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Unformatted text preview: Higher order derivatives 1 Chapter 3 Higher order derivatives You certainly realize from singlevariable calculus how very important it is to use derivatives of orders greater than one. The same is of course true for multivariable calculus. In particular, we shall definitely want a “second derivative test” for critical points. A. Partial derivatives First we need to clarify just what sort of domains we wish to consider for our functions. So we give a couple of definitions. DEFINITION. Suppose x ∈ A ⊂ R n . We say that x is an interior point of A if there exists r > 0 such that B ( x ,r ) ⊂ A . A DEFINITION. A set A ⊂ R n is said to be open if every point of A is an interior point of A . EXAMPLE. An open ball is an open set. (So we are fortunate in our terminology!) To see this suppose A = B ( x,r ). Then if y ∈ A , k x y k < r and we can let ² = r k x y k . Then B ( y,² ) ⊂ A . For if z ∈ B ( y,² ), then the triangle inequality implies k z x k ≤ k z y k + k y x k < ² + k y x k = r, so that z ∈ A . Thus B ( y,² ) ⊂ B ( x,r ) (cf. Problem 1–33). This proves that y is an interior point of A . As y is an arbitrary point of A , we conclude that A is open. PROBLEM 3–1. Explain why the empty set is an open set and why this is actually a problem about the logic of language rather than a problem about mathematics. PROBLEM 3–2. Prove that if A and B are open subsets of R n , then their union A ∪ B and their intersection A ∩ B are also open. 2 Chapter 3 PROBLEM 3–3. Prove that the closed ball B ( x,r ) is not open. PROBLEM 3–4. Prove that an interval in R is an open subset ⇐⇒ it is of the form ( a,b ), where∞ ≤ a < b ≤ ∞ . Of course, ( a,b ) = { x ∈ R  a < x < b } . PROBLEM 3–5. Prove that an interval in R is never an open subset of R 2 . Specifically, this means that the xaxis is not an open subset of the x y plane. More generally, prove that if we regard R k as a subset of R n , for 1 ≤ k ≤ n 1, by identifying R k with the Cartesian product R k × { } , where 0 is the origin in R n k , then no point of R k is an interior point relative to the “universe” R n . NOTATION. Suppose A ⊂ R n is an open set and A f→ R is differentiable at every point x ∈ A . Then we say that f is differentiable on A . Of course, we know then that in particular the partial derivatives ∂f/∂x j all exist. It may happen that ∂f/∂x j itself is a differentiable function on A . Then we know that its partial derivatives also exist. The notation we shall use for the latter partial derivatives is ∂ 2 f ∂x i ∂x j = ∂ ∂x i ∂f ∂x j ¶ . Notice the care we take to denote the order in which these differentiations are performed. In case i = j we also write ∂ 2 f ∂x 2 i = ∂ 2 f ∂x i ∂x i = ∂ ∂x i ∂f ∂x i ¶ ....
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This note was uploaded on 09/28/2008 for the course MATH 1220 taught by Professor Hatcher during the Fall '08 term at Cornell.
 Fall '08
 Hatcher
 Calculus, Derivative, Multivariable Calculus

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