Orthogonal Curvilinear Coordinates 28.3 Introduction The derivatives div, grad and curl from Section 29.2 can be carried out using coordinate systems other than the rectangular cartesian coordinates. This Section shows how to calculate these derivatives in other coordinate systems. Two coordinate systems - cylindrical polar coordinates and spherical polar coordinates - will be illustrated. Prerequisites Before starting this Section you should... Learning Outcomes After completing this Section you should be able to... 1 be able to find the gradient, divergence and curl of a field in cartesian coordinates. 2 be familiar with polar coordinates be able to find the divergence, gradient or curl of a vector or scalar field expressed in terms of orthogonal curvilinear coordinates.
1. Orthogonal Curvilinear Coordinates The results shown in Section 29.2 have been given in terms of the familiar cartesian (x, y, z) coordinate system. However, other coordinate systems can be used to better describe some physical situations. A set of coordinates u u(x, y, z), v v(x, y, z) and w w(x, y, z) where the directions at any point indicated by u, v and w are orthogonal (perpendicular) to each other is referred to as a set of orthogonal curvilinear coordinates. With each coordinate is associated ( a scale factor h u, h v or h w respectively where h u x ) 2 ( u + y ) 2 ( u + z 2 u) (with similar expressions for h v and h w ). The scale factor gives a measure of how a change in the coordinate changes the position of a point. Two commonly-used sets of orthogonal curvilinear coordinates are cylindrical polar coordinates and spherical polar coordinates. These are similar to the plane polar coordinates introduced in Section 17.2 but represent extensions to three dimensions. Cylindrical Polar Coordinates This corresponds to plane polar (, φ) coordinates with an added z coordinate directed out of the xy plane. Normally the variables and φ are used instead of r and θ to give the three coordinates, φ and z. A cylinder has equation constant. The relationship between the coordinate systems is given by x cos φ y sin φ z z (i.e. the same z is used by the two coordinate systems). See Figure 1. z (x, y, z) φ y x, Figure 1 The scale factors h, h φ and h z are given as follows (x ) 2 ( ) 2 ( ) 2 y z h + + (cos φ) 2 +(sin φ) 2 + 01 (x ) 2 ( ) 2 ( ) 2 y z h φ + + ( sin φ) φ φ φ 2 +(cos φ) 2 +0 HELM (VERSION 1: April 14, 2004): Workbook Level 1 2
