Integral of a 2-Form Over a Certain Region of Integration












0














This is a rather simple problem, but one that I'm struggling with nonetheless. I'm given a 2-Form, $beta = zdx wedge dy-x^2dy wedge dz$ that I need to integrate over the surface S : $z=4-x^2-y^2 ge 0 $.



Parametrising the surface, I have:



$G(x,y) = < x,y,4-x^2-y^2>$ and the Jacobian: $$ begin{bmatrix} DG end{bmatrix} = begin {bmatrix}1 & 0 \0 & 1 \ -2x & -2y end{bmatrix}$$



Evaluating the 2-Form, I have $int_Sbeta= iint_{z=4-x^2-y^2} (4-x^2-y^2) begin{vmatrix}1&0 \ 0&1 end {vmatrix} - x^2 begin{vmatrix} 0 & 1\ -2x & -2y end{vmatrix}dxdy$



This simplifies to: $int_Sbeta = iint_{z=4-x^2-y^2} 4 -x^2-y^2-2x^3dxdy$



In determining the region of integration, though, I get stuck. My gut is telling me to go with polar coordinates. If this were the case, I know $0 le theta le 2{pi}$, but I'm uncertain of the limits of integration for $r$. Would it be $0 le r le 2$? I don't think this is correct but am uncertain of any other way you'd do it.










share|cite|improve this question






















  • Your intuition is correct - if you want to make sure, just plot all the points in the integration region, and you will recognize it is a circle of radius 2, so the above parameterization would work.
    – Neeyanth Kopparapu
    Dec 1 at 1:35










  • So basically this is saying that r spans from 0 to 2. But here, my region of integration is a paraboloid. Say my region of integration was a circle of radius 2. The limits of integration would be the same as they are for my paraboloid, but they are clearly different shapes.
    – Jackson Joffe
    Dec 1 at 18:21
















0














This is a rather simple problem, but one that I'm struggling with nonetheless. I'm given a 2-Form, $beta = zdx wedge dy-x^2dy wedge dz$ that I need to integrate over the surface S : $z=4-x^2-y^2 ge 0 $.



Parametrising the surface, I have:



$G(x,y) = < x,y,4-x^2-y^2>$ and the Jacobian: $$ begin{bmatrix} DG end{bmatrix} = begin {bmatrix}1 & 0 \0 & 1 \ -2x & -2y end{bmatrix}$$



Evaluating the 2-Form, I have $int_Sbeta= iint_{z=4-x^2-y^2} (4-x^2-y^2) begin{vmatrix}1&0 \ 0&1 end {vmatrix} - x^2 begin{vmatrix} 0 & 1\ -2x & -2y end{vmatrix}dxdy$



This simplifies to: $int_Sbeta = iint_{z=4-x^2-y^2} 4 -x^2-y^2-2x^3dxdy$



In determining the region of integration, though, I get stuck. My gut is telling me to go with polar coordinates. If this were the case, I know $0 le theta le 2{pi}$, but I'm uncertain of the limits of integration for $r$. Would it be $0 le r le 2$? I don't think this is correct but am uncertain of any other way you'd do it.










share|cite|improve this question






















  • Your intuition is correct - if you want to make sure, just plot all the points in the integration region, and you will recognize it is a circle of radius 2, so the above parameterization would work.
    – Neeyanth Kopparapu
    Dec 1 at 1:35










  • So basically this is saying that r spans from 0 to 2. But here, my region of integration is a paraboloid. Say my region of integration was a circle of radius 2. The limits of integration would be the same as they are for my paraboloid, but they are clearly different shapes.
    – Jackson Joffe
    Dec 1 at 18:21














0












0








0







This is a rather simple problem, but one that I'm struggling with nonetheless. I'm given a 2-Form, $beta = zdx wedge dy-x^2dy wedge dz$ that I need to integrate over the surface S : $z=4-x^2-y^2 ge 0 $.



