chap03 - hiorder derivatives

chap03 - hiorder derivatives - Higher order derivatives 1...

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Unformatted text preview: Higher order derivatives 1 Chapter 3 Higher order derivatives You certainly realize from single-variable 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 x-axis 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.

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chap03 - hiorder derivatives - Higher order derivatives 1...

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