Engineering Calculus Notes 438

Engineering Calculus Notes 438 - mappings allows us to...

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426 CHAPTER 4. MAPPINGS AND TRANSFORMATIONS In particular, when −→ v is the j th element of the standard basis for R 3 , this gives us the velocity of the image of the j th coordinate axis, and as a column this consists of the j th partial derivatives of the component functions. But this column is the j th column of the matrix representative of DF −→ x 0 , giving us 5 Remark 4.2.3. The matrix representative of the derivative DF −→ x 0 of F : R 3 R 3 is the matrix of partial derivatives of the component functions of F : [ DF ] = ∂f 1 /∂x ∂f 1 /∂y ∂f 1 /∂z ∂f 2 /∂x ∂f 2 /∂y ∂f 2 /∂z ∂f 3 /∂x ∂f 3 /∂y ∂f 3 /∂z . The matrix above is called the Jacobian matrix of F , and denoted 6 JF . As a special case, we note the following, whose (easy) proof is left to you (Exercise 3 ): Remark 4.2.4. If F : R 3 R 3 is linear, then it is diFerentiable and DF −→ x 0 = F for every −→ x 0 R 3 . In particular, the linearization (at any point) of an a±ne map is the map itself. The Chain Rule We have seen several versions of the Chain Rule before. The setting of
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Unformatted text preview: mappings allows us to formulate a single uniFed version which includes the others as special cases. In the statements below, we assume the dimensions m , n and p are each 1, 2 or 3. Theorem 4.2.5 (General Chain Rule) . If F : R n → R m is diFerentiable at −→ y ∈ R n and G : R p → R n is diFerentiable at −→ x ∈ R p where −→ y = G ( −→ x ) , then the composition F ◦ G : R p → R m is diFerentiable at −→ x , and its derivative is the composition of the derivatives of G (at −→ x ) and F (at −→ y = G ( −→ x ) ): D ( F ◦ G ) −→ x = ( DF −→ y ) ◦ ( DG −→ x ); 5 Again, the analogue when the domain or the target or both are the plane instead of space is straightforward. 6 An older, but sometimes useful notation, based on viewing F as an m-tuple of func-tions, is ∂ ( f 1 ,f 2 ,f 3 ) ∂ ( x,y,z ) ....
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