Calculating Surface and Volume Integrals on a Sphere: A Problem-Based Approach

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Homework Help Overview

The discussion revolves around calculating surface and volume integrals on a sphere, specifically focusing on the surface integral of the vector r and the volume integral of the divergence of r. Participants are exploring the mathematical implications of these integrals in the context of vector calculus.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants are considering the integration of the vector r and discussing the relationship between the surface integral and the volume integral. There are questions about the correct interpretation of r and its representation in different forms, such as in Cartesian coordinates versus spherical coordinates.

Discussion Status

The discussion is ongoing, with participants raising questions about the integration process and the distinction between the vector r and the scalar r. Some guidance has been offered regarding the integration with respect to the unit normal vector, and there is a suggestion that the first part of the problem may not require an integral at all.

Contextual Notes

Participants are navigating potential confusion between the vector and scalar representations of r, and there is an emphasis on understanding the geometric interpretation of the integrals involved. The problem context includes specific constraints related to the sphere's radius and center.

Noone1982
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"Find the surface integral of r over a surface of a sphere of radius and center at the origin. Also find the volume integral of Gradient•R and compare your results"


Do I just integrate r to get (1/2)r^2 and plug some limits in since the r-hats equal one?
 
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Noone1982 said:
"Find the surface integral of r over a surface of a sphere of radius and center at the origin. Also find the volume integral of Gradient•R and compare your results"


Do I just integrate r to get (1/2)r^2 and plug some limits in since the r-hats equal one?

Well, it must give you what the vector r is right?

[tex]\iint\limits_{S}\mathbf{F}dS[/tex]

where S is the surface area of the sphere and F=r.
 
In xyz, r would equal R = x x^ + y y^ + z z^ but I'm just using R = r r^r to be simpler.
 
Be sure to distinguish the VECTOR r from the variable r. The vector r is the vector from the origin to the point on the sphere (of radius R?). You will actually be integrating that with the unit normal vector. Since the unit normal vector to the surface of the sphere and r are in the same direction, that is just the length of r. Hmmm, for the first part of this problem you don't actually have to do an integral at all!
 

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