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Engineering Calculus Notes 228

Engineering Calculus Notes 228 - coordinatewise Using this...

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216 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION or −→ x = ( x,y,z ) = ( x 1 ,x 2 ,x 3 ) and simply write our function as f ( −→ x ) . A third notation which is sometimes useful is that of mappings: we write f : R n R (with n = 2 or n = 3) to indicate that f has inputs coming from R n and produces outputs that are real numbers. 1 In much of our expositon we will deal explicitly with the case of three variables, with the understanding that in the case of two variables one simply ignores the third variable; conversely, we will in some cases concentrate on the case of two variables and if necessary indicate how to incorporate the third variable. Much of what we will do has a natural extension to any number of input variables, and we shall occasionally comment on this. In this chapter, we consider the definition and use of derivatives in this context. 3.1 Continuity and Limits Recall from § 2.3 that a sequence of vectors converges if it converges
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Unformatted text preview: coordinatewise Using this notion, we can deFne continuity of a real-valued function of three (or two) variables f ( −→ x ) by analogy to the deFnition for real-valued functions f ( x ) of one variable: Defnition 3.1.1. A real-valued function f ( −→ x ) is continuous on a subset D ⊂ R 3 of its domain if whenever the inputs converge in D (as points in R 3 ) the corresponding outputs also converge (as numbers): −→ x k → −→ x ⇒ f ( −→ x k ) → f ( −→ x ) . It is easy, using this deFnition and basic properties of convergence for sequences of numbers, to verify the following analogues of properties of continuous functions of one variable. ±irst, the composition of continuous functions is continuous (Exercise 3 ): 1 When the domain is an explicit subset D ⊂ R n we will write f : D → R ....
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