Please confirm problem in Spivak

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SUMMARY

The discussion centers on a specific problem from Spivak's "Calculus on Manifolds," particularly problem 2-33, which questions the necessity of the continuity of the first partial derivative D1f^{i} at point 'a' in relation to Theorem 2-8. Participants clarify that the problem is not a typo, indicating that the continuity of any partial derivative is not required. This conclusion emphasizes the robustness of the theorem under consideration, allowing for more flexibility in the continuity conditions of the derivatives involved.

PREREQUISITES
  • Understanding of multivariable calculus concepts, particularly partial derivatives.
  • Familiarity with the content and structure of Spivak's "Calculus on Manifolds."
  • Knowledge of theorems related to continuity and differentiability in mathematical analysis.
  • Ability to interpret mathematical notation and terminology used in advanced calculus.
NEXT STEPS
  • Review the implications of Theorem 2-8 in Spivak's text.
  • Study the properties of continuity in multivariable functions.
  • Explore examples of theorems that relax continuity conditions for derivatives.
  • Practice solving similar problems from "Calculus on Manifolds" to reinforce understanding.
USEFUL FOR

Students of advanced calculus, mathematicians, and educators seeking clarity on continuity conditions in multivariable analysis, particularly those using Spivak's "Calculus on Manifolds" as a reference.

krcmd1
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Working my way through Spivak "Calculus on Manifolds."

On p. 34, problem 2-33, the problem asks "show that the continuity of D1f^{i} at a can be eliminated from the hypothesis of Theorem 2-8.

Is this a typo? Is he saying that there is no need for continuity of ONE partial derivative, or no need for continuity of ANY of the partial derivatives?

If the latter, please don't give me the answer yet; I'll struggle with it some more.

I've been assuming this is a typo, and should be Di and not D1; haven't been able to do it yet though.


Thank you!

Ken Cohen
 
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I apologize for the ambiguity in my original posting.

Is it then that there was a typo?

thanks

Ken Cohen
 
[edit] deleted

not a typo i don't think. try ..um...
 

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