International Journal of Modeling and Optimization, Vol. 5, No. 3, June 2015

Divergence and Curl Operators in Skew Coordinates

Rajai S. Alassar and Mohammed A. Abushoshah

 where r is the position vector in any three dimensional space, Abstract—The divergence and curl operators appear in i.e. r r(,,) u1 u 2 u 3 . numerous differential equations governing engineering and physics problems. These operators, whose forms are well  r Note that is a tangent vector to the ui curve where known in general orthogonal coordinates systems, assume  ui different casts in different systems. In certain instances, one the other two coordinates variables remain constant. A unit needs to custom-make a coordinates system that my turn out to be skew (i.e. not orthogonal). Of course, the known formulas for tangent vector in this direction, therefore, is the divergence and curl operators in orthogonal coordinates are not useful in such cases, and one needs to derive their rr counterparts in skew systems. In this note, we derive two uuii r formulas for the divergence and curl operators in a general ei  or  hiei (3) coordinates system, whether orthogonal or not. These formulas r hi ui generalize the well known and widely used relations for ui orthogonal coordinates systems. In the process, we define an orthogonality indicator whose value ranges between zero and unity. The Divergence of a vector field over a control volume V bounded by the surface S , denoted by  F , is defined by Index Terms—Coordinates systems, curl, divergence, Laplace, skew systems.  F n dS F lim S (4) I. INTRODUCTION V 0 V The curl and divergence operators play significant roles in physical relations. They arise in fluid mechanics, elasticity where n is an outward unit normal vector to the surface S . theory and are fundamental in the theory of electromagnetism, The divergence actually measures the net outflow of a vector [1], [2]. field from an infinitesimal volume around a given point (or The physical significance of the Curl of a vector field F , how much a vector field "converges to" or "diverges from" a denoted by F , is that it measures the amount of rotation given point). The general expression of the divergence for or angular momentum of the contents of a given region of arbitrary orthogonal curvilinear coordinates is given by space. If the value of the curl is zero then the field is said to be irrotational. The curl is defined in an arbitrary orthogonal 1   F  () F1 h 2 h 3  curvilinear coordinates (,,)u u u as h1 h 2 h 3 u 1 1 2 3 (5)  ()()F h h F h h he h e h e 2 1 3 3 1 2  11 2 2 3 3 uu23 1    F  (1) h1 h 2 h 3 u 1  u 2  u 3 The Laplacian operator of a scalar  (,,)u1 u 2 u 3 , 2 F1 h 1 F 2 h 2 F 3 h 3 denoted by   is defined as the divergence of the gradient of  ; that is where e is a unit vector in the direction of u , and i i FFFF1e13  2 e 2  3 e . The length of the tangent vector in 1  hh  2  ()  23  the direction of u is known as scale factor, h , and is defined   i i h1 h 2 h 3 u 1 h 1 u 1 by (6) hh13  hh12     u h  u  u h  u  r 2 2 2 3 3 3  h  (2) i u where i 1 1   1    e  e  e (7) Manuscript received March 29, 2015; revised May 19, 2015. This work h u1 h  u 2 h  u 3 was supported by King Fahd University of Petroleum & Minerals. 1 1 2 2 3 3 The authors are with the Department of Math and Stat, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia (e-mail: Equations (1) and (5) are valid for orthogonal systems [email protected], [email protected]).

