Differential for surface of revolution

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Discussion Overview

The discussion revolves around the differential of a function defined on a surface of revolution generated by revolving a curve in a plane around a line. Participants explore the implications of the differential being non-zero on the surface and challenge the assumptions made in a referenced text regarding the conditions under which the differential is zero or non-zero.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • Some participants question the claim that the differential dg is never zero on the surface of revolution M, suggesting that it may be zero on the submanifold defined by the level curve of the function g.
  • Others argue that dg can be non-zero when acting on vectors not tangent to the surface, while it may return zero on tangent vectors.
  • A participant presents a specific case where the curve is defined by y=h(x) and claims that dg is non-zero everywhere under certain conditions.
  • Counter-examples are provided to challenge the assertion that dg is always non-zero, with participants emphasizing that the conditions of the function f are crucial to the outcome.
  • Some participants assert that the theorem referenced in the text is not incorrect, but the implications drawn from it regarding dg and the surface of revolution are debated.
  • There is a discussion about the application of the chain rule and its implications for the differential dg in the context of the surface of revolution.
  • Participants express concern that if the surface of revolution does not always yield a non-zero differential, it could undermine other material presented in the referenced text.

Areas of Agreement / Disagreement

Participants do not reach a consensus; multiple competing views remain regarding the behavior of the differential dg on the surface of revolution and the validity of the claims made in the referenced text.

Contextual Notes

Limitations include the dependence on the specific definitions of the function f and the assumptions about the curve's relationship with the axes. The discussion highlights unresolved mathematical steps and varying interpretations of the theorem regarding the differential and surfaces.

  • #31
Regarding post #30, yes I see that xy=0 gives you the union of the x and y axes, hardly a smooth curve.
 

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