Spivak Calculus on Manifolds and Epsilon delta proofs

[unintuit]

I am currently having some issue understanding, what you may find trivial, epsilon-delta proofs. I have worked through Apostol Vol.1 and ran through Spivak and I found Apostol just uses neighborhoods in proofs instead of the epsilon-delta approach, while nesting neighborhoods is 'acceptable' I would like to correct the weakness of understanding the epsilon-delta approach, to be even more specific I would like to be able to calculate with it and do manipulations. I have found Spivak to be inadequate for teaching this and was hoping you would know specifically if, say Ross or Pugh's book teaches this. I have read the first two chapters of Rudin's P.M.A. and found the exercises in the first chapter extremely difficult and decided I had some gaps in my knowledge to correct. As of this moment I would like to remedy this deficiency and then move on the Spivak's Calculus on Manifolds but I am interested in working through Rudin's P.M.A. prior. I am also currently working through Artin's Algebra book as well for whatever that is worth.

Also I have one question about Spivak's Calculus on Manifolds book. I have not learned directional derivatives and understand that these are left as exercises in his book, which would make one think these are not that important whereas he focuses on total derivatives or what you may name them. Therefore I am asking if it is worthwhile to pursue the topic of directional derivatives from another source or just learn solely from Spivak's book?

Thank you for your time

 

[Stephen Tashi]

I would like to correct the weakness of understanding the epsilon-delta approach, to be even more specific I would like to be able to calculate with it and do manipulations.

I'm unfamilar with the books you mention, but I'm curious about what you mean by "calculate and do manipulations". Do you understand how to translate a proof written in terms of neighborhoods into a proof using epsilon-delta's? I suggest you try that exercise a few times. (If you do, you'll start to see why proofs using neigborhoods are often the simpler method of writing - and the simpler method of thinking.

The elementary type of epsilon-delta proof begins "Suppose we are given \epsilon > 0" and the body of the proof is sometimes written backwards (-very often written backwards by students). Working backwards means "solving for \delta". In sophisticated proofs, it is often not possible to solve for \delta by algebraic manipulations. Sophisticated proofs can rely on the clever use of inequalities instead of the solution of equations (or inequations).

If you are reading epsilon-delta proofs, you can't expect them all to take the "solve for \delta" approach. Some wil say things ("out of the blue") like "Let \delta = \min( \frac{ \epsilon^2}{a} , \epsilon +2 ). Then they prove this choice of \delta works. There need be no story of how the choice of \delta was deduced.

 

[mathwonk]

directional derivatives are important but he treats the special case of partial derivatives in the text. and in spivak, things in the exercises are not unimportant.