Chapter 1 Topics in Analytic Geometry
Chapter 1 Topics in Analytic Geometry Chapter 1 Topics in Analytic Geometry
MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 583.8 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.
MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 59In spherical coordinates the surfaces represented by equations of the formwhere ρ 0 ,θ 0 , and φ 0 are constant.ρ = ρ 0 , θ = θ 0 , and φ = φ 0• The surface ρ = ρ 0 consists of all points whose distance ρ from the origin is ρ 0 .Assuming ρ 0 to be nonnegative, this is a sphere of radius ρ 0 centered at the origin.• As in cylindrical coordinates, the surface θ = θ 0 is a half-plane attached along thez-axis, making an angle θ 0 with the positive x-axis.• The surface φ = φ 0 consists of all point from which a line segment to the origin makesan angle of φ 0 with the positive z-axis. If 0 ≤ φ 0 < π/2, this will be the nappe ofa cone opening up, while if π/2 < φ 0 < π, this will be the nappe of a cone openingdown. (If φ 0 = π/2, then the cone is flat, and the surface is the xy-plane.)Converting CoordinatesJust as we needed to convert between rectangular and polar coordinates in 2-space, so wewill need to be able to convert between rectangular, cylindrical, and spherical coordinatesin 3-space. The following Table provides formulas for making these conversions.ConversionFormulasCylindrical to rectangular (r,θ,z) → (x,y,z) x = rcosθ,y = rsinθ,z = zRectangular to cylindrical (x,y,z) → (r,θ,z) r = √ x 2 +y 2 ,tanθ = y/x,z = zSpherical to cylindrical (ρ,θ,φ) → (r,θ,z) r = ρsinφ,θ = θ,z = ρcosφCylindrical to spherical (r,θ,z) → (ρ,θ,φ) ρ = √ r 2 +z 2 ,θ = θ,tanφ = r/zSpherical to rectangular (ρ,θ,φ) → (x,y,z) x = ρsinφcosθ,y = ρsinφsinθ,z = ρcosφRectangular to spherical (x,y,z) → (ρ,θ,φ) ρ = √ x 2 +y 2 +z 2 ,tanθ = y/x,cosφ = z/ √ x 2 +y 2 +z 2Note that r ≥ 0, ρ ≥ 0, 0 ≤ θ < 2π, 0 ≤ φ ≤ π.The diagrams in the following Figure will help you to understand how the formulas inTable are derived.xzθyr•Pz{(x,y,z)(r,θ,z)(r,θ,0)yzφθρrφ•Pz{(ρ,θ,φ)(r,θ,z)yx(a)x(b)
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MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 59In spherical coord<strong>in</strong>ates the surfaces represented by equations of the formwhere ρ 0 ,θ 0 , and φ 0 are constant.ρ = ρ 0 , θ = θ 0 , and φ = φ 0• The surface ρ = ρ 0 consists of all po<strong>in</strong>ts whose distance ρ from the orig<strong>in</strong> is ρ 0 .Assum<strong>in</strong>g ρ 0 to be nonnegative, this is a sphere of radius ρ 0 centered at the orig<strong>in</strong>.• As <strong>in</strong> cyl<strong>in</strong>drical coord<strong>in</strong>ates, the surface θ = θ 0 is a half-plane attached along thez-axis, mak<strong>in</strong>g an angle θ 0 with the positive x-axis.• The surface φ = φ 0 consists of all po<strong>in</strong>t from which a l<strong>in</strong>e segment to the orig<strong>in</strong> makesan angle of φ 0 with the positive z-axis. If 0 ≤ φ 0 < π/2, this will be the nappe ofa cone open<strong>in</strong>g up, while if π/2 < φ 0 < π, this will be the nappe of a cone open<strong>in</strong>gdown. (If φ 0 = π/2, then the cone is flat, and the surface is the xy-plane.)Convert<strong>in</strong>g Coord<strong>in</strong>atesJust as we needed to convert between rectangular and polar coord<strong>in</strong>ates <strong>in</strong> 2-space, so wewill need to be able to convert between rectangular, cyl<strong>in</strong>drical, and spherical coord<strong>in</strong>ates<strong>in</strong> 3-space. The follow<strong>in</strong>g Table provides formulas for mak<strong>in</strong>g these conversions.ConversionFormulasCyl<strong>in</strong>drical to rectangular (r,θ,z) → (x,y,z) x = rcosθ,y = rs<strong>in</strong>θ,z = zRectangular to cyl<strong>in</strong>drical (x,y,z) → (r,θ,z) r = √ x 2 +y 2 ,tanθ = y/x,z = zSpherical to cyl<strong>in</strong>drical (ρ,θ,φ) → (r,θ,z) r = ρs<strong>in</strong>φ,θ = θ,z = ρcosφCyl<strong>in</strong>drical to spherical (r,θ,z) → (ρ,θ,φ) ρ = √ r 2 +z 2 ,θ = θ,tanφ = r/zSpherical to rectangular (ρ,θ,φ) → (x,y,z) x = ρs<strong>in</strong>φcosθ,y = ρs<strong>in</strong>φs<strong>in</strong>θ,z = ρcosφRectangular to spherical (x,y,z) → (ρ,θ,φ) ρ = √ x 2 +y 2 +z 2 ,tanθ = y/x,cosφ = z/ √ x 2 +y 2 +z 2Note that r ≥ 0, ρ ≥ 0, 0 ≤ θ < 2π, 0 ≤ φ ≤ π.The diagrams <strong>in</strong> the follow<strong>in</strong>g Figure will help you to understand how the formulas <strong>in</strong>Table are derived.xzθyr•Pz{(x,y,z)(r,θ,z)(r,θ,0)yzφθρrφ•Pz{(ρ,θ,φ)(r,θ,z)yx(a)x(b)