(x ) 2 h z + z ( ) 2 y + z ( ) 2 z (0 z 2 +0 2 +1 2 )1 Spherical Polar Coordinates In this system a point is referred to by its distance from the origin r and two angles φ and θ. The angle θ is the angle between the positive z-axis and the line from the origin to the point. The angle φ is the angle from the x-axis to the projection of the point in the xy plane. A useful analogy is of latitude, longitude and height on Earth. The variable r plays the role of height (but height measured above the centre of Earth rather than from the surface). The variable θ plays the role of latitude but is modified so that θ 0represents the North Pole, θ 90 π 2 represents the equator and θ 180 π represents the South Pole. The variable φ plays the role of longitude. A sphere has equation r constant. The relationship between the coordinate systems is given by x r sin θ cos φ y r sin θ sin φ z r cos θ. See Figure 2. z (x, y, z) θ r φ y x, Figure 2 The scale factors h r, h θ and h φ are given by (x ) 2 ( ) 2 ( ) 2 y z h r + + (sin θ cos φ) r r r 2 +(sin θ sin φ) 2 + (cos θ) 2 1 (x ) 2 ( ) 2 ( ) 2 y z h θ + + (r cos θ cos φ) θ θ θ 2 +(rcos θ sin φ) 2 +( rsin θ) 2 r (x ) 2 ( ) 2 ( ) 2 y z h φ + + ( r sin θ sin φ) φ φ φ 2 +(rsin θ sin φ) 2 +0rsin θ 3 HELM (VERSION 1: April 14, 2004): Workbook Level 1
2. Vector Derivatives in Orthogonal Coordinates Given an orthogonal coordinate system u, v, w with unit vectors û, ˆv and ŵ and scale factors, h u, h v and h w,itispossible to find the derivatives f, F and F. It is found that grad f f f h u uû + f h v v ˆv + f h w wŵ If F F u û + F vˆv + F w ŵ then div F F h u h v h w u (F uh v h w )+ v (F vh u h w )+ ] w (F wh u h v ) Also if F F u û + F vˆv + F w ŵ then h u û h vˆv h w ŵ 1 curl F F h u h v h w u v w h u F u h v F v h w F w Key Point In orthogonal curvilinear coordinates, the vector derivatives f, F and F are influenced by the scale factors h u, h v and h w. 3. Cylindrical Polar Coordinates In cylindrical polar coordinates (, φ, z), the 3 unit vectors are ˆ, ˆφ and ẑ with scale factors h 1,h φ, h z 1. The quantities and φ are related to x and y by x cos φ and y sin φ. The unit vectors are ˆ cos φi + sin φj and ˆφ sin φi + cos φj. Incylindrical polar coordinates, grad f f f ˆ + f φ ˆφ + f z ẑ The scale factor is necessary in the φ-component because the derivatives with respect to φ are distorted by the distance from the axis 0. If F F ˆ + F φ ˆφ + Fz ẑ then div F F 1 (F )+ φ (F φ)+ ] z (F z) HELM (VERSION 1: April 14, 2004): Workbook Level 1 4
curl F F 1 ˆ ˆφ ẑ φ z F F φ F z. Example In cylindrical polar coordinates, find f for (a) f 2 + z 2 (b) f 3 sin φ (c) Show that the result for (b) is consistent with that found working in cartesian coordinates. Solution (a) If f 2 + z 2 then f 2,f φ 0andf 2z so f 2ˆ +2zẑ. z (b) If f 3 sin φ then f 32 sin φ, f φ 3 cos φ and f 0and hence, z f 3 2 sin φˆ + 2 cos φ ˆφ. (c) f 3 sin φ 2 sin φ (x 2 + y 2 )y x 2 y + y 3 so f 2xyi +(x 2 +3y 2 )j. Using cylindrical polar coordinates, from (b) we have f 3 2 sin φˆ + 3 cos φ ˆφ 3 2 sin φ(cos φi + sin φj)+ 2 cos φ( sin φi + cos φj) 3 2 sin φ cos φ 2 sin φ cos φ ] i + 3 2 sin 2 φ + 2 cos 2 φ ] j 2 2 sin φ cos φ ] i + 3 2 sin 2 φ + 2 cos 2 φ ] j 2xyi +(3y 2 + x 2 )j So the results using cartesian and cylindrical polar coordinates are consistent. Example Find F for F F ˆ + F φ ˆφ + Fz ẑ 3 ˆ + z ˆφ + z sin φẑ. Show that the results are consistent with those found using cartesian coordinates. 5 HELM (VERSION 1: April 14, 2004): Workbook Level 1
