Unformatted text preview: 424 CHAPTER 4. MAPPINGS AND TRANSFORMATIONS 4.2 Differentiable Mappings We have seen several versions of the notion of a derivative in previous sections: for a realvalued function f of one real variable, the derivative is a number, which gives the slope of the line tangent to the graph y = f ( x ) at the given point; for a vectorvalued function of one real variable, the derivative is a vector, giving the velocity of the motion described by the function, or equivalently giving the coefficients of the “time” variable in the natural parametrization of the tangent line; for a realvalued function of a vector variable, the derivative is the linear part of an affine function making firstorder contact with the function at the given point. We can combine these last two interpretations to formulate the derivative of a vectorvalued function of a vector variable. Extending our terminology from realvalued functions (as in § 3.2 ) to (vectorvalued) mappings, we define an affine mapping to be a mapping of the form...
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 Spring '08
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 Calculus, Derivative, Transformations, Slope, Continuous function, real variable, vector variable

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