Parametrising the surface, I have:



$G(x,y) = < x,y,4-x^2-y^2>$ and the Jacobian: $$ begin{bmatrix} DG end{bmatrix} = begin {bmatrix}1 & 0 \0 & 1 \ -2x & -2y end{bmatrix}$$



Evaluating the 2-Form, I have $int_Sbeta= iint_{z=4-x^2-y^2} (4-x^2-y^2) begin{vmatrix}1&0 \ 0&1 end {vmatrix} - x^2 begin{vmatrix} 0 & 1\ -2x & -2y end{vmatrix}dxdy$



This simplifies to: $int_Sbeta = iint_{z=4-x^2-y^2} 4 -x^2-y^2-2x^3dxdy$



In determining the region of integration, though, I get stuck. My gut is telling me to go with polar coordinates. If this were the case, I know $0 le theta le 2{pi}$, but I'm uncertain of the limits of integration for $r$. Would it be $0 le r le 2$? I don't think this is correct but am uncertain of any other way you'd do it.










share|cite|improve this question













This is a rather simple problem, but one that I'm struggling with nonetheless. I'm given a 2-Form, $beta = zdx wedge dy-x^2dy wedge dz$ that I need to integrate over the surface S : $z=4-x^2-y^2 ge 0 $.



Parametrising the surface, I have:



$G(x,y) = < x,y,4-x^2-y^2>$ and the Jacobian: $$ begin{bmatrix} DG end{bmatrix} = begin {bmatrix}1 & 0 \0 & 1 \ -2x & -2y end{bmatrix}$$



Evaluating the 2-Form, I have $int_Sbeta= iint_{z=4-x^2-y^2} (4-x^2-y^2) begin{vmatrix}1&0 \ 0&1 end {vmatrix} - x^2 begin{vmatrix} 0 & 1\ -2x & -2y end{vmatrix}dxdy$



This simplifies to: $int_Sbeta = iint_{z=4-x^2-y^2} 4 -x^2-y^2-2x^3dxdy$



In determining the region of integration, though, I get stuck. My gut is telling me to go with polar coordinates. If this were the case, I know $0 le theta le 2{pi}$, but I'm uncertain of the limits of integration for $r$. Would it be $0 le r le 2$? I don't think this is correct but am uncertain of any other way you'd do it.







calculus integration multivariable-calculus polar-coordinates vector-fields






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 1 at 0:14









Jackson Joffe

575




575












  • Your intuition is correct - if you want to make sure, just plot all the points in the integration region, and you will recognize it is a circle of radius 2, so the above parameterization would work.
    – Neeyanth Kopparapu
    Dec 1 at 1:35










  • So basically this is saying that r spans from 0 to 2. But here, my region of integration is a paraboloid. Say my region of integration was a circle of radius 2. The limits of integration would be the same as they are for my paraboloid, but they are clearly different shapes.
    – Jackson Joffe
    Dec 1 at 18:21


















  • Your intuition is correct - if you want to make sure, just plot all the points in the integration region, and you will recognize it is a circle of radius 2, so the above parameterization would work.
    – Neeyanth Kopparapu
    Dec 1 at 1:35










  • So basically this is saying that r spans from 0 to 2. But here, my region of integration is a paraboloid. Say my region of integration was a circle of radius 2. The limits of integration would be the same as they are for my paraboloid, but they are clearly different shapes.
    – Jackson Joffe
    Dec 1 at 18:21
















Your intuition is correct - if you want to make sure, just plot all the points in the integration region, and you will recognize it is a circle of radius 2, so the above parameterization would work.
– Neeyanth Kopparapu
Dec 1 at 1:35




Your intuition is correct - if you want to make sure, just plot all the points in the integration region, and you will recognize it is a circle of radius 2, so the above parameterization would work.
– Neeyanth Kopparapu
Dec 1 at 1:35












So basically this is saying that r spans from 0 to 2. But here, my region of integration is a paraboloid. Say my region of integration was a circle of radius 2. The limits of integration would be the same as they are for my paraboloid, but they are clearly different shapes.
– Jackson Joffe
Dec 1 at 18:21




So basically this is saying that r spans from 0 to 2. But here, my region of integration is a paraboloid. Say my region of integration was a circle of radius 2. The limits of integration would be the same as they are for my paraboloid, but they are clearly different shapes.
– Jackson Joffe
Dec 1 at 18:21















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%2f3020839%2fintegral-of-a-2-form-over-a-certain-region-of-integration%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown






























active

oldest

votes













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














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3020839%2fintegral-of-a-2-form-over-a-certain-region-of-integration%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

Berounka

Sphinx de Gizeh

Different font size/position of beamer's navigation symbols template's content depending on regular/plain...