DOI: 10.7763/IJMO.2015.V5.461 198 International Journal of Modeling and Optimization, Vol. 5, No. 3, June 2015 only (i.e. e1 , e2 , and e3 are orthogonal), [3]-[5].  ee31e2 w 3 h 1 h 2 du 1 du 2 du 3 (11) u3 II. DIVERGENCE FOR GENERAL COORDINATES SYSTEMS Using the same argument, the net flux through the The Divergence Theorem relates the flow, or the flux, of a remaining two pairs of surfaces are: vector field through a surface to the behavior of the vector field inside the surface. More precisely, it states that the  outward flux of a vector field through a closed surface is ee31e2 w 2 h 1 h 3 du 1 du 2 du 3 equal to the volume integral of the divergence over the region u2 inside the surface, i.e.  ee31e2 w 1 h 2 h 3 du 1 du 2 du 3 u1 w n dS  wdv (8) SD w n dS on the left hand side of equation (8) becomes where D is a closed bounded region with piecewise smooth boundary S, n is an outer unit vector normal to the surface    S , w we  w e  w e , wn represents the component ewhh3 1 2 ewhh 2 1 3  ewhh 1 2 3  11 2 2 3 3 u3  u 2  u 1 (12) of w in the direction of n , and dv is the volume bounded du1 du 2 du 3 by the region D.

where eeee31 2 is an orthognality indicator of the system, 01e  . The volume element, dv, is given by

dv h3e3 du 3  h 1 e 1 du 1 h 2 e 2 du 2  (13)

The right hand side of equation (8), then, is Fig. 1. An infinitesimal control volume bounded by the surface. wdv   w he du  h e du  h e du  Considering the bottom shaded surface dS of the given 33 3 1 1 1 2 2 2 control volume, Fig. 1, one can find that  wee31  e2 hhhdududu 1 2 3 1 2 3  (14)

 we hhh1 2 3 dududu 1 2 3  rr  uu12 rr In the limit, by equating (12) and (14), an expression for n  , and dS du12 du (9) rr uu the divergence of a vector field w in a general coordinates  12 uu12 system, whether orthogonal or not, is given by

Hence 1   w  () e w1 h 2 h 3  eh1 h 2 h 3 u 1 rr (15)   uu12rr ()()e w2 h 1 h 3 e w 3 h 1 h 2  w n dS  w  du12 du uu23 rruu  12 uu 12 It is easy to see that when e 1 (orthogonal system),  ()w1e1  w 2 e 2  w 3 e 3 h 1 e 1  h 2 e 2  du 1 du 2 (10) equation (15) reduces to (5).

w3e3 e 1× e 2  h 1 h 2 du 1 du 2

III. CURL FOR GENERAL COORDINATES SYSTEMS On the upper surface, using Taylor series, the outward flux is

w3e 3 e12 e  h 1 h 2 du 1 du 2  we e e  h h du du du ... u 3 312 1 2 1 2 3 3

The net flux through the upper and lower surfaces, then, is Fig. 2. An infinitesimal surface enclosed by the closed path c.

199 International Journal of Modeling and Optimization, Vol. 5, No. 3, June 2015

Stokes' Theorem relates the surface integral of the curl of a 1  vector field over a surface to the line integral of the vector ()[][]w1  h 3 wee32  h 2 w (21) e h h u u field over the surface boundary, or 3 2 2 3

Equations (19), (20) and (21) can be written in the concise w n dS   w dr (16) form Sc

he h e h e Considering the bottom shaded surface of the given 11 2 2 3 3 1    control volume as shown in Fig. 2, w  (22) This surface is bounded by the curve c which is traced eh1 h 2 h 3 u 1  u 2  u 3 counterclockwise. This curve is comprised of the four h1 we1 h 2 w e 2 h 3 w e 3 segments c , c , c , and c . We evaluate the right hand side 1 2 3 4 and the left hand side of (16) over this surface and two other Obviously, the difference between orthogonal and surfaces not opposite to each other. On the shown shaded nonorthogonal coordinates systems lies in the introduction of surface, one can write the scalar triple product eeee into the expressions of 31 2 w n dS  F , and  F . This product returns a value of unity if the coordinates system is orthogonal. ()()()we   w e   w e  11 2 2 3 3  hhee 1212hee h du du (17) 112 2 1 2 IV. EXAMPLE hh12ee12 Consider the simple right-handed coordinates system () we e  e h h du du  33  1 2  1 2 1 2 (,,)u v z which is related to the Cartesian system by the

