- #31
Gene Naden
- 320
- 64
Regarding post #30, yes I see that xy=0 gives you the union of the x and y axes, hardly a smooth curve.
Undergrad Differential for surface of revolution
- Thread starter Gene Naden
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The discussion centers on a proposition from O'Neill's Elementary Differential Geometry regarding the differential of a function defining a surface of revolution. The participants debate whether the differential \( dg \) can be zero on the surface \( M \) generated by revolving a curve \( C \) around an axis \( A \). They argue that while \( dg \neq 0 \) is necessary for \( M \) to be a surface, it is not sufficient, as counterexamples exist where \( dg = 0 \) on level surfaces. Additionally, the conversation highlights the importance of the function \( f \) defining the curve, noting that not all functions yielding the same level curves will maintain a non-zero differential. The conclusion emphasizes that the surface of revolution is valid, but the conditions for \( dg \) must be carefully considered.
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