Vector Calculus Question about Surface Integrals

Click For Summary

Discussion Overview

The discussion revolves around the evaluation of surface integrals of different force fields, specifically comparing a force field defined as z^2 and another defined as z. Participants explore the implications of these definitions on the resulting flux when integrated over a sphere of radius a.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions why the flux calculated for the force field z^2 over a sphere yields the same result as for the force field z, despite the apparent difference in the force definitions.
  • Another participant emphasizes that the force field must be a vector function, suggesting that the original question may have misrepresented the nature of the force fields involved.
  • There is confusion regarding the terminology used, with one participant asking for clarification on whether "polar coordinates" were meant to refer to spherical or cylindrical coordinates.
  • Concerns are raised about the physical interpretation of the flux, with a participant asking for clarification on the direction of the force field and the physical quantity being calculated.

Areas of Agreement / Disagreement

Participants express differing views on the nature of the force fields and the implications for the flux calculations. There is no consensus on the correctness of the original question or the assumptions made about the force fields.

Contextual Notes

Participants note potential misunderstandings regarding the definitions of force fields and the coordinate systems used in the calculations, which may affect the interpretation of the results.

Conaissance99
Messages
2
Reaction score
0
Why is it that when the force field is z^2 and you take the surface integral over a sphere of radius a using spherical coordinates, that yields the flux to be (4pi a^3 )/ 3


BUT in a calculus book, the force field is z instead of z^2 evaluated using polar coordinates and it yields the same amount of flux, (4pi a^3 )/ 3.

How can this be when the force is different (z^2 instead of z?) Isn't it when you for example, get five times the force, like 5z you would get the answer multiplied by a factor of 5. When you square z it should come out to be different shouldn't it?

Any help greatly appreciated. Thanks in advance
 
Last edited:
Physics news on Phys.org
Please note that this is not a homework question. Simply a question that if you change the value of the force in your surface integral in this case, shouldn't the answer be different?
 
Sorry I don't understand the question. What exactly is the integral being calculated? Alternatively, what exactly is the physical quantity being calculated? If it is the flux of a force field across the sphere, then what is the force field? You need to say what direction it is pointing.
 
Conaissance99 said:
Why is it that when the force field is z^2 and you take the surface integral over a sphere of radius a using spherical coordinates, that yields the flux to be (4pi a^3 )/ 3


BUT in a calculus book, the force field is z instead of z^2 evaluated using polar coordinates and it yields the same amount of flux, (4pi a^3 )/ 3.

How can this be when the force is different (z^2 instead of z?) Isn't it when you for example, get five times the force, like 5z you would get the answer multiplied by a factor of 5. When you square z it should come out to be different shouldn't it?

Any help greatly appreciated. Thanks in advance
First, this doesn't make sense. Force is a vector quantity and the force field must be a vector function, not scalar. I will assume you mean something like <0, 0, z>. In that case, "the force field is z instead of z^2 evaluated using polar coordinates and it yields the same amount of flux, (4pi a^3 )/ 3" is incorrect. The integral over the top part of the sphere, z> 0, will cancel the integral over the bottom part, t< 0, and the integral is 0.
 
Conaissance99 said:
Why is it that when the force field is z^2 and you take the surface integral over a sphere of radius a using spherical coordinates, that yields the flux to be (4pi a^3 )/ 3


BUT in a calculus book, the force field is z instead of z^2 evaluated using polar coordinates and it yields the same amount of flux, (4pi a^3 )/ 3.

How can this be when the force is different (z^2 instead of z?) Isn't it when you for example, get five times the force, like 5z you would get the answer multiplied by a factor of 5. When you square z it should come out to be different shouldn't it?

Any help greatly appreciated. Thanks in advance

What do you mean by polar coordinates? Did you mean spherical, cylindrical?
 

Similar threads

Undergrad Struggling with vector calculus identity used in E&M derivation
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Confirming my knowledge on surface integrals
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Equivalence of alternative definitions of conservative vector fields and line integrals in different metric spaces
  • · Replies 3 ·
Replies
3
Views
3K
MHB Evaluate the surface integral $\iint\limits_{\sum}f\cdot d\sigma$
  • · Replies 1 ·
Replies
1
Views
3K
Insights Demystifying Parameterization and Surface Integrals
  • · Replies 3 ·
Replies
3
Views
4K
Undergrad Question about Integrals to Determine Volume vs Surface Area
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Is interchanging the order of the surface and volume integrals valid here?
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Determine the flux of the vector field trough the surface
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Substitution in a definite integral
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad Stokes' theorem and surface integrals
  • · Replies 5 ·
Replies
5
Views
2K