What Is the Role of Divergence in Vector Calculus?

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Discussion Overview

The discussion revolves around the role of divergence in vector calculus, particularly in the context of a mathematical exercise involving geometric shapes like spheres and cubes. Participants explore the implications of simplifying the problem and seek clarification on the application of divergence in these scenarios.

Discussion Character

  • Homework-related
  • Exploratory
  • Technical explanation

Main Points Raised

  • Some participants question whether it is permissible to simplify a problem involving divergence to a sphere centered at the origin, as opposed to a more complex shape like a cube.
  • There is a discussion about the transformation of vectors, with participants expressing uncertainty about the specifics of the transformation and its implications for applying the divergence operation.
  • One participant mentions having resolved their queries but seeks general background information rather than specific solutions to the exercise.
  • Another participant notes a relationship between volume increase and divergence, suggesting that they found the volume increase to be expressed as \(1 + \text{div} \, \vec{h}\), but seeks further validation of this understanding.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and uncertainty regarding the application of divergence in the context of the exercise. There is no clear consensus on the simplifications allowed or the implications of the divergence operation.

Contextual Notes

Participants have not fully resolved the mathematical steps involved in applying divergence to the given problem, and there are indications of missing assumptions regarding the geometric configurations discussed.

Physgeek64
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Got it. Thank you ;)
 
Last edited:
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Hi there,

Looks a bit like homework, for which we have a nice template !

Is it allowed to simplify this exercise to a sphere with the origin at the center ? Because if it is about a cube with the origin way outside of the cube, things seem complicated to me...

In the simple case ##\vec r ## goes to ## \vec r'= {\rm \ ?} ## so that ## \vec h = \vec r'- \vec r = {\rm \ ?} ## and that you can subject to the divergence operation (from its definition) , I should hope ?
 
BvU said:
Hi there,

Looks a bit like homework, for which we have a nice template !

Is it allowed to simplify this exercise to a sphere with the origin at the center ? Because if it is about a cube with the origin way outside of the cube, things seem complicated to me...

In the simple case ##\vec r ## goes to ## \vec r'= {\rm \ ?} ## so that ## \vec h = \vec r'- \vec r = {\rm \ ?} ## and that you can subject to the divergence operation (from its definition) , I should hope ?

Its okay, I have sorted out my queries. But thank you. I didn't know whether to post this in homework as it is not technically homework, as I am doing it a while in advance so I was hoping more for general background information than for specifics on how to do the question. :)
 
Please don't delete the original post -- it makes the thread incomprehensible !

Was your answer ##3\alpha## ?

What kind of background info did you have in mind ?
 
BvU said:
Please don't delete the original post -- it makes the thread incomprehensible !

Was your answer ##3\alpha## ?

What kind of background info did you have in mind ?
ahh I am sorry- I just don't know how to delete the post.
Kind of- I got the volume increase to be 1+divh, which is the same thing?
 
Physgeek64 said:
ahh I am sorry- I just don't know how to delete the post.
Well, effectively you did by overwriting it. Perhaps you can restore the original ?

Kind of- I got the volume increase to be 1+divh, which is the same thing?
Relative, I assume. Sounds good. Can you underpin it ?
 

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