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Engineering Calculus Notes 177

Engineering Calculus Notes 177 - −→ f t = x t,y t,z 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 differentiation. We shall often use the Newtonian “dot” notation for the derivative of a vector-valued function interchangeably with “prime”. Definition 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 ) = vector f ( t 0 ) = d dt vextendsingle vextendsingle vextendsingle vextendsingle t = t 0 bracketleftBig −→ f bracketrightBig = lim h 0 1 h bracketleftBig −→ f ( t 0 + h ) −→ f ( t 0 ) bracketrightBig provided it exists. (If not, the function is not differentiable at t = t 0 .) Again, using Lemma 2.3.5 , we connect this with differentiation of the
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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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