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Orthogonal Curvilinear Coordinate Systems A Orthogonal Curvilinear Coordinate Systems A A-l Curvilinear Coordinates The location of a point in three-dimensional 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. It is often more convenient to describe the position of the point by another set of coordinates more appropriate to the problem at hand, common examples being spherical and cylindrical coordinates. ,These are but special cases of curvilinear coordinate systems, whose general properties we propose to ex­ amine in detail. Suppose that ql> q2> and qa are independent functions of position such that ql = ql (x, y, z), q2 = q2(X, y, z), qa = qa(x, y, z) (A-1.l) or, in terms of the position vector, qk = qk(R) (k = 1,2,3) In regions where the Jacobian determinant, oql oql oql ox oy oz o(ql> q2' qa) = Oq2 Oq2 ~ (A-1.2) o(x,y, z) ox oy oz oqa oqa oqa ox oy oz 474 Orthogonol Curvilinear Coordinate Systems 475 is different from zero this system of equations can be solved simultaneously for x, y, and z, giving y = y(qj, q2, q3), or R = R(qj, q2' q3) A vanishing of the Jacobian implies that qj, q2' and q3 are not independent functions but, rather, are connected by a functional relationship of the form f(qj, q2, q3) = O. In accordance with (A-I.3), the specification of numerical values for qj, q2, and q3 leads to a corresponding set of numerical values for x, y, z; that is, it locates a point (x, y, z) in space. Thus, we come to regard the set of three numbers (qj, q2' q3) as the curvilinear coordinates of a point in space. It is natural in dealing with physical problems to restrict attention to systems of curvilinear coordinates in which each point in space may be represented at least once by letting qj, q2, and q3 vary over all possible values. Curvilinear coordinates have a simple geometric interpretation. If for the moment we ascribe some constant value to qk> we have qk(X, y, z) = constant (k = 1,2,3) which describes a surface in space. By assigning a series of different values to qk, we generate a family of surfaces on which qk is constant .. If the func­ tions have been properly chosen there is at least one surface belonging to each of the three families which passes through any arbitrary point P in space. Thus, a point in space is characterized by the intersection of the three surfaces, qj = constant, q2 = constant, q3 = constant (see Fig. A-I.I), termed coordinate surfaces. The coordinate surface is named for that co­ ordinate which is constant, the other two coordinates being variable along q3 - coordinate curve \ \ ~oordinate surface Coordinate surface qz = constant ql = constant ,,~ Coordinate surface qz - coordinate curve ql - coordinate q3 = constant curve Figure A-1.1. Curvilinear coordinates. 476 Appendix A that surface. The intersection of any two coordinate surfaces results in a skew curve termed a coordinate curve. For example, the intersection of the q2- and q3-coordinate surfaces results in the coordinate curve labeled qt. Since this curve lies simultaneously on the surfaces q2 = constant and q3 = constant, only q[ varies as we move along the curve; hence, the designation q[-coordinate curve. In the special case of cartesian coordinates the coordinate surfaces con­ sist of three mutually perpendicular planes; the coordinate curves consist of three mutually perpendicular lines. In cartesian coordinates the differentials dx, dy, and dz correspond to distances measured along each of the three cartesian coordinate curves. The analogous differentials in curvilinear coordinates, dq[, dq2' and dqs, do not necessarily have a similar interpretation. As in Fig. A-l.1, let dl[ be the distance measured along the q[-coordinate curve from the point P(q[, q2, qs) to the neighboring point (q[ + dq[, q2, q3). Similar definitions apply to dl2 and dis. We define the three quantities* (k = 1,2,3) (A-1.4) These quantities (or simple variations q3 - coordinate curve thereof) are termed metrical coeffi­ cients. They are intrinsic properties of any particular system of curvilinear coordinates and, in general, are func­ tions of position, hk = hk(qt> q2, qs) Let it, i2 and is be unit vectors drawn tangent to the q[-, q2- and qs­ q2 - coordinate coordinate curves, respectively, in the q1- coordinate curve directions of algebraically increasing curve qk'S (Fig. A-1.2). It is evident that these Figure A.1.2. Unit tangent vectors. unit tangent vectors are given by (k = 1,2,3) (A-1.5) Whereas the magnitudes of these unit vectors are necessarily constant, (A-1.6) it does not follow