Engineering Calculus Notes 20

Engineering Calculus Notes 20 - r,θ ) of the projection P...

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8 CHAPTER 1. COORDINATES AND VECTORS θ P P xy z r Figure 1.6: Cylindrical Coordinates the origin as follows (Figure 1.6 ): if P is not on the z -axis, then this axis together with the line O P determine a (vertical) plane, which can be regarded as the xz -plane rotated so that the x -axis moves θ radians counterclockwise (in the horizontal plane); we take as our coordinates the angle θ together with the abcissa and ordinate of P in this plane. The angle θ can be identi±ed with the polar coordinate of the projection P xy of P on the horizontal plane; the abcissa of P in the rotated plane is its distance from the z -axis, which is the same as the polar coordinate r of P xy ; and its ordinate in this plane is the same as its vertical rectangular coordinate z . We can think of this as a hybrid: combine the polar coordinates (
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Unformatted text preview: r,θ ) of the projection P xy with the vertical rectangular coordinate z of P to obtain the cylindrical coordinates ( r,θ,z ) of P . Even though in principle r could be taken as negative, in this system it is customary to con±ne ourselves to r ≥ 0. The relation between the cylindrical coordinates ( r,θ,z ) and the rectangular coordinates ( x,y,z ) of a point P is essentially given by Equation ( 1.3 ): x = r cos θ y = r sin θ z = z. (1.6) We have included the last relation to stress the fact that this coordinate is the same in both systems. The inverse relations are given by ( 1.4 ), ( 1.5 ) and the trivial relation z = z . The name “cylindrical coordinates” comes from the geometric fact that the...
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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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