Engineering Calculus Notes 439

Engineering Calculus Notes 439 - 7 M such that b L ( −→...

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4.2. DIFFERENTIABLE MAPPINGS 427 in matrix language, the Jacobian matrix of the composition is the product of the Jacobian matrices: J ( F G )( −→ x 0 ) = JF ( −→ y 0 ) · JG ( −→ x 0 ) . Proof. We need to show that the “afne approximation” we get by assuming that the derivative oF F G is the composition oF the derivatives, say T ( −→ x 0 + −→ x ) = ( F G )( −→ x 0 ) + ( DF −→ y 0 DG −→ x 0 )( −→ x ) has ±rst-order contact at −→ x = −→ 0 with ( F G )( −→ x 0 + −→ x ). The easiest Form oF this to work with is to show that For every ε > 0 there exists δ > 0 such that ( F G )( −→ x 0 + −→ x ) = ( F G )( −→ x 0 )+( DF −→ y 0 DG −→ x 0 )( −→ x )+ E ( −→ x ) (4.2) such that bE ( −→ x ) b < ε b△ −→ x b whenever b△ −→ x b < δ . To carry this out, we need ±rst to establish an estimate on how much the length oF a vector can be increased when we apply a linear mapping. Claim: If L : R n R m is linear, then there exists a number
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Unformatted text preview: 7 M such that b L ( −→ x ) b ≤ M b −→ x b for every n ∈ R . This number can be chosen to satisfy the estimate M ≤ mna max where a max is the maximum absolute value of entries in the matrix representative [ L ] . This is an easy application oF the triangle inequality. Given a vector −→ x = ( v 1 ,... ,v n ), let v max be the maximum absolute value oF the components oF −→ x , and let a ij be the entry in row i , column j oF [ L ], The i th component oF L ( −→ v ) is ( L ( −→ v )) i = a i 1 v 1 + ··· + a in v n so we can write | ( L ( −→ v )) i | ≤ | a 1 i || v 1 | + ··· + | a in || v n | ≤ na max v max . 7 The least such number is called the operator norm of the mapping, and is denoted b L b...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Spring '08 term at University of Florida.

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