Solution Here, F 3, F φ z and F z z sin φ so F 1 (F )+ φ (F φ)+ ] z (F z) 1 (4 )+ φ (z)+ ] z (2 z sin φ) 1 4 3 +0+ 2 sin φ ] 4 2 + sin φ Converting to cartesian coordinates, F F ˆ + F φ ˆφ + Fz ẑ 3 ˆ + z ˆφ + z sin φẑ 3 (cos φi + sin φj)+z( sin φi + cos φj)+z sin φk ( 3 cos φ z sin φ)i +( 3 sin φ + z cos φ)+zk 2 ( cos φ) sin φz ] i + 2 ( sin φ)+ cos φz ] j + sin φzk (x 2 + y 2 )x yz ] i + (x 2 + y 2 )y + xz ] j + yzk (x 3 + xy 2 yz)i +(x 2 y + y 3 + xz)j + yzk So F x (x3 + xy 2 yz)+ y (x2 y + y 3 + xz)+ z (yz) (3x 2 + y 2 )+(x 2 +3y 2 )+y 4x 2 +4y 2 + y 4(x 2 + y 2 )+y 4 2 + sin φ So F is the same in both coordinate systems Example Find F for F 2 ˆ + z sin φ ˆφ +2z cos φẑ HELM (VERSION 1: April 14, 2004): Workbook Level 1 6
Solution F ˆ ˆφ ẑ φ z 1 F F φ F z ˆ 1 1 φ 2z cos φ z sin φ z ˆ ˆφ ẑ φ z 2 z sin φ 2z cos φ ] + ˆφ ˆ( 2z sin φ sin φ)+ ˆφ(0) + ẑ(z sin φ) (2z sin φ) ˆ + z sin φ ẑ z 2 2z cos φ ] ] +ẑ ]] z sin φ φ 2 Example A magnetic field B is given by B 2 ˆφ + kz. Find B and B. Solution B 1 0+ φ 2 + ] z k 1 0 + 0 +0]0 ˆ ˆφ ẑ ˆ ˆφ ẑ B φ z φ z B B φ B z 2 0 k 0 All magnetic fields satisfy B 0i.e. an absence of magnetic monopoles. There is a class of magnetic fields known as potential fields that satisfy B O (a) Using cylindrical polar coordinates, find f for f r 2 z sin θ (b) Using cylindrical polar coordinates, find f for f z sin 2θ 7 HELM (VERSION 1: April 14, 2004): Workbook Level 1
(a) 2z sin θˆ + z cos θ ˆφ + 2 sin θẑ (b) 2 z cos 2θ ˆφ + sin 2θẑ Find F for F cos φˆ sin φˆ + zẑ (a) Find the derivatives F ], (b) Combine these to find F φ F φ], z F z] HELM (VERSION 1: April 14, 2004): Workbook Level 1 8
(a) 2 cos φ, cos φ, 2 (b) F 1 1 1 (F )+ φ (F φ)+ ] z (F z) (2 cos φ)+ φ ( sin φ)+ z (2 z) ] 2 cos φ cos φ + 2 ] cos φ + 2 Find F for F F ˆ + F φ ˆφ + Fz ẑ 3 ˆ + z ˆφ + z sin φẑ. Show that the results are consistent with those found using cartesian coordinates. (a) Find the curl F (b) Find F in cartesian coordinates. (c) Hence find F in cartesian coordinates. (d) Using ˆr cos φi + sin φj and ˆθ sin φi + cos φj show that the solution to part (a) is equal to the solution for part (c). (a) (z cos φ )ˆ 2 sin φ ˆφ +2zẑ (b) (x 3 + xy 2 yz)i +(x 2 y + y 3 + xz)+yzk (c) (z x)i yj +2zk 9 HELM (VERSION 1: April 14, 2004): Workbook Level 1
(a) For F ˆ +(sin θ + z) ˆφ + zẑ, find F and F. (b) For f r 2 z 2 cos2φ, find ( f). (a) 1+cos θ +, ˆ z cos θ ˆφ +(2 sin φ + z)ẑ (b) 0 4. Spherical Polar Coordinates In spherical polar coordinates (r, θ, φ), the 3 unit vectors are ˆr, ˆθ and ˆφ with scale factors h r 1, h θ r, h φ r sin θ. The quantities r, θ and φ are related to x, y and z by x r sin θ cos φ, y r sin θ sin φ and z r cos θ. Inspherical polar coordinates, grad f f f r ˆr + f r θ ˆθ + f φ ˆφ If F F rˆr + F θ ˆθ + Fφ ˆφ then div F F r (r2 sin θf r )+ θ (r sin θf θ)+ ] φ (rf φ) HELM (VERSION 1: April 14, 2004): Workbook Level 1 10
