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Are Directional Derivatives Worth Pursuing in Spivak's Calculus on Manifolds?
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[QUOTE="mathwonk, post: 4966420, member: 13785"] 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. [/QUOTE]
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Are Directional Derivatives Worth Pursuing in Spivak's Calculus on Manifolds?
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