How to Evaluate Surface Integrals Using Divergence Theorem?

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Discussion Overview

The discussion revolves around evaluating surface integrals using the Divergence Theorem, specifically in the context of two geometric shapes: a unit cube and a sphere. Participants express confusion regarding the approach to surface integrals and seek clarification on the application of the Divergence Theorem.

Discussion Character

  • Homework-related
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about surface integrals and requests an overview and guidance on specific problems involving a cube and a sphere.
  • Another participant suggests that knowing the equations for the surfaces may not be necessary and emphasizes the use of the Divergence Theorem and basic space geometry.
  • A participant indicates they have not started the problems and is unsure how to express the planes of the cube as equations, questioning whether they need to calculate the divergence or take a different approach.
  • Another participant reassures that only the Divergence Theorem is needed and advises setting up the integral without the need for parametrization of the surface.

Areas of Agreement / Disagreement

Participants generally agree on the relevance of the Divergence Theorem for solving the problems, but there is disagreement on the necessity of knowing the equations for the surfaces and how to approach the problems, indicating a lack of consensus on the best method to proceed.

Contextual Notes

Participants express uncertainty regarding the initial steps to take in evaluating the surface integrals, including the need for surface equations and the application of the Divergence Theorem. There are unresolved questions about the participants' understanding of the concepts involved.

Who May Find This Useful

This discussion may be useful for students struggling with surface integrals, the Divergence Theorem, and those seeking clarification on geometric interpretations in multivariable calculus.

Hoofbeat
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Please help! I'm soo confused with surface integrals and have several to do by tues for my tutorial. I don't really understand how to approach surface integrals! :cry: Could someone give me an over-view and help me through the question below - hopefully then I can manage the rest myself :cool: (Btw $ = integral sign)

Q. If n is the unit normal to the surface S, evaluate $$ r.n dS over:
a) the unit cube bounded by the coordinate planes and the the planes x=1, y=1 & z=1;
b) the surface of a sphere of radius a centred on the origin.

I *think* that I have to start by finding an equation for the cube (how?!) and then using divergence theorem but I really have no idea what I'm doing (I couldn't follow the lectures and books confuse me with their notation! :rolleyes:

Thanks
 
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I guess you needn't really know the "equations" for the surfaces of the cube and the sphere.Just some calculus,Gauss-Ostrogradski theorem & some basic space geometry...

How about posting your work...?

Daniel.
 
dextercioby said:
How about posting your work...?

That's the thing, I haven't done any of it! I'm genuinely confused as to even start the problem. I'm thinking I have to consider each face of the cube, but I really don't work well in 3D and can't even think how to express each plane as an equation! Then how do I progress? Do I have to calculate the Div or do it a different way :rolleyes: I thought I was following the lectures ok as the first few were fine and I could do the respective problem sheets but with the last few lectures on surface integrals & Divergance I've been so confused! :cry:
 
All you need is the Divergence (or Gauss's/Ostrogradksy's) theorem.
Try setting up the integral first. You don't have to parametrize the surface.
 

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