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” φ 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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