Engineering Calculus Notes 177

Engineering Calculus Notes 177 - −→ f ( t ) = ( x ( t )...

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2.3. CALCULUS OF VECTOR-VALUED FUNCTIONS 165 Derivatives of Vector-Valued Functions When we think of a function −→ f : R R 3 as describing a moving point, it is natural to ask about its velocity, acceleration and so on. For this, we need to extend the notion of di±erentiation. We shall often use the Newtonian “dot” notation for the derivative of a vector-valued function interchangeably with “prime”. Defnition 2.3.13. The derivative of the function −→ f : R R 3 at an interior point t 0 of its domain is the limit ˙ −→ f ( t 0 ) = v f ( t 0 ) = d dt v v v v t = t 0 b −→ f B = lim h 0 1 h b −→ f ( t 0 + h ) −→ f ( t 0 ) B provided it exists. (If not, the function is not diFerentiable at t = t 0 .) Again, using Lemma 2.3.5 , we connect this with di±erentiation of the component functions: Remark 2.3.14. The vector-valued function
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Unformatted text preview: −→ f ( t ) = ( x ( t ) ,y ( t ) ,z ( t )) is diFerentiable at t = t precisely if all of its component functions are diFerentiable at t = t , and then v f ′ ( t ) = ( x ′ ( t ) ,y ′ ( t ) ,z ′ ( t )) . In particular, every diFerentiable vector-valued function is continuous. When −→ p ( t ) describes a moving point, then its derivative is referred to as the velocity of −→ p ( t ) −→ V ( t ) = ˙ −→ p ( t ) and the derivative of velocity is acceleration −→ a ( t ) = ˙ −→ V ( t ) = ¨ −→ p ( t ) . The magnitude of the velocity is the speed , sometimes denoted d s dt = b −→ V ( t ) b ....
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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