lecture04

lecture04 - Lecture 4 Ch14 Calculus of vector-valued...

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2006 Fall MATH 100 Lecture 4 1 Lecture 4 Ch14 Calculus of vector-valued functions 14.1 Introduction 14.2 Calculus of vector-valued functions 14.3 Arc length and change of variables
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2006 Fall MATH 100 Lecture 4 2 Vector-valued Functions ( ) ( ) ( ) () ( ) () () () () () () () k t z j t y i t x t z t y t x t r j t y i t x t y t x t r G G G G G G G + + = = + = = , , space, - 3 In , space, - 2 In () () () π t t z t t y t t x k t j t i t t r cos , 2 , ln : functions component has cos 2 ln function valued - ctor ve : Example 2 2 = + = = + + + = G G G G
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2006 Fall MATH 100 Lecture 4 3 Vector-valued functions and its parametric equations () ( )( ) () () () t r t r k t z j t y i t x k z j y i x t z z t y y t x x : functions valued - Single : functions valued - Vector : scalars and ctors between ve ation Differenti , , G G G G G G G 8 + + = + + = = =
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2006 Fall MATH 100 Lecture 4 4 Graph of vector-valued function ( ) t r G of tip the of Trace () helix circular , sin , cos are sin cos of equations Parametric : Example = = = + + = ct z t a y t a x k ct j t a i t a t r G G G G
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2006 Fall MATH 100 Lecture 4 5 1 circle the to correspond All sin cos and sin cos ation parametriz different have can curve single a : Remark 2 2 2 2 = + = = = = y x t y t x t y t x Graph of vector-valued function
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2006 Fall MATH 100 Lecture 4 6 Calculus of Vector-valued Functions ( ) ( ) ( ) ( ) ( ) ( ) ( ) exist components of limits if exists limit lim lim lim lim : limits of n Deifinitio k t z j t y i t x t r a t a t a t a t G G G G + + = ( ) () () ( ) 0 0 0 0 0 0 at continuous are components at continuous is lim . 3 exists lim 2. defined is 1. if at continuous is : Continuity 0 0 t t t r t r t r t r t r t t r t t t t = G G G G G G
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2006 Fall MATH 100 Lecture 4 7 Definition of derivatives () ( ) ( ) h t r h t r t r h G G
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This note was uploaded on 09/17/2011 for the course MATH 100 taught by Professor Qt during the Fall '09 term at HKUST.

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lecture04 - Lecture 4 Ch14 Calculus of vector-valued...

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