Coordination change for integrals and determinant of the jacobian matrix












1












$begingroup$


so I am currently learning about flow integrals and my book had an example which confused me. (Example 1) As you see, in Example 1, they don't care about the determinant of the jacobian of the coordination change. They just switch coordinats without transforming the integral properly. In Example 2 they actually use the determinant of the jacobian of the coordination change.



Let me make two examples:



Example 1:
Calculate the flow of $v=(0,0,1-z)$ from bottom up through



$$H={(x,y,z)in mathbb R^3 | x^2+y^2+z^2=1, z>0}$$



Parametrization:



$$Phi:[0,2pi]times[0,pi/2]tomathbb R^3, quad (u,v)mapsto begin{pmatrix}sin vcos u \ sin vsin u\ cos vend{pmatrix}$$



Normal vector:
$$Phi_vtimes Phi_u = begin{pmatrix}sin^2 vcos u \ sin vsin^2 u\ sin v cos vend{pmatrix}$$



Flow integraL:
$int_H vcdot ndo = int_0^{2pi}duint_0^{pi/2}dv begin{pmatrix}0\0\1-cos vend{pmatrix}cdotbegin{pmatrix}sin^2 vcos u \ sin vsin^2 u\ sin v cos vend{pmatrix}$



$=2piint_0^{pi/2}dv sin v cos v(1-cos v)$



$=2piint_0^{pi/2}dv sin v cos v - 2pi int_0^{pi/2}dv sin v cos^2 v=...=pi/3$



Example 2: We want to calculate the flow of $v=(xz,z,y)$ through the unit ball, centered around $O=(0,0,0)$ using Gauss's Divergene Theorem.



We have to calculate $int_S vcdot n do = int_{B_1} div(v) dmu$



$div(v)=z$



We use spherical coordinates. So: $dxdydz=r^2drsinphi dphi dvarphi$



$int_s v cdot n do = int_{B_1} zdmu = int_0^1 r^2 dr int_0^pi sinphi dphi int_0^{2pi}dvarphi cdot rcos phi=...=0$










share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    The jacobian is absorbed into $Phi_vtimes Phi_u$. If you go through the derivation of the formula, you will see where it arises
    $endgroup$
    – Matematleta
    Dec 31 '18 at 15:45










  • $begingroup$
    Yeah I just went through everything again and the normal vector$n$ absorbs it. So if we calculate a given flow using Gauss's Divergence Theorem, we actually have to take it into account but if we use the normal definition, we don't - right?
    $endgroup$
    – xotix
    Dec 31 '18 at 15:59
















1












$begingroup$


so I am currently learning about flow integrals and my book had an example which confused me. (Example 1) As you see, in Example 1, they don't care about the determinant of the jacobian of the coordination change. They just switch coordinats without transforming the integral properly. In Example 2 they actually use the determinant of the jacobian of the coordination change.



Let me make two examples:



Example 1:
Calculate the flow of $v=(0,0,1-z)$ from bottom up through



$$H={(x,y,z)in mathbb R^3 | x^2+y^2+z^2=1, z>0}$$



Parametrization:



$$Phi:[0,2pi]times[0,pi/2]tomathbb R^3, quad (u,v)mapsto begin{pmatrix}sin vcos u \ sin vsin u\ cos vend{pmatrix}$$



Normal vector:
$$Phi_vtimes Phi_u = begin{pmatrix}sin^2 vcos u \ sin vsin^2 u\ sin v cos vend{pmatrix}$$



Flow integraL:
$int_H vcdot ndo = int_0^{2pi}duint_0^{pi/2}dv begin{pmatrix}0\0\1-cos vend{pmatrix}cdotbegin{pmatrix}sin^2 vcos u \ sin vsin^2 u\ sin v cos vend{pmatrix}$



$=2piint_0^{pi/2}dv sin v cos v(1-cos v)$



$=2piint_0^{pi/2}dv sin v cos v - 2pi int_0^{pi/2}dv sin v cos^2 v=...=pi/3$



Example 2: We want to calculate the flow of $v=(xz,z,y)$ through the unit ball, centered around $O=(0,0,0)$ using Gauss's Divergene Theorem.



We have to calculate $int_S vcdot n do = int_{B_1} div(v) dmu$



$div(v)=z$



We use spherical coordinates. So: $dxdydz=r^2drsinphi dphi dvarphi$



$int_s v cdot n do = int_{B_1} zdmu = int_0^1 r^2 dr int_0^pi sinphi dphi int_0^{2pi}dvarphi cdot rcos phi=...=0$










share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    The jacobian is absorbed into $Phi_vtimes Phi_u$. If you go through the derivation of the formula, you will see where it arises
    $endgroup$
    – Matematleta
    Dec 31 '18 at 15:45










  • $begingroup$
    Yeah I just went through everything again and the normal vector$n$ absorbs it. So if we calculate a given flow using Gauss's Divergence Theorem, we actually have to take it into account but if we use the normal definition, we don't - right?
    $endgroup$
    – xotix
    Dec 31 '18 at 15:59














1












1








1





$begingroup$


so I am currently learning about flow integrals and my book had an example which confused me. (Example 1) As you see, in Example 1, they don't care about the determinant of the jacobian of the coordination change. They just switch coordinats without transforming the integral properly. In Example 2 they actually use the determinant of the jacobian of the coordination change.



