Introducing polar coordinates

In order to investigate spherically symmetric problems, it is adequate to introduce polar coordinates (r, r), ip) which are related to the cartesian coordinates of the same point by 

Note that can be defined for xi = 0 by the limit of the expression above, but there is always an ambiguity on the half-plane xi 0, X2 = 0 (which includes the a 3-axis). (This is because coordinates on a sphere cannot be defined globally. If necessary, one has to choose another set of local spherical coordinates). 

Given a function (x), we may substitute x = (a i,a 2,a 3) by the functions expressing the cartesian coordinates in terms of polar coordinates and define a new function 

Performing this substitution in the normalization integral gives 

Here L S ) denotes the set of functions of and v that are square-integrable on the sphere S. The Hilbert space H has a basis which consists of wave functions of the form 

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In comparing the algebraic and differential approach in Chapter 2 it will turn out to be useful to consider also the case of two dimensions. Introducing polar coordinates in the plane,... 

It is convenient to introduce polar coordinates, k and [Pg.171]

Due to the simple product form of the Maxwell-Boltzmann distribution, the derivations given above are easily generalized to the expression for the relative velocity in three dimensions. Since the integrand in Eq. (2.18) (besides the Maxwell-Boltzmann distribution) depends only on the relative speed, we can simplify the expression in Eq. (2.18) further by integrating over the orientation of the relative velocity. This is done by introducing polar coordinates for the relative velocity. The full three-dimensional probability distribution for the relative speed is... 

Next we consider the vector (Ru R2, R3) for which we and -x 2 such that can introduce polar coordinates... 

The integral over r is evaluated by introducing polar coordinates (r, 9, reference axis of 6 being taken along the vector k), giving... 

Introducing polar coordinates, x = pcosO and y = psin8 and rescaling as... 

To describe the fiber cross-section, we introduce polar coordinates r and ff, as shown in Fig. 23-2(b), whence (r, ) form a set of orthogonal coordinates... 

Introducing polar coordinates (x = rcosfi, y = rsinfi), and omitting the time factor, one obtains for the amplitude distribution at the entrance aperture ... 

Traditional hydrogenic orbitals used in atomic and molecular physics as expansion bases belong to the nlm) representation, which in configuration space corresponds to separation in polar coordinates, and in momentum space to a separation in spherical coordinates on the (Fock s) hypersphere [1], The tilm) basis will be called spherical in the following. Stark states npm) have also been used for atoms in fields and correspond to separation in parabolic coordinates an ordinary space and in cylindrical coordinates on (for their use for expanding molecular orbitals see ref. [2]). A third basis, to be termed Zeeman states and denoted nXm) has been introduced more recently by Labarthe [3] and has found increasing applications [4]. 

It is convenient to introduce a coordinate system where the positive z direction coincides with e, (Fig. 13.11). Without loss of generality we may take the scattering plane to be the yz plane, in which instance the scattering amplitudes for two orthogonal polarization states of the incident light are (see Chapter 3, particularly Fig. 3.3 and Section 3.4)... 

Another fracture criterion has been introduced by Irwin, based on the stress intensity existing at a point of polar coordinates, r and 9 from the tip of a sharp crack of length 2a in a body uniformly stressed by an applied stress, a0. For regions close to the crack tip, the components of the stress tensor at the considered point take the form ... 

Equation (11) is written in the form of Newton s second law and states that the mass times acceleration of a fluid particle is equal to the sum of the forces causing that acceleration. In flow problems that are accelerationless (Dx/Dt = 0) it is sometimes possible to solve Eq. (11) for the stress distribution independently of any knowledge of the velocity field in the system. One special case where this useful feature of these equations occurs is the case of rectilinear pipe flow. In this special case the solution of complex fluid flow problems is greatly simplified because the stress distribution can be discovered before the constitutive relation must be introduced. This means that only a first-order differential equation must be solved rather than a second-order (and often nonlinear) one. The following are the components of Eq. (11) in rectangular Cartesian, cylindrical polar, and spherical polar coordinates ... 

To make the shield tensor DSij(d) explicit, we introduce spherical coordinate systems (r, 6, tp) with the polar axis in the x, direction, i = 1,2, 3 (Fig. 1). It then follows from Eq. (23) that DSij d) — Sy(d) — <)ijSrr has the values... 

Introducing the polar coordinates (p, 6) on the initial z-plane, we can rewrite Eq. (4) as follows ... 

In a treatment using polar coordinates, it may be convenient to introduce a basis of the form... 

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