Differentiation

Piotr Mikusiński and Michael D. Taylor
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Piotr Mikusiński: University of Central Florida, Department of Mathematics
Michael D. Taylor: University of Central Florida, Department of Mathematics

Chapter 3 in An Introduction to Multivariable Analysis from Vector to Manifold, 2002, pp 75-112 from Springer

Abstract: Abstract Let $$ \mathbb{R}^{\rm N} \to \mathbb{R} $$ . By the partial derivative of f with respect to its i th variable we mean the function $$ Dif(x) = \mathop {\lim }\limits_{\lambda \to 0} \frac{{f(x + \lambda ei) - f(x)}} {\lambda } $$ Remember that ei is the vector with 1 in the ith coordinate and 0 everywhere else.This is also denoted by the symbol \frac{{\partial (x)}} {{\partial x_i }} The domain of this function is, of course, the set of all x for which the limit exists. We recall from calculus that in terms of Computing a partial derivative from a given function, we simply regard all variables except the ith one as constants and apply standard differentiation rules.

Keywords: Linear Transformation; Inverse Function; Open Ball; Chain Rule; Implicit Function Theorem (search for similar items in EconPapers)
Date: 2002
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4612-0073-4_3

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DOI: 10.1007/978-1-4612-0073-4_3

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