Engineering Calculus Notes 171

Engineering Calculus Notes 171 - 2.3. CALCULUS OF...

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2.3. CALCULUS OF VECTOR-VALUED FUNCTIONS 159 Then δ dist( P,Q ) δ 3 . (2.20) Proof. On one hand, clearly each of ( x ) 2 , ( y ) 2 and ( z ) 2 is less than or equal to their sum ( x ) 2 + ( y ) 2 + ( z ) 2 , since all three are non-negative. Thus δ 2 dist( P,Q ) 2 and taking square roots, δ dist( P,Q ) . On the other hand, ( x ) 2 + ( y ) 2 + ( z ) 2 δ 2 + δ 2 + δ 2 = 3 δ 2 and taking square roots we have dist( P,Q ) δ 3 . In particular, we clearly have dist( P,Q ) = 0 ⇐⇒ P = Q. (2.21) The next important property is proved by a calculation which you do in Exercise 4 . Lemma 2.3.2 (Triangle Inequality) . For any three points P,Q,R R 3 , dist( P,Q ) dist( P,R ) + dist( R,Q ) . (2.22) With these properties in hand, we consider the notion of convergence for a
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