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Engineering Calculus Notes 183

Engineering Calculus Notes 183 - 171 2.3 CALCULUS OF...

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2.3. CALCULUS OF VECTOR-VALUED FUNCTIONS 171 each of the component functions f i ( t ), i = 1 , 2 , 3 is integrable over [ a,b ], and then integraldisplay b a −→ f ( t ) dt = parenleftbiggintegraldisplay b a f 1 ( t ) dt, integraldisplay b a f 2 ( t ) dt, integraldisplay b a f 3 ( t ) dt parenrightbigg . A direct consequence of this and the Fundamental Theorem of Calculus is that the integral of (vector) velocity is the net (vector) displacement: Lemma 2.3.19. If −→ v ( t ) = ˙ −→ p ( t ) is continuous on [ a,b ] , then integraldisplay b a −→ v ( t ) dt = −→ p ( b ) −→ p ( a ) . The proof of this is outlined in Exercise 11 . In Appendix C , we discuss the way in which the calculus of vector-valued functions can be used to reproduce Newton’s derivation of the Law of Universal Gravitation from Kepler’s Laws of Planetary Motion.
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