2433-Ch13-slides

# 2433-Ch13-slides - Vector Functions; Calculus: 13.2) (13.1,...

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Vector Functions; Calculus: (13.1, 13.2) Given functions f 1 ( t ) ,f 2 ( t ) 3 ( t ) de- Fned on some t -interval I . The vector f ( t )=( f 1 ( t ) 2 ( t ) 3 ( t )) = f 1 ( t ) i + f 2 ( t ) j + f 3 ( t ) k is called vector-valued function or vec- tor function . 1

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Examples: 1. f ( t )=( f 1 ( t ) ,f 2 ( t ) 3 ( t )) f 1 ( t ) = 2+3 t, f 2 ( t ) = 1+4 t, f 3 ( t )=3 - 2 t, I : -∞ <t< 2. f ( t f 1 ( t ) 2 ( t ) 3 ( t )) f 1 ( t ) = 3sin 2 t, f 2 ( t ) = 4cos 2 t, f 3 ( t )=1 I :0 t 2 π 2
3. f ( t )=( f 1 ( t ) ,f 2 ( t ) 3 ( t )) f 1 ( t ) = 2cos t, f 2 ( t ) = 2 sin t, f 3 ( t )=2 t I :0 t 2 π NOTE: as t increases on I the tip of f traces out an oriented curve C in ”space.” 3

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Calculus of Vector Functions I. Limits: Let c I . lim t c f ( t )= L = L 1 i + L 2 j + L 3 k if and only if lim t c ± f ( t ) - L ± =0 if and only if lim t c f 1 ( t L 1 , lim t c f 2 ( t L 2 , lim t c f 3 ( t L 3 That is: limits are calculated ”component- wise.” 4
Examples: f ( t ) = 2cos t i + 2sin t j +2 t k lim t 2 π/ 3 f ( t )= f ( t sin 2 t t i + t ln t j + e 2 t - 1 4 t k lim t 0 f ( t 5

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Arithmetic of limits: Let f and g be vector functions and let u be a scalar function with: f ( t ) L , g ( t ) M ,u ( t ) A . 1. f ( t )+ g ( t ) L + M 2. f ( t ) - g ( t ) L - M 3. f ( t ) · g ( t ) L · M 6
4. f ( t ) × g ( t ) L × M 5. α f ( t ) α L 6. u ( t ) f ( t ) A L

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II. Continuity f ( t )= f 1 ( t ) i + f 2 ( t ) j + f 3 ( t ) k is con- tinuous at t = c if and only if lim t c f ( t f ( c ) . f is continuous on an interval I if and only if it is continuous at each point c I . 7
f is a continuous vector function if and only if each of its components f 1 ,f 2 3 is a continuous function. 8

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III. Diﬀerentiation: f ( t )= f 1 ( t ) i + f 2 ( t ) j + f 3 ( t ) k is dif- ferentiable at t if and only if lim h 0 1 h [ f ( t + h ) - f ( t )] exists . f ( t f 1 ( t ) i + f 2 ( t ) j + f 3 ( t ) k is dif- ferentiable at t if and only if each of f 1 ,f 2 3 is diﬀerentiable at t and, if so, f ± ( t f ± 1 ( t ) i + f ± 2 ( t ) j + f ± 3 ( t ) k 9

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Examples: 1. f ( t )=( t 3 - 2 t ) i + t j + 1 t k f ± ( t )=(3 t 2 - 2) i + 1 2 t j - 1 t 2 k 2. f ( t )= e 2 t i + cos 3 t j +ln t k f ± ( t )=2 e 2 t i - 3 sin 3 t j + 1 t k 10
Diﬀerentiation formulas: f and g are diﬀerentiable vector func- tions and u is a diﬀerentiable scalar function.

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## This note was uploaded on 05/19/2010 for the course MATH 18427 taught by Professor Etgen during the Spring '10 term at University of Houston.

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2433-Ch13-slides - Vector Functions; Calculus: 13.2) (13.1,...

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