Engineering Calculus Notes 21

Engineering - containing the z-axis and the line O P and the angle φ between the(positive z-axis and the line O P(±igure 1.7 Of course the

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1.1. LOCATING POINTS IN SPACE 9 locus of the equation r = c (which in polar coordinates gives a circle of radius c about the origin) gives a vertical cylinder whose axis of symmetry is the z -axis with radius c . Cylindrical coordinates carry the ambiguities of polar coordinates: a point on the z -axis has r = 0 and θ arbitrary, while a point oF the z -axis has θ determined up to adding even multiples of π (since r is taken to be positive). ±or example, the point P with rectangular coordinates ( 2 3 , 2 , 4) has cylindrical coordinates r = 4 θ = 5 π 6 + 2 z = 4 . Spherical Coordinates Another coordinate system in space, which is particularly useful in problems involving rotations around various axes through the origin (for example, astronomical observations, where the origin is at the center of the earth) is the system of spherical coordinates . Here, a point P is located relative to the origin O by measuring the distance of P from the origin ρ = |O P | together with two angles: the angle θ between the xz -plane and the plane
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Unformatted text preview: containing the z-axis and the line O P , and the angle φ between the (positive) z-axis and the line O P (±igure 1.7 ). Of course, the spherical coordinate θ of P is identical to the cylindrical coordinate θ , and we use the same letter to indicate this identity. While θ is sometimes allowed to take on all real values, it is customary in spherical coordinates to restrict φ to 0 ≤ φ ≤ π . The relation between the cylindrical coordinates ( r,θ,z ) and the spherical coordinates ( ρ,θ,φ ) of a point P is illustrated in ±igure 1.8 (which is drawn in the vertical plane determined by θ ): 2 r = ρ sin φ θ = θ z = ρ cos φ. (1.7) 2 Be warned that in some of the engineering and physics literature the names of the two spherical angles are reversed, leading to potential confusion when converting between spherical and cylindrical coordinates....
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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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