Engineering Calculus Notes 437

Engineering Calculus Notes 437 - 4.2. DIFFERENTIABLE...

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Unformatted text preview: 4.2. DIFFERENTIABLE MAPPINGS 425 The linear part is called the derivative or differential 4 of F at x and denoted either d x F or DF x ; we shall use the derivative terminology and the D notation. The full affine map will be denoted T x F , in keeping with the notation for Taylor polynomials: this is sometimes called the linearization of F at x . Thus, the linearization of the differentiable mapping F : R n R m at x is T x F ( x ) = F ( x ) + DF x ( x x ) = F ( x ) + DF x ( x ) . To calculate the derivative, let us fix a point x and a velocity vector v . If we write a mapping F with values in space as a column of functions F ( x ) = f 1 ( x ) f 2 ( x ) f 3 ( x ) then we can consider the action of the differentials at x of the various component functions f i on v : recall from...
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