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Light-Matter InteractionPhysics and Engineering at the Nanoscale$

John Weiner and Frederico Nunes

Print publication date: 2012

Print ISBN-13: 9780198567653

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780198567653.001.0001

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(p.235) Appendix C Gradient, Divergence, and Curl in Cylindrical and Polar Coordinates

(p.235) Appendix C Gradient, Divergence, and Curl in Cylindrical and Polar Coordinates

Source:
Light-Matter Interaction
Publisher:
Oxford University Press

Although Cartesian coordinates are the most familiar and serve many purposes, they are not the only orthogfinal coordinate system that can be used to define a set of basis vectors. Often we can take advantage of the natural symmetry of a problem to express vectors in some other orthogfinal curvilinear coordinate system. The most common of these are the cylindrical and polar coordinates because they are appropriate for many practical problems.

In general we can expand a vector V in basis vectors of the Cartesian system or some other system with basis vectors {q},

(C.1)
V = x ^ V x + y ^ V y + z ^ V z or V = q ^ 1 V 1 + q ^ 2 V 2 + q ^ 3 V 3

However, we cannot treat the position vector with this general rule. In this special case

(C.2)
r = x x ^ + y y ^ + z z ^ Cartesian coordinates
(C.3)
r = ρ ρ ^ + z z ^ cylindrical coordinates
(C.4)
r = r r ^ spherical coordinates

In the general case we convert from the Cartesian system to another through the relations

x = x ( q 1 , q 2 , q 3 ) y = y ( q 1 , q 2 , q 3 ) z = z ( q 1 , q 2 , q 3 )

The complete differential in x can be written

d x = x q 1 d q 1 + y q 2 d q 2 + y q 3 d q 3
and similarly for y and z. The differential of the position vector r in the Cartesian basis is
d r = d x ^ + d y ^ + d z ^

(p.236) and the square of the distance between two neighboring points can then be written

d s 2 = d r · d r = d x 2 + d y 2 + d z 2

But the differential of dr can also be expanded in the curvilinear coordinate system

(C.5)
d r = r q 1 d q 1 + r q 2 d q 2 + r q 3 d q 3
and
(C.6)
d s 2 = d r · d r = i j r q i · r q j d q i d q j

Remembering that we have supposed the curvilinear coordinates orthogfinal,

q ^ i · q ^ j = δ i j
we write Eq. C.6
d s 2 = i 2 r q i 2 d q i 2 = g 11 d q 1 2 + g 22 d q 2 2 + g 33 d q 3 2 = i ( h i d q i ) 2
where we have identified g i i = h i 2 . We see from Eq. C.6 that hi is a scale factor such that the differential length segment dsi can be written
d s i = h i d q i and r q i = h i q ^ i
and from Eq. C.5 the position vector differential dr can be expanded in terms of these scale factors in the curvilinear basis as
(C.7)
d r = h 1 d q 1 q ^ 1 + h 2 d q 2 q ^ 2 + h 3 d q 3 q ^ 3

From the line segment dsi we can easily write the surface and volume elements σ, τ in the curvilinear system.

d σ i j = d s i d s j = h i h j d q i d q j i j
and
d τ = d s 1 d s 2 d s 3 = h 1 h 2 h 3 d q 1 d q 2 d 3

We can expand an area vector σ in the curvilinear basis set analogous to the position vector r in Eq. C.7

(C.8)
d σ = d s 2 d s 3 d q ^ 1 + d s 3 d s 1 q ^ 2 + d s 1 d s 2 q ^ 3 = h 2 h 3 d q 2 d q 3 q ^ 1 + h 3 h 1 d q 3 d q 1 q ^ 2 + h 1 h 2 d q 1 d q 2 q ^ 3

(p.237) For example, we can take an ordinary vector quantity F and expand it in Cartesian coordinates or in spherical coordinates.

(C.9)
F = F x x ^ + F y y ^ + F z z ^
(C.10)
F = F r r ^ + F θ θ ^ + F φ φ ^

The unit vectors r̂, θ̂, φ̂ in terms of the Cartesian unit vectors are

(C.11)
r ^ = sin θ cos φ x ^ + sin θ sin φ y ^ + cos θ z ^
(C.12)
θ ^ = cos θ cos φ x ^ + cos θ sin φ y ^ sin θ z ^
(C.13)
φ ^ = sin φ x ^ + cos φ y ^

The differential length vector along the surface in spherical coordinates is then

(C.14)
d l = d r r ^ + r d θ θ ^ + r sin θ d φ φ ^
and the differential volume element in spherical coordinates is obtained from the product of ds 1, ds 2, ds 3 with
(C.15)
ds 1 =h 1 ,dq 1 =dr
(C.16)
d s 2 = h 2 d q 2 = r d θ
(C.17)
d s 3 = h 3 d q 3 = r sin θ d φ

The differential volume element in spherical coordinates is thus

(C.18)
d V = d s 1 d s 2 d s 3 = d r r d θ r sin θ d φ = r 2 sin θ d θ d φ d r

The expressions for dr and dσ in Eqs C.7, C.8 give us the tools to write vector line and surface integrals in the curvilinear coordinates

(C.19)
V · d r = i V i h i d q i
(C.20)
V · d σ = i V i h j h k d q j d q k j k i

C.1 The Gradient in Curvilinear Coordinates

The gradient is a vector operator that operates on a scalar point function ψ. In Cartesian coordinates we write

(C.21)
ψ = ( x x ^ + y y ^ + z z ^ ) ψ ( x , y , z )

(p.238) An alternative integral definition is more convenient for finding the expression for the gradient in some curvilinear coordinate system.

(C.22)
ψ = lim d τ 0 ψ d σ d τ

The gradient in general is defined such that it yields the maximum spatial rate of change of the scalar function, and this maximum spatial change is independent of the coordinate system in which it is described. Therefore we should be able to express it in coordinate systems other than Cartesian. Remembering that the length segments dx, dy, dz in Cartesian coordinates go over into ds 1, ds 2, ds 3 in our chosen curvilinear system, we can write the gradient in our new coordinates as

(C.23)
ψ ( q 1 , q 2 , q 3 ) = q ^ 1 ψ s 1 + q ^ 2 ψ s 2 + q ^ 3 ψ s 3
and also remembering that dsi = hi dqi we have
(C.24)
ψ ( q 1 , q 2 , q 3 ) = q ^ 1 1 h 1 ψ q 1 + q ^ 2 1 h 2 ψ q 2 + q ^ 3 1 h 3 ψ q 3

C.2 The Divergence in Curvilinear Coordinates

The divergence ∇ · V of some vector field V can be expressed as a differential in Cartesian coordinates as

(C.25)
( x x ^ + y y ^ + z z ^ ) · ( V x x ^ + V y y ^ + V z z ^ )

Similar to the gradient operation, the divergence can be generalized to curvilinear coordinates by considering it as the result of taking the limit of a differential vector field surface integral divided by the differential volