Differential for surface of revolution

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SUMMARY

The discussion centers on the proposition from O'Neill's "Elementary Differential Geometry" regarding the differential of a function defined on a surface of revolution. The participants debate the validity of the claim that the differential dg is never zero on the surface M generated by revolving a profile curve C around an axis A. Counterexamples are provided, demonstrating that dg can indeed equal zero for certain functions, particularly when the function is constant in some regions. The conclusion is that while the theorem states that dg ≠ 0 implies M is a surface, the converse does not hold true, as dg can be zero on a level surface.

PREREQUISITES
  • Understanding of differential geometry concepts, particularly surfaces of revolution.
  • Familiarity with the chain rule in multivariable calculus.
  • Knowledge of level surfaces and their properties in R3.
  • Basic understanding of parametrization of curves and surfaces.
NEXT STEPS
  • Study the implications of the chain rule in multivariable calculus, particularly in the context of differential forms.
  • Explore the properties of level surfaces and conditions under which dg can be zero.
  • Investigate the parametrization of surfaces of revolution and their geometric interpretations.
  • Review O'Neill's "Elementary Differential Geometry" for further insights on the definitions and theorems related to surfaces.
USEFUL FOR

Mathematicians, students of differential geometry, and anyone interested in the properties of surfaces and their differentials, particularly in the context of surfaces of revolution.

  • #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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