Chapter 1 Topics in Analytic Geometry

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MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 58Solution .........Elliptic Cones A sketch of the elliptic conez 2 = x2a 2 + y2b 2 (a > 0,b > 0) (3.39)can be obtained by first sketching the elliptical traces in the planes z = ±1 and thensketching the linear traces that connect the endpoints of the axes of the ellipses.Example 3.47 Sketch the graph of the elliptic coneSolution .........z 2 = x 2 + y24Elliptic Paraboloids A sketch of the elliptic paraboloidz = x2a 2 + y2b 2 (a > 0,b > 0) (3.40)canbe obtainedby first sketching the elliptical traces in the planes z = 1 and then sketchingthe parabolic traces in the vertical coordinate planes to connect the origin to the ends ofthe axes of the ellipses.Example 3.48 Sketch the graph of the elliptic paraboloidSolution .........z = x24 + y29Hyperbolic Paraboloids A sketch of the hyperbolic paraboloidz = y2b 2 − x2a 2 (a > 0,b > 0) (3.41)can be obtained by first sketching the two parabolic traces that pass through the origin(one in the plane x = 0 and the other in the plane y = 0). After the parabolic traces aredraw, sketch the hyperbolic traces in the planes z = ±1 and then fill in any missing edges.Example 3.49 Sketch the graph of the hyperbolic paraboloidSolution .........z = y24 − x29
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.
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Chapter 1 Topics in Analytic Geometry

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