VECTOR OPERATORS IN ORTHOGONAL CURVILINEAR COORDINATES

9. VECTOR OPERATORS IN ORTHOGONAL CURVILINEAR COORDINATES

We have previously (Chapter 6, Sections 6 and 7) defined the gradient (∇u), the divergence (∇ · V), the curl (∇ × V), and the Laplacian (∇ 2 u) in rectangular coordinates x, y, z. Since in many practical problems it is better to use some other coordinate system (cylindrical or spherical, for example), we need to see how to

express the vector operators in terms of general orthogonal coordinates x 1 ,x 2 ,x 3 . (We consider only orthogonal coordinate systems here; see Section 10 for the more general case.) We shall outline proofs of the formulas; some of the details of the proofs are left to the problems.

Gradient, ∇u. In Chapter 6, Section 6, we showed that the directional derivative du/ds in a given direction is the component of ∇u in that direction.

Example 1. In cylindrical coordinates, if we go in the r direction (θ and z constant), then by (8.5) ds = dr. Thus the r component of ∇u is du/ds when ds = dr, that is, ∂u/∂r. Similarly, the θ component of ∇u is du/ds when ds = r dθ, that is, (1/r)(∂u/∂θ). Thus ∇u in cylindrical coordinates is

In general orthogonal coordinates x 1 ,x 2 ,x 3 , the component of ∇u in the x 1 direction (x 2 and x 3 constant) is du/ds if ds = h 1 dx 1 [from (8.11)]; that is, the component of ∇u in the direction e 1 is (1/h 1 )(∂u/∂x 1 ). Similar formulas hold for the other components and we have

h 1 ∂x 1 h 2 ∂x 2 h 3 ∂x 3 (9.2)

3 1 ∂u

h i ∂x i

i=1

526 Tensor Analysis Chapter 10

Divergence, ∇ · V Let (9.3)

V=e 1 V 1 +e 2 V 2 +e 3 V 3

be a vector with components V 1 ,V 2 ,V 3 in an orthogonal system. We can prove (Problem 1) that

h 1 h 2 h 1 h 3 h 2 h 3 Let us write (9.3) as

We find ∇ · V by taking the divergence of each term on the right side of (9.5). Using (7.6) of Chapter 6, namely

∇· (φv) = v · (∇φ) + φ∇ · v,

with φ = h 2 h 3 V 1 and v = e 1 /h 2 h 3 , we find that the divergence of the first term on the right side of (9.5) is

By (9.4), the last term in (9.7) is zero. In the first term on the right side of (9.7), the dot product of e 1 with ∇(h 2 h 3 V 1 ) is the first component of ∇(h 2 h 3 V 1 ). By (9.2), this is

1 ∂ (h 2 h 3 V 1 ).

h 1 ∂x 1

Calculating the divergence of the other terms of (9.5) in a similar way, we get ∇·

Example 2. In cylindrical coordinates, h 1 = 1, h 2 = r, h 3 = 1. By (9.8), the divergence in cylindrical coordinates is

V=

(rV r )+

(V θ )+

(rV z )

r ∂r

∂z

1 ∂V θ

∂V z

(rV r )+

r ∂r

r ∂θ

∂z

Section 9 Vector Operators in Orthogonal Curvilinear Coordinates 527

2 2 Laplacian, ∇ 2 u. Since ∇ u = ∇ · ∇u we can find ∇ u by combining (9.2) and (9.8) with V = ∇u. We get

Example 3. In cylindrical coordinates, the Laplacian is then

Curl, ∇ × V. By methods similar to those used in finding ∇ · V we can find ∇×

V (Problem 2). The result is

Example 4. In cylindrical coordinates, we find

PROBLEMS, SECTION 9

1. Prove (9.4) in the following way. Using (9.2) with u = x 1 , show that ∇x 1 =e 1 /h 1 . Similarly, show that ∇x 2 =e 2 /h 2 and ∇x 3 =e 3 /h 3 . Let e 1 ,e 2 ,e 3 in that order form a right-handed triad (so that e 1 × e 2 =e 3 , etc.) and show that ∇x 1 ×∇ x 2 = e 3 /(h 1 h 2 ). Take the divergence of this equation and, using the vector identities (h) and (b) in the table at the end of Chapter 6, show that ∇ · (e 3 /h 1 h 2 ) = 0. The other parts of (9.4) are proved similarly.

528 Tensor Analysis Chapter 10

2. Derive the expression (9.11) for curl V in the following way. Show that ∇x 1 =e 1 /h 1

and ∇ × (∇x 1 ) = ∇ × (e 1 /h 1 ) = 0. Write V in the form e 1 e 2 e 3

and use vector identities from Chapter 6 to complete the derivation. 3. Using cylindrical coordinates write the Lagrange equations for the motion of a par-

ticle acted on by a force F = −∇V , where V is the potential energy. Divide each Lagrange equation by the corresponding scale factor so that the components of F (that is, of −∇V ) appear in the equations. Thus write the equations as the com- ponent equations of F = ma, and so find the components of the acceleration a. Compare the results with Problem 8.2.

4. Do Problem 3 in spherical coordinates; compare the results with Problem 8.3. 5. Write out ∇U , ∇ · V, ∇ 2 U , and ∇ × V in spherical coordinates.

Do Problem 3 for the coordinate systems indicated in Problems 6 to 9. Compare the results with Problems 8.11 to 8.14.

6. Parabolic cylinder 7. Elliptic cylinder 8. Parabolic

9. Bipolar

Do Problem 5 for the coordinate systems indicated in Problems 10 to 13. 10. Parabolic cylinder

11. Elliptic cylinder 12. Parabolic

13. Bipolar

In each of the following coordinate systems, find the scale factors h u and h v ; the basis vectors e u and e v ; the u and v Lagrange equations, and from them the acceleration components (see Problem 3).

14. x = u − v,

√ x = uv, 15. √

y=2 uv. y=u 1−v 2 . Use equations (9.2), (9.8), and (9.11) to evaluate the following expressions

16. In cylindrical coordinates, ∇ · e r ,∇·e θ ,∇×e r ,∇×e θ . 17. In spherical coordinates, ∇ · e r ,∇·e θ ,∇×e θ ,∇×e φ . 18. In cylindrical coordinates, ∇ × k ln r, ∇ ln r, ∇ · (re r + ze z ). 19. In spherical coordinates, ∇ × (re θ ), ∇(r cos θ), ∇ · r. 20. 2 2 In cylindrical coordinates, ∇ 2 r, ∇ (1/r), ∇ ln r.

21. 2 2 (r 2 In spherical coordinates, ∇ 2 r, ∇ ), ∇ (1/r 2 2 ), ∇ ikr e cos θ .

Section 10 Non-Cartesian Tensors 529