Let me make two examples:



Example 1:
Calculate the flow of $v=(0,0,1-z)$ from bottom up through



$$H={(x,y,z)in mathbb R^3 | x^2+y^2+z^2=1, z>0}$$



Parametrization:



$$Phi:[0,2pi]times[0,pi/2]tomathbb R^3, quad (u,v)mapsto begin{pmatrix}sin vcos u \ sin vsin u\ cos vend{pmatrix}$$



Normal vector:
$$Phi_vtimes Phi_u = begin{pmatrix}sin^2 vcos u \ sin vsin^2 u\ sin v cos vend{pmatrix}$$



Flow integraL:
$int_H vcdot ndo = int_0^{2pi}duint_0^{pi/2}dv begin{pmatrix}0\0\1-cos vend{pmatrix}cdotbegin{pmatrix}sin^2 vcos u \ sin vsin^2 u\ sin v cos vend{pmatrix}$



$=2piint_0^{pi/2}dv sin v cos v(1-cos v)$



$=2piint_0^{pi/2}dv sin v cos v - 2pi int_0^{pi/2}dv sin v cos^2 v=...=pi/3$



Example 2: We want to calculate the flow of $v=(xz,z,y)$ through the unit ball, centered around $O=(0,0,0)$ using Gauss's Divergene Theorem.



We have to calculate $int_S vcdot n do = int_{B_1} div(v) dmu$



$div(v)=z$



We use spherical coordinates. So: $dxdydz=r^2drsinphi dphi dvarphi$



$int_s v cdot n do = int_{B_1} zdmu = int_0^1 r^2 dr int_0^pi sinphi dphi int_0^{2pi}dvarphi cdot rcos phi=...=0$










share|cite|improve this question









$endgroup$




so I am currently learning about flow integrals and my book had an example which confused me. (Example 1) As you see, in Example 1, they don't care about the determinant of the jacobian of the coordination change. They just switch coordinats without transforming the integral properly. In Example 2 they actually use the determinant of the jacobian of the coordination change.



Let me make two examples:



Example 1:
Calculate the flow of $v=(0,0,1-z)$ from bottom up through



$$H={(x,y,z)in mathbb R^3 | x^2+y^2+z^2=1, z>0}$$



Parametrization:



$$Phi:[0,2pi]times[0,pi/2]tomathbb R^3, quad (u,v)mapsto begin{pmatrix}sin vcos u \ sin vsin u\ cos vend{pmatrix}$$



Normal vector:
$$Phi_vtimes Phi_u = begin{pmatrix}sin^2 vcos u \ sin vsin^2 u\ sin v cos vend{pmatrix}$$



Flow integraL:
$int_H vcdot ndo = int_0^{2pi}duint_0^{pi/2}dv begin{pmatrix}0\0\1-cos vend{pmatrix}cdotbegin{pmatrix}sin^2 vcos u \ sin vsin^2 u\ sin v cos vend{pmatrix}$



$=2piint_0^{pi/2}dv sin v cos v(1-cos v)$



$=2piint_0^{pi/2}dv sin v cos v - 2pi int_0^{pi/2}dv sin v cos^2 v=...=pi/3$



Example 2: We want to calculate the flow of $v=(xz,z,y)$ through the unit ball, centered around $O=(0,0,0)$ using Gauss's Divergene Theorem.



We have to calculate $int_S vcdot n do = int_{B_1} div(v) dmu$



$div(v)=z$



We use spherical coordinates. So: $dxdydz=r^2drsinphi dphi dvarphi$



$int_s v cdot n do = int_{B_1} zdmu = int_0^1 r^2 dr int_0^pi sinphi dphi int_0^{2pi}dvarphi cdot rcos phi=...=0$







calculus integration






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 31 '18 at 15:23









xotixxotix

291411




291411








  • 1




    $begingroup$
    The jacobian is absorbed into $Phi_vtimes Phi_u$. If you go through the derivation of the formula, you will see where it arises
    $endgroup$
    – Matematleta
    Dec 31 '18 at 15:45










  • $begingroup$
    Yeah I just went through everything again and the normal vector$n$ absorbs it. So if we calculate a given flow using Gauss's Divergence Theorem, we actually have to take it into account but if we use the normal definition, we don't - right?
    $endgroup$
    – xotix
    Dec 31 '18 at 15:59














  • 1




    $begingroup$
    The jacobian is absorbed into $Phi_vtimes Phi_u$. If you go through the derivation of the formula, you will see where it arises
    $endgroup$
    – Matematleta
    Dec 31 '18 at 15:45










  • $begingroup$
    Yeah I just went through everything again and the normal vector$n$ absorbs it. So if we calculate a given flow using Gauss's Divergence Theorem, we actually have to take it into account but if we use the normal definition, we don't - right?
    $endgroup$
    – xotix
    Dec 31 '18 at 15:59








1




1




$begingroup$
The jacobian is absorbed into $Phi_vtimes Phi_u$. If you go through the derivation of the formula, you will see where it arises
$endgroup$
– Matematleta
Dec 31 '18 at 15:45




$begingroup$
The jacobian is absorbed into $Phi_vtimes Phi_u$. If you go through the derivation of the formula, you will see where it arises
$endgroup$
– Matematleta
Dec 31 '18 at 15:45












$begingroup$
Yeah I just went through everything again and the normal vector$n$ absorbs it. So if we calculate a given flow using Gauss's Divergence Theorem, we actually have to take it into account but if we use the normal definition, we don't - right?
$endgroup$
– xotix
Dec 31 '18 at 15:59




$begingroup$
Yeah I just went through everything again and the normal vector$n$ absorbs it. So if we calculate a given flow using Gauss's Divergence Theorem, we actually have to take it into account but if we use the normal definition, we don't - right?
$endgroup$
– xotix
Dec 31 '18 at 15:59










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


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3057803%2fcoordination-change-for-integrals-and-determinant-of-the-jacobian-matrix%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.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3057803%2fcoordination-change-for-integrals-and-determinant-of-the-jacobian-matrix%23new-answer', 'question_page');
}
);

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







Popular posts from this blog

Probability when a professor distributes a quiz and homework assignment to a class of n students.

Aardman Animations

Are they similar matrix