for the polar system and e 1 and e 2 for the Cartesian system The point P in

For the polar system and e 1 and e 2 for the

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for the polar system and e 1 and e 2 for the Cartesian system. The point P in the figure is identified simultaneously by ( x, y ) coordinates in the Cartesian system and by ( ρ, φ ) coordinates in the polar system where these coordinates are related through the two sets of Eqs. 9 and 10. mutually perpendicular and they are defined to be of unit length . Figure 4 is a graphical illustration of the spherical coordinate system and its basis vectors with a cor- responding reference rectangular Cartesian system in a standard position. The transformation from the Cartesian coordinates ( x, y, z ) of a particular point in the space to the spherical coordinates ( r, θ, φ ) of that point, where the two systems are in a standard position, is performed by the following equations: [9] r = p x 2 + y 2 + z 2 θ = arccos z p x 2 + y 2 + z 2 ! φ = arctan y x (11) [9] Again, arctan ( y x ) in the third equation should be selected consistent with the signs of x and y .
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1.3.1 Coordinate Systems 21 x 1 x 2 x 3 e r e φ φ P O θ r e θ Figure 4: Spherical coordinate system, superimposed on a rectangular Cartesian system in a standard position, and its basis vectors e r , e θ and e φ in a 3D space. The point P in the figure is identified simultaneously by ( x, y, z ) coordinates in the Cartesian system and by ( r, θ, φ ) coordinates in the spherical system where these coordinates are related through the two sets of Eqs. 11 and 12. while the opposite transformation from the spherical to the Cartesian coordinates is performed by the following equations: x = r sin θ cos φ y = r sin θ sin φ z = r cos θ (12) E. General Curvilinear Coordinate System The general curvilinear coordinate system is characterized by having coordinate axes which
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1.3.1 Coordinate Systems 22 are curved in general. Also, its basis vectors are generally position-dependent and hence they are variable in magnitude and direction throughout the system. Consequently, the basis vectors are not necessarily of unit length or mutually orthogonal . A graphic demonstration of the general curvilinear coordinate system at a particular point in the space with its covariant basis vectors (see § 2.6.1) is shown in Figure 5. x 3 x 1 x 2 E 1 E 3 E 2 Figure 5: General curvilinear coordinate system and its covariant basis vectors E 1 , E 2 and E 3 (see § 2.6.1) in a 3D space, where x 1 , x 2 and x 3 are the labels of the coordinates. F. General Orthogonal Curvilinear Coordinate System The general orthogonal curvilinear coordinate system is a subset of the general curvilinear coordinate system as described above. It is distinguished from the other subsets of the general curvilinear system by having coordinate axes and basis vectors which are mutually orthogonal throughout the system. The cylindrical and spherical systems are examples of orthogonal curvilinear coordinate systems.
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1.3.2 Vector Algebra and Calculus 23 1.3.2 Vector Algebra and Calculus This subsection provides a short introduction to vector algebra and calculus, a subject that is closely related to tensor calculus. In fact many ideas and methods of tensor calculus
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  • Summer '20
  • Rajendra Paramanik
  • Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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