Engineering Calculus Notes 363

Engineering Calculus Notes 363 - 351 3.7. HIGHER...

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3.7. HIGHER DERIVATIVES 351 where −→ x = −→ x −→ a . Taylor’s Theorem with Lagrange Remainder for functions of one variable (( Calculus Deconstructed , Theorem 6.1.7)) tells us that g ( t ) = g (0) + tg (0) + t 2 2 g ′′ ( s ) (3.28) for some 0 s t . By the Chain Rule (Proposition 3.3.6 ) g ( t ) = −→ f ( −→ a + t −→ x ) · △ −→ x = s j ∂f ∂x j ( −→ a + t −→ x ) j −→ x and so g ′′ ( s ) = s i s j 2 f ∂x i ∂x j ( −→ a + s −→ x ) i −→ x j −→ x . This is a homogeneous polynomial of degree two, or quadratic form , in the components of −→ x . By analogy with our notation for the total diFerential, we denote it by d 2 −→ a f ( −→ x ) = s i s j 2 f ∂x i ∂x j ( −→ a ) i −→ x j −→ x . We shall refer to this particular quadratic form—the analogue of the
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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