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Engineering Calculus Notes 368

# Engineering Calculus Notes 368 - 356 CHAPTER 3 REAL-VALUED...

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356 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION and so | f ( −→ x ) T 2 f ( −→ a ) −→ x | bardbl −→ x −→ a bardbl 2 1 2 3 summationdisplay i =1 3 summationdisplay j =1 vextendsingle vextendsingle vextendsingle vextendsingle 2 f ∂x i ∂x j ( −→ a ) 2 f ∂x i ∂x j ( vectors ) vextendsingle vextendsingle vextendsingle vextendsingle |△ x i x j | −→ x 2 n 2 2 max i,j vextendsingle vextendsingle vextendsingle vextendsingle 2 f ∂x i ∂x j ( −→ a ) 2 f ∂x i ∂x j ( vectors ) vextendsingle vextendsingle vextendsingle vextendsingle max i,j |△ x i x j | −→ x 2 . (3.29) By an argument analogous to that giving Equation ( 2.20 ) on p. 159 (Exercise 7 ), we can say that max i,j |△ x i x j | bardbl△ −→ x bardbl 2 1 and by continuity of the second-order partials, for each i and j lim −→ x −→ a 2 f ∂x i ∂x j ( −→ x ) = 2 f ∂x i ∂x j ( −→ a ) . Together, these arguments show that the right-hand side of Equation ( 3.29 ) goes to zero as −→ x −→ a (since also vectors −→ a ), proving our claim. We note in passing that higher-order “total” derivatives, and the
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