The basis vectors of the cylindrical system e ρ e φand e z are pointing in the

The basis vectors of the cylindrical system e ρ e

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The basis vectors of the cylindrical system ( e ρ , e φ and e z ) are pointing in the direction of increasing ρ , φ and z respectively. Hence, while e z is constant in magnitude and direction throughout the system, e ρ and e φ are coordinate-dependent as they vary in direction from point to point. All these basis vectors are mutually perpendicular and they are defined to be of unit length . Figure 2 is a graphical illustration of the cylindrical coordinate system and its basis vectors with a corresponding reference Cartesian system in a standard position. The transformation from the Cartesian coordinates ( x, y, z ) of a particular point in
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1.3.1 Coordinate Systems 18 x 1 x 2 x 3 e z e φ e ρ φ ρ z P O Figure 2: Cylindrical coordinate system, superimposed on a rectangular Cartesian system in a standard position, and its basis vectors e ρ , e φ and e z in a 3D space. The point P in the figure is identified simultaneously by ( x, y, z ) coordinates in the Cartesian system and by ( ρ, φ, z ) coordinates in the cylindrical system where these coordinates are related through the two sets of Eqs. 7 and 8. the space to the cylindrical coordinates ( ρ, φ, z ) of that point, where the two systems are in a standard position, is performed through the following equations: [8] ρ = p x 2 + y 2 φ = arctan y x z = z (7) while the opposite transformation from the cylindrical to the Cartesian coordinates is [8] In the second equation, arctan ( y x ) should be selected consistent with the signs of x and y to be in the right quadrant.
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1.3.1 Coordinate Systems 19 performed by the following equations: x = ρ cos φ y = ρ sin φ z = z (8) C. Plane Polar Coordinate System The plane polar coordinate system is the same as the cylindrical coordinate system with the absence of the z coordinate, and hence all the equations of the polar system are obtained by setting z = 0 in the equations of the cylindrical coordinate system, that is: ρ = p x 2 + y 2 φ = arctan y x (9) and x = ρ cos φ y = ρ sin φ (10) This system is illustrated graphically in Figure 3 where the point P is located by ( ρ, φ ) in the polar system and by ( x, y ) in the corresponding 2D Cartesian system. D. Spherical Coordinate System The spherical coordinate system is defined by three parameters: r, θ and φ which range over: 0 r < , 0 θ π and 0 φ < 2 π . These parameters identify the coordinates of a point P in a 3D space where r represents the distance from the origin of coordinates O to P , θ is the angle from the positive x 3 -axis of the corresponding Cartesian system to the line connecting the origin of coordinates O to P , and φ is the same as in the cylindrical coordinate system. The basis vectors of the spherical system ( e r , e θ and e φ ) are pointing in the direction of increasing r, θ and φ respectively. Hence, all these basis vectors are coordinate- dependent as they vary in direction from point to point. All these basis vectors are
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1.3.1 Coordinate Systems 20 x 1 x 2 e 1 φ ρ P O e ρ e 2 e φ x = ρ cos φ y = ρ sin φ Figure 3: Plane polar coordinate system, superimposed on a 2D rectangular Cartesian system in a standard position, and their basis vectors e ρ and e φ
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  • Summer '20
  • Rajendra Paramanik
  • Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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