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Unformatted text preview: 7 Curvilinear coordinates Read: Boas sec. 5.4, 10.8, 10.9. 7.1 Review of spherical and cylindrical coords. First I’ll review spherical and cylindrical coordinate systems so you can have them in mind when we discuss more general cases. 7.1.1 Spherical coordinates Figure 1: Spherical coordinate system. The conventional choice of coordinates is shown in Fig. 1. θ is called the “polar angle”, φ the “azimuthal angle”. The transformation from Cartesian coords. is x = r sin θ cos φ y = r sin θ sin φ z = r cos θ. (1) In the figure the unit vectors pointing in the directions of the changes of the three spherical coordinates r, θ, φ are also shown. Any vector can be expressed in terms of them: ~ A = A x ˆ x + A y ˆ y + A z ˆ z = A r ˆ r + A θ ˆ θ + A φ ˆ φ. (2) Note the qualitatively new element here: while both ˆ x, ˆ y, ˆ z and ˆ r, ˆ θ, ˆ φ are three mutually orthogonal unit vectors, ˆ x, ˆ y, ˆ z are fixed in space but ˆ r, ˆ θ, ˆ φ point in different directions according to the direction of vector ~ r . We now ask by how large 1 a distance ds the head of the vector ˆ r changes if infinitesimal changes dr, dθ, dφ are made in the three spherical directions: ds r = dr , ds θ = rdθ , ds φ = r sin θdφ, (3) as seen from figure 2 (only the ˆ θ and ˆ φ displacements are shown). Figure 2: Geometry of infinitesimal changes of ~ r . So the total change is d~s = dr ˆ r + rdθ ˆ θ + r sin θdφ ˆ φ. (4) The volume element will be dτ = ds r ds θ ds φ = r 2 sin θ dr dθ dφ, (5) and the surface measure at constant r will be...
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This note was uploaded on 04/17/2008 for the course PHZ 3113 taught by Professor Hirshfeld during the Fall '07 term at University of Florida.
- Fall '07