31 Coordinate Systems 16 plane polar system The most general type of coordinate

31 coordinate systems 16 plane polar system the most

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1.3.1 Coordinate Systems 16 plane polar system. The most general type of coordinate system is the general curvilinear. A subset of the latter is the orthogonal curvilinear. These types of coordinate system are briefly investigated in the following subsections. A. Orthonormal Cartesian Coordinate System This is the simplest and the most commonly used coordinate system. It consists, in its simplest form, of three mutually orthogonal straight axes that meet at a common point called the origin of coordinates O . The three axes, assuming a 3D space, are scaled uniformly and hence they all have the same unit length. Each axis has a unit vector oriented along the positive direction of that axis. [6] These three unit vectors are called the basis vectors or the bases of the system. These basis vectors are constant in magnitude and direction throughout the system. [7] This system with its basis vectors ( e 1 , e 2 and e 3 ) is depicted in Figure 1. The three axes, as well as the basis vectors, are usually labeled according to the right hand rule , that is if the index finger of the right hand is pointing in the positive direction of the first axis and its middle finger is pointing in the positive direction of the second axis then the thumb will be pointing in the positive direction of the third axis. B. Cylindrical Coordinate System The cylindrical coordinate system is defined by three parameters: ρ, φ and z which range over: 0 ρ < , 0 φ < 2 π and -∞ < z < . These parameters identify the coordinates of a point P in a 3D space where ρ represents the perpendicular distance from the point to the x 3 -axis of a corresponding rectangular Cartesian system, φ represents the angle between the x 1 -axis and the line connecting the origin of coordinates O to the [6] As indicated before, these features are what qualify a Cartesian system to be described as orthonormal. [7] In fact, the basis vectors are constant only in rectangular Cartesian and oblique Cartesian coordinate systems. As indicated before, the oblique Cartesian systems are the same as the rectangular Cartesian but with the exception that their axes are not mutually orthogonal. Also, labeling the oblique as Cartesian may be controversial.
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1.3.1 Coordinate Systems 17 x 1 x 2 x 3 O v v 3 v 1 v 2 O x 1 x 2 x 3 e 1 e 2 e 3 Figure 1: Orthonormal right-handed Cartesian coordinate system and its basis vectors e 1 , e 2 and e 3 in a 3D space (left frame) with the components of a vector v in this system (right frame). perpendicular projection of the point on the x 1 - x 2 plane of the corresponding Cartesian system, and z is the same as the third coordinate of the point in the reference Cartesian system. The sense of the angle φ is given by the right hand twist rule, that is if the fingers of the right hand curl in the sense of rotation from the x 1 -axis towards the line of projection, then the thumb will be pointing in the positive direction of the x 3 -axis.
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  • Summer '20
  • Rajendra Paramanik
  • Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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