Chapter 1 Topics in Analytic Geometry

MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 593.8 Cylindrical and Spherical CoordinatesIn this section we will discuss two new types of coordinate systems in 3-space that are oftenmore useful than rectangular coordinate systems for studying surfaces with symmetries.Cylindrical and Spherical Coordinate SystemsThree coordinates are required to establish the location of a point in 3-space. We havealready done this using rectangular coordinates. However, the following Figure shows twoother possibilities: part (a) of the figure shows the rectangular coordinates (x,y,z) ofa point P, part (b) shows the cylindrical coordinates (r,θ,z) of P, and part (c) showsthe spherical coordinates (ρ,θ,φ) of P.zzzy• P(x,y,z)zyxθr• P(r,θ,z)zyφρθ• P(ρ,θ,φ)yxRectangular coordinates(x,y,z)xCylindrical coordinates(r,θ,z)(r ≥ 0,0 ≤ θ < 2π)xSpherical coordinates(ρ,θ,φ)(ρ ≥ 0,0 ≤ θ < 2π,0 ≤ φ ≤ π)Constant SurfacesIn rectangular coordinates the surfaces represented by equations of the formx = x 0 , y = y 0 , and z = z 0wherex 0 ,y 0 ,andz 0 areconstant, areplanesparalleltotheyz-plane, xz-plane, andxy-plane,respectively.In cylindrical coordinates the surfaces represented by equations of the formwhere r 0 ,θ 0 , and z 0 are constant.r = r 0 , θ = θ 0 , and z = z 0• The surface r = r 0 is a right circular cylinder of radius r 0 centered on the z-axis.• The surface θ = θ 0 is a half-plane attached along the z-axis and making an angle θ 0with the positive x-axis.• The surface z = z 0 is a horizontal plane.