curl F F 1 ˆr rˆθ ˆφ r θ φ F r rf θ F φ Example In spherical polar coordinates, find f for (a) f r (b) f 1 r (c) f r 2 sin(φ + θ) Solution (a) (b) (c) f f r ˆr + f r θ ˆθ + 1 (r) r ˆr + 1 r 1ˆr ˆr f φ ˆφ (r) θ ˆθ + 1 f f r ˆr + f r θ ˆθ + 1 ( 1 r ) r ˆr + 1 r 1 r 2 ˆr f f r ˆr + f r θ ˆθ + 1 (r2 sin(φ + θ) ˆr + r r f φ ˆφ ( 1) r θ ˆθ + 1 (r) φ ˆφ ( 1 r ) φ ˆφ f φ ˆφ (r 2 sin(φ + θ) ˆθ + 1 θ (r 2 sin(φ + θ) φ 2r sin(φ + θ)ˆr + 1 r r2 cos(φ + θ)ˆθ + 1 r2 cos(φ + θ) ˆφ 2r sin(φ + θ)ˆr + r cos(φ + θ)ˆθ + cos(φ + θ) sin θ ˆφ ˆφ 11 HELM (VERSION 1: April 14, 2004): Workbook Level 1
Example Using spherical polar coordinates, find F for the following vector functions. (a) F rˆr (b) F ˆr (c) F r sin θ ˆr +r 2 sin φ ˆθ +r cos θ ˆφ Solution (a) F r (r2 sin θf r )+ θ (r sin θf θ)+ ] φ (rf φ) r (r2 sin θ r)+ ] (r sin θ 0) + (r 0) θ φ r (r3 sin θ)+ θ (0) + ] φ (0) 1 3 +0+0 ] 3 Note :- in cartesian coordinates, the corresponding vector is F xi + yj + zk with F 1 + 1 + 1 3(hence consistency). (b) F r (r2 sin θf r )+ θ (r sin θf θ)+ ] φ (rf φ) r (r2 sin θ)+ ] (r sin θ 0) + (r 0) θ φ r (r4 sin 2 θ)+ θ (0) + ] φ (0) 1 4r 3 sin 2 θ +0+0 ] 4rsin θ (c) F r (r2 sin θf r )+ θ (r sin θf θ)+ ] φ (rf φ) r (r2 sin θrsin θ)+ θ (r sin θ r2 sin φ)+ ] (r r cos θ) φ r (r3 sin 2 θ)+ θ (r3 sin θ sin φ)+ ] φ (r2 cos φ) 1 3r 2 sin 2 θ + r 3 cos θ sin φ +0 ] 3sin θ + r cot θ sin φ Example Find F for the following vector fields F. (a) F r kˆr, where k is a constant (b) F r 2 cos θ ˆr + sin θ ˆθ + sin 2 θ ˆφ HELM (VERSION 1: April 14, 2004): Workbook Level 1 12
Solution (a) F ˆr rˆθ ˆφ 1 r θ φ F r rf θ F φ ˆr rˆθ ˆφ 1 r θ φ r k r 0 0 ( θ (0) ) ( φ (0) ˆr + φ (rk ) ) r (0) rˆθ ( + r (0) ) ] θ (rk ) ˆφ 0 ˆr +0 ˆθ +0 ˆφ 0 (b) ˆr rˆθ ˆφ ˆr rˆθ ˆφ 1 F 1 r θ φ r θ φ F r rf θ F φ r 2 cos θ r sin θ sin 2 θ ( θ (r2 sin 3 θ) ) ( (r sin θ) ˆr + φ φ (r2 cos θ) ) r (r2 sin 3 θ) rˆθ ( + r (r sin θ) ) ] θ (r2 cos θ) ˆφ 1 (3r 2 sin 2 θ cos θ +0 ) ˆr + ( 0 2r sin 3 θ ) rˆθ + ( sin θ + ) ˆφ] 3sin θ cos θ ˆr 2 sin 2 ˆθ +(1+r 2 ) sin θ ˆφ 13 HELM (VERSION 1: April 14, 2004): Workbook Level 1
(a) Using spherical polar coordinates, find f for i. f r 4 ii. f r r 2 +1 iii. f r 2 sin 2θ cos φ (b) For F r sin θˆr + r cos φˆθ + r sin φ ˆφ, find F and F. (c) For F r 4 cos θˆr + r 4 sin θˆθ, find F and F. (d) For F r 2 cos θˆr + cos φˆθ find ( F ). (a) (i) 4r 3ˆr, (ii) 1 r 2 (1 + r 2 ) ˆr, (iii) 2r sin 2θ cos φˆr +2rcos 2θ cos φˆθ 2r cos θ sin φ ˆφ 2 (b) cos φ(cot θ + cosechθ)+3sin θ,cot θ 2 sin φˆr 2 sin φˆθ +(2cos φ cos θ) ˆφ HELM (VERSION 1: April 14, 2004): Workbook Level 1 14
(c) 0, 2r 5 sin θ ˆφ (d) 0 15 HELM (VERSION 1: April 14, 2004): Workbook Level 1