# Spivak Calculus on Manifolds

• unintuit
In summary, the topic of directional derivatives is left as exercises in Spivak's Calculus on Manifolds book, leading one to believe they are not important. However, it is recommended to supplement Spivak's book with other sources, such as the book "Multivariable Calculus" by James Stewart, which provides exercises and applications of directional derivatives. These derivatives, while a generalization of partial derivatives, still hold independent significance, such as determining the direction of maximum increase for a function.

#### unintuit

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 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?
If you would like further exercises on directional derivatives, including when they exist and why they are useful in applications, I highly recommend accompanying Spivak with https://www.amazon.com/dp/0130414085/?tag=pfamazon01-20. It will, at the very least, concretize many of the highly theoretical exercises in Spivak.

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Joker93
the most important directions are the unit directions along the axes, and in this case the directional derivatives are called "partial derivatives". these are treated carefully in spivak and play a major role, since they can be calculated and give one a way both to prove the existence of the total derivative and to calculate its matrix, see theorem 2-8 of spivak. derivatives in other directions are a generalization of these partial derivatives, but they do have some independent importance, e.g. it is of interest to know in which direction a numerical valued function is increasing fastest, the "gradient" direction.

## What is "Spivak Calculus on Manifolds"?

"Spivak Calculus on Manifolds" is a textbook written by Michael Spivak that covers the topic of calculus on manifolds, which is a branch of mathematics that deals with multivariate calculus in a more abstract setting.

## What makes "Spivak Calculus on Manifolds" different from other calculus textbooks?

"Spivak Calculus on Manifolds" is unique in that it focuses on developing the fundamental concepts and techniques of calculus on manifolds in a rigorous and intuitive manner, rather than just providing a collection of formulas and techniques to memorize.

## Do I need to have a strong background in mathematics to understand "Spivak Calculus on Manifolds"?

Yes, "Spivak Calculus on Manifolds" is intended for students who have already taken courses in single and multivariate calculus, linear algebra, and introductory analysis. It is not recommended for self-study without a strong foundation in these subjects.

## What are some real-world applications of calculus on manifolds?

Calculus on manifolds has many applications in fields such as physics, engineering, and computer science. It is used to study motion in curved spaces, to solve optimization problems, and to model and analyze complex systems.

## Is "Spivak Calculus on Manifolds" suitable for self-study?

While "Spivak Calculus on Manifolds" is a highly regarded textbook, it is not recommended for self-study unless you have a strong mathematical background. It is best used as a supplement to a course or as a reference for those already familiar with the subject.