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Unformatted text preview: The Derivative Ryan T. Moore 14 September 2004 Introduction This short paper clarifies several definitions of “the derivative” one encounters in introductory vector calculus. General Definitions The derivative of a single function of n variables is a linear transformation from < n → < 1 . It maps each point in < n onto the “slope” of the n-dimensional “plane” tangent at that point. When several functions are defined (say, m of them), the derivative is a linear transformation from < n → < m that maps each point in < n onto the “slopes” of the m n-dimensional tangent “planes”. One “plane” is tangent to each function. The Derivative of a Vector of k Functions of One Variable Define ~x as a vector of functions of one variable, t . That is, ~x = x 1 ( t ) x 2 ( t ) . . . x k ( t ) 1 The derivative of a vector ~x , which can be written several ways, is D ~x = d ~x d t = ...
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This note was uploaded on 05/12/2010 for the course APPLIED ST 2010 taught by Professor Various during the Spring '10 term at Universidad Nacional Agraria La Molina.
- Spring '10