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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.

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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.unintuit said:

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"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.

"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.

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.

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.

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.

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