(  w )3 e h 1 h 2 du 1 du 2 relations

x u v where ()w is the component of w in the direction i yu (23) of e . i zz The term w dr in equation (16), over the four segments comprising the curve c, is This system is composed of the planes: yu which are planes parallel to the xz plane, w dr w dr  w dr  w dr  w dr y x v which are planes that are extrusions of the straight c1 c 2 c 3 c 4 w dr  w dr  w dr  w dr lines y x u into the z -direction, and planes that are c1 c 3 c 4 c 2 parallel to the xy plane. w he du  111 The level curves of the coordinates system on the xy  w h1ee11 du 1   w h 1 du 1 du 2   plane are shown in Fig. 3. u2

w h22e2 du   w h2ee 2 du2 w h 2 2 du 2 du 1 u1

Or

 wdr wh2ee21  wh 1 dudu 1 2  (18) uu12 Fig. 3. Example of a non-orthogonal coordinates system. In the limit, after equating (17) and (18), Obviously, the coordinates system is not orthogonal. Simple calculations reveal that 1  ()[][]w3  h 2 wee21  h 1 w (19) e h1 h 2 u 1 u 2 r hh1 u   2 u Similar results can be obtained for the other two surfaces. r hh   1 (24) 2 v v 1  ()[][]w  h wee  h w (20) r 2 1 1 3 3 hh3 z   1 e h3 h 1 u 3 u 1 z

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The corresponding unit vectors are obtained. These formulas generalize the well known and widely used relations for orthogonal coordinates systems. rr The difference between orthogonal and nonorthogonal e /  (ij  ) / 2 1 uu coordinates systems lies in the introduction of the orthogonality indicator eeee which returns a value rr 31 2 e /  i (25) 2 vv of unity if the coordinates system is orthogonal. rr e / k ACKNOWLEDGMENT 3 zz The authors would like to thank King Fahd University of Petroleum & Minerals (KFUPM) for supporting this research Now, the orthogonality indicator is under grant IN131060.

0 0 1 REFERENCES 1 1 1 [1] R. S. Alassar and H. M. Badr, “Analytical solution of oscillating eeee31 2 0  (26) inviscid flow over oblate spheroids with spheres and flat disks as 2 2 2 special cases,” Ocean Engng, vol. 24, no. 3, pp. 217-225, 1997. 1 0 0 [2] R. S. Alassar and B. J. Alminshawy, “Heat conduction from two spheres,” Aiche Journal, vol. 56, no. 9, pp. 2248-2256, 2010. [3] O. Moon and D. Spencer, Field Theory for Engineers (The Van The value of the orthogonality indicator being different Nostrand Series in Electronics and Communications), D. Van from unity indicates that the system is indeed skew. Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1961, One can now find the corresponding operator (15) as pp. 1-48. [4] G. Arfken, Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, 1985, pp. 100-152. [5] P. Morse and H. Feshbach, Methods of Theoretical Physics, Part I, New 1 FF12F3 F    (27) York: McGraw-Hill, 1953, pp. 660-690. 2 u1  u 2  u 3

and the three components of the curl in (22) to be Rajai S. Alassar is a professor in the Department of Mathematics and Statistics at King Fahd University of Petroleum & Minerals (KFUPM). He was born in w (ww / 2) (w )  2 3  21 (28) Gaza in 1965. He holds a PhD in hydrodynamics. 1 vz He has published many papers in his field of interest  and has authored 2 textbooks in Arabic on the subject of Math Olympiad. He has also translated the famous book of Zill on differential equations to Arabic. (ww / 2) w (w )  2 12  3 (29) 2 zu

Mohammed Abushoshah is a lecturer in the ( w1 / 2  w 2 )  ( w 1  w 2 / 2) (w )3   2 (30) Department of Mathematics and Statistics at King Fahd uv University of Petroleum & Minerals (KFUPM). He was born in Jeddah in 1987. He holds an MSc in applied mathematics. He has published several papers in applied V. CONCLUSION mathematics with concentration on fluid mechanics and Two formulas for the divergence and curl operators in any thermal sciences. coordinates systems whether orthogonal or not have been

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