Orthogonal coordinate system pdf

2019-10-14 01:56

Orthogonal Curvilinear Coordinates 1 De nitions Let x (x 1; x 2; x 3) be the Cartesian coordinates of a point M with respect to a frame of reference de ned by the unit vectors e 3); j 1; 3; (1) in a region R. The equation u j c j, where c j is a constant, represents a surface. The system of two equations u 2 c 2 and u 3 c 3 representVector operators in curvilinear coordinate systems In a Cartesian system, take x 1 x, x 2 y, and x 3 z, then an element of arc length ds2 is, ds2 dx2 1 dx 2 2 dx 2 3 In a general system of coordinates, we still have x orthogonal coordinate system pdf

, all other orthogonal coordinate systems have their own set of vectors. In polar coordinates, these vectors are r, f, z; in spherical polar coordinates, they are r, q, f. We are interested in expressing these basis vectors in terms of x, y, z, and also to express x, y, z in terms of the basis vectors in other coordinate systems.

coordinate system can be de ned starting from the orthogonal cartesian one. If x, y, zare the cartesian coordinates, the curvilinear ones, u, v, w, can be expressed as smooth functions of x, The choice of a speci c coordinate system is decided by the geometry of the given problem. There are 8 orthogonal coordinate systems, namely 1. Cartesian Coordinate System 2. Cylindrical Coordinate System 3. Spherical Coordinate System 4. Parabolic Cylindrical Coordinate System 5. Conical Coordinate System 6. Prolate Spheroidal Coordinate System 7. orthogonal coordinate system pdf Orthogonal Curvilinear Coordinate Systems Al Curvilinear Coordinates A The location of a point in threedimensional space (with respect to some origin) is usually specified by giving its three cartesian coordinates (x, y, z) or, what is equivalent, by specifying the position vector R of the point.

Of the orthogonal coordinate systems, there are several that are in common use for the description of the physical world. Certainly the most common is the Cartesian or rectangular coordinate system (xyz). Probably the second most common and of paramount importance for astronomy is the system orthogonal coordinate system pdf Orthogonal Curvilinear Coordinates. 571. S AND. This is the general expression for the gradient operator, valid for any orthogonal, curvi linear coordinate system. Several identities involving. Ui, hi' and e i are useful in deriving expressions for the other differential operators. From Eq.

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