that their directions remain constant from point to point, so the unit vectors are, in general, functions of position, ik = ik(q[, q2' qs) *Some authors prefer to define these symbols to be the reciprocals of the values given here. Orthogonal Curvilinear Coordinate Systems 477 The three non-coplanar unit vectors ilo i2, ia are said to constitute a base set of unit vectors for the particular system of curvilinear coordinates. Any arbitrary vector, D, may be uniquely expressed in terms of them by a relation of the form D = il UI + i2 u2 + iaua. A simple calculation shows that Upon combining Eqs. (A-I.4) and (A-1.5), we obtain ik = hk oR (A-I. 7) Oqk which provides an alternative formula for determining the metrical coef­ ficients, (A-I. 8) The utility of this particular expression lies in the calculation of metrical coefficients for systems of curvilinear coordinates defined explicitly by Eq. (A-1.3). Thus, if we put R=ix+jy+kz it is evident that and hence, -\- = (OX)2 + (Oy)2 + (~)2 (k = 1,2,3) (A-1.9) hk Oqk Oqk Oqk For example, in cartesian coordinates where ql = x, q2 = y, qa = z, we find that hI = h2 = ha = I. A-2 Orthogonal Curvilinear Coordinates If (qlo q20 qa) are the curvilinear coordinates of a point P whose position vector is Rand (ql + dqlo q2 + dq2, qa + dqa) those of a neighboring point Q whose position vector is R + dR, then PQ = dR = oR dql + oR dq2 + oR dqa oql Oq2 oqa -1_ • -dql +. 1 -dq2 +.la- dqa (A-2.1) 1 hI 2 h2 ha Thus, the distance dl between these adjacent points is given by df2 = IdRI 2 = dq~ + dq~ + dq~ + 2(i .i) dql dq2 h~ h~ h~ I 2 hlh2 (A-2.2) ' .) dq2 dqa 2(' .) dqa dql + 2(12'la --+ la'll -- h2ha hshl 478 Appendix A When the system of curvilinear coordinates is such that the three co­ ordinate surfaces are mutually perpendicular at each point, it is termed an orthogonal curvilinear coordinate system. In this event the unit tangent vectors to the coordinate curves are also mutually perpendicular at each point and thus ijoik = 0 (j, k = 1,2,3) (j -=F k) (A-2.3) whereupon dl2 = dqi + dql + ~qJ (A-2.4) hi h~ hi Therefore, an essential attribute of orthogonal systems is that mixed terms of the form dqjdqk (j -=F k) do not appear in the expression for the distance dl. This condition is not only necessary for orthogonality but is sufficient as well; for ql, q2, and q3 in Eq. (A-2.2) are independent variables. In consequence of Eq. (A-I.7) the necessary and sufficient conditions for orthogonality may also be expressed by the relations oR 0 oR = 0 (j, k = 1,2,3) (j-=F k) (A-2.S) oqj Oqk or, putting R = ix + jy + kz, ox ox + oy oy + ~ ~ = 0 (j, k = 1,2,3) (j -=F k) (A-2.6) oqj Oqk oqj Oqk oqj Oqk which provides a useful test of orthogonality for systems of curvilinear coordinates defined explicitly by the relations x = x(ql' q2' q3), y = y(ql' q2' q3), z = z(ql' q2, q3) If, instead, the system of coordinates is defined explicitly by the equations ql = ql (x, y, z), q2 = q2(X, y, z), q3 = q3(X, y, z) the computation of the partial derivatives required in Eq. (A-2.6) can be a tedious chore, and it is best to proceed somewhat differently. In consequence of the general properties of the gradient operator, each of the vectors Vqk (k = 1,2,3) is necessarily perpendicular to the corresponding coordinate surface qk = constant. Thus, the necessary and sufficient conditions for orthogonality are equally well expressed by the relations Vqjo Vqk = 0 (j, k = 1,2,3) (j -=F k) (A-2.7) or, expressing V in cartesian coordinates, oqj Oqk + oqj Oqk + oqj Oqk = 0 ox ox oy oy oz OZ (A-2.8) (j, k = 1,2,3) (j -=F k) These may be contrasted with Eqs. (A-2.6). If the system of coordinates does prove orthogonal we can avail ourselves of still another method for computing the metrical coefficients. For, in this Orthogonal Curvilinear Coordinate Systems 479 event, the q2- and qa-coordinate surfaces are perpendicular to the ql-co­ ordinate surfaces. But since the ql-coordinate curves lie simultaneously on each of the former surfaces, these curves must be perpendicular to the sur­ faces ql = constant. In general, then, the qk-coordinate curves lie normal to the surfaces on which qk is constant. The general properties of the V operator are such that the vector V qk is normal to the surfaces on which qk is constant and points in the direction of increasing qk' Consequently, the unit tangent vector, ik> to the qk-coordinate curve passing through a particular point in space is identical to the unit normal vector, Ok, to the qk-coordinate surface passing through the point in question.
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