Engineering Calculus Notes 170

Engineering Calculus Notes 170 - parametrizations via...

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158 CHAPTER 2. CURVES (b) A wheel of radius 1 in the plane rolls along the outer edge of the disc of radius 3 centered at the origin. Parametrize the motion of a point on the rim. 8. A vertical plane P through the z -axis makes an angle θ radians with the xz -plane counterclockwise (seen from above). The torus T consists of all points in R 3 at distance 1 from the circle x 2 + y 2 = 9, z = 0 in the xy -plane. Parametrize the intersection P ∩ T of these surfaces. ( Hint: It is a circle.) 9. Parametrize the path in space of a point on the wheel of a unicycle of radius b which is ridden along a circular path of radius a centered at the origin. ( Hint: Note that the plane of the unicycle is vertical and contains, at any moment, the line tangent to the path at the point of contact with the wheel. Note also that as the wheel turns, it travels along the path a distance given by the amount of rotation (in radians) times the radius of the wheel.) 2.3 Calculus of Vector-Valued Functions To apply methods of calculus to curves in R 2 or R 3 or equivalently to their
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Unformatted text preview: parametrizations via vector-valued functions, we must Frst reformulate the basic notion of convergence, as well as di±erentiation and integration, in these contexts. Convergence of Sequences of Points The convergence of sequences of points { −→ p i } in R 2 or R 3 is a natural extension of the corresponding idea for numbers, or points on the line R . We will state everything in terms of R 3 , but the corresponding statements and/or proofs for R 2 are easy modiFcations of the R 3 versions. Before formulating a geometric deFnition of convergence, we note a few properties of the distance function on R 3 . The Frst property will allow us to use estimates on coordinates to obtain estimates on distances, and vice-versa . Lemma 2.3.1. Suppose P,Q ∈ R 3 have respective (rectangular) coordinates ( x 1 ,y 1 ,z 1 ) and ( x 2 ,y 2 ,z 2 ) . Let δ := max( |△ x | , |△ y | , |△ z | ) (where △ x := x 2 − x 1 , etc.)...
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