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Spivak Calculus on Manifolds

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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?
 

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  • #2
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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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mathwonk
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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.
 

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