# Spivak Calculus on Manifolds and Epsilon delta proofs

• Calculus
• unintuit
In summary, the person is looking for a book or course that will teach him how to use epsilon-delta proofs and also how to calculate with them. They are looking for a more in-depth understanding of the approach rather than just a cursory understanding. They have read Apostol Vol.1 and Spivak and found them to be inadequate, and are currently working through Rudin's P.M.A. to remedy the deficiency. They are also working through Artin's Algebra book.

#### 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?

unintuit said:
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.

• vanhees71
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.

## 1. What is Spivak Calculus on Manifolds?

Spivak Calculus on Manifolds is an advanced mathematics textbook written by Michael Spivak, which covers multivariable calculus, differential forms, and integrals on manifolds. It is commonly used in upper-level undergraduate and graduate mathematics courses.

## 2. What are Epsilon delta proofs?

Epsilon delta proofs are a method of mathematical proof used to show the limit of a function as it approaches a certain value. The "epsilon" represents a small distance from the limit value, and the "delta" represents a corresponding distance from the input value. By choosing a small enough epsilon, it can be shown that the function values will always be within that distance of the limit value as long as the input values are within a certain distance delta.

## 3. Is Spivak Calculus on Manifolds suitable for beginners?

No, Spivak Calculus on Manifolds is not suitable for beginners. It assumes a strong foundation in single-variable calculus, linear algebra, and basic mathematical proof techniques. It is typically used in advanced undergraduate or graduate level courses.

## 4. What topics are covered in Spivak Calculus on Manifolds?

The textbook covers topics such as multivariable calculus, differential forms, manifolds, vector fields, integration, and Stokes' theorem. It also includes a variety of exercises and problems to reinforce understanding of the material.

## 5. How can I use Spivak Calculus on Manifolds in my research or career?

Spivak Calculus on Manifolds is a valuable resource for mathematicians, physicists, and engineers who work with advanced mathematical concepts and need a solid understanding of multivariable calculus and differential forms. It can also be used as a reference for those studying topics such as differential geometry, topology, and mathematical physics.