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Unformatted text preview: ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 Introduction VECTORS AND VECTOR ANALYSES Three dimensional Space Coordinates & Transformations: Cartesian Coordinates Cylindrical Coordinates Spherical Coordinates ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 Cartesian Coordinates x y z P(x,y,z) (x,y,0) (0,0,z) (x,0,0) (x,0,z) (0,y,z) Mutually Orthogonal Axes: Ox, Oy, Oz Righthanded system: A rightthreaded screw pointing along Oz will advance when twisted from Ox to Oy O yz  plane: => x = 0 xy – plane => z = 0 xz –plane => y = 0 These planes divide the space into 8 octants First octant: All positive ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 Cylindrical Coordinates x y z O P(r,θ,z) x y z r θ x = r cos(θ) r 2 = x 2 + y 2 y = r sin(θ) tan(θ) = y / x z = z r = 0 => represents zaxis r = constant => represents a cylinder θ = constant => represents a plane Cylindrical coordinates are useful when there is an axis of symmetry ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 x y z O Spherical Coordinates y r θ x P(r,θ,z) z Φ ρ ρ = OP ≥ 0 distance from the origin 0 ≤ Φ ≤ π and 0 ≤ θ ≤ 2 π ρ = constant => represents a sphere Φ = constant => represents a cone θ = constant => represents a plane r = ρ sin Φ x = r cos θ x = ρ sin Φ cos θ z = ρ cos Φ y = r sin θ y = ρ sin Φ sin θ θ = θ z = z z = ρ cos Φ Spherical coordinates are useful when there is a point of symmetry ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 Example: Describe the locus of points that satisfy the following pairs of simultaneous equations in Cartesian coordinate system: 1. y = constant (any x, any z) and z = constant (any x, any y) x y z y = constant is a plane x = constant is a plane Intersection line between two planes ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 Example: Describe the locus of points that satisfy the following pairs of simultaneous equations in Cartesian coordinate system: 2. x = 2 (any y, any z) and y 2 + z 2 = 9 (any x) x y z x = 2 is a plane y 2 + z 2 = 9 is a cylinder Locus is the full circle with r = 3 ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 Example: Describe the locus of points that satisfy the following pairs of simultaneous equations in Cartesian coordinate system: 3. y = 0 (any x, any z) and x 2 /a 2 + z 2 /b 2 = 1 (any y) x y z y = 0 is the xz plane x 2 /a 2 + z 2 /b 2 = 1 is the full ellipsoid Locus is the ellipse ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 Example: Describe the locus of points that satisfy the following pairs of simultaneous equations in Cylindrical coordinate system: 1. r = 2 (any z) and z = 3 (any x, any y) x y z z = 3 is a plane r = 2 is a cylinder Locus is the full circle ME – 210...
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 Spring '09
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