Patch of a surface in spherical coordinates?

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Homework Help Overview

The discussion revolves around the calculation of surface area in spherical coordinates, specifically focusing on the integral expression for surface area and the relationship between surface area and volume in the context of infinitesimal thickness.

Discussion Character

  • Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the integral expression for surface area and question the relationship between surface area and volume when considering infinitesimal thickness. There is a discussion about the implications of dividing by thickness as it approaches zero.

Discussion Status

Some participants have provided guidance on the relationship between surface area and volume, suggesting that the original poster may have miscalculated by not properly dividing by the thickness. Multiple interpretations of the relationship between volumetric shells and surface area are being explored.

Contextual Notes

There is a mention of an accidental notation in the original poster's work, which may have contributed to confusion. The discussion also touches on the concept of limits as thickness approaches zero.

theBEAST
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Homework Statement


I am currently trying to prove:

S = ∫∫a2sinΦdΦdθ

Here is my work (note that in my work I use dS instead of S, this is an accident):
92zNaWh.jpg


I end up with:

S = ∫∫a*da2sinΦdΦdθ

Where da is the infinitesimal thickness of the surface.

Why am I getting the wrong answer?
 
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theBEAST said:

Homework Statement


I am currently trying to prove:

S = ∫∫a2sinΦdΦdθ

Here is my work (note that in my work I use dS instead of S, this is an accident):
92zNaWh.jpg


I end up with:

S = ∫∫a*da2sinΦdΦdθ

Where da is the infinitesimal thickness of the surface.

Why am I getting the wrong answer?

You haven't computed S. You've computed the volume V=S*da. To get S, divide by the da.
 
Dick said:
You haven't computed S. You've computed the volume V=S*da. To get S, divide by the da.

I thought that if the volumetric shell thickness approaches zero you end up with the surface area?
 
theBEAST said:
I thought that if the volumetric shell thickness approaches zero you end up with the surface area?

If the shell thickness approaches zero then you end up with zero volume. To get the surface area divide by the thickness as it approaches zero. Maybe that's what 'volumetric shell' means.
 
Last edited:

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