Engineering Calculus Notes 436

Engineering Calculus Notes 436 - 424 CHAPTER 4 MAPPINGS AND...

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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 real-valued 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 vector-valued 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 real-valued function of a vector variable, the derivative is the linear part of an affine function making first-order contact with the function at the given point. We can combine these last two interpretations to formulate the derivative of a vector-valued function of a vector variable. Extending our terminology from real-valued functions (as in § 3.2 ) to (vector-valued) mappings, we define an affine mapping to be a mapping of the form...
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