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 ) , f 3 ( t ) de- fined on some t -interval I . The vector f ( t ) = ( f 1 ( t ) , f 2 ( t ) , f 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 ) , f 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 ) , f 2 ( t ) , f 3 ( t )) f 1 ( t ) = 3 sin 2 t, f 2 ( t ) = 4 cos 2 t, f 3 ( t ) = 1 I : 0 t 2 π 2
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3. f ( t ) = ( f 1 ( t ) , f 2 ( t ) , f 3 ( t )) f 1 ( t ) = 2 cos 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
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Examples: f ( t ) = 2 cos t i + 2 sin 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
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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
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f is a continuous vector function if and only if each of its components f 1 , f 2 , f 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 , f 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.
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