%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%2f3055001%2fgausss-divergence-theorem-and-change-of-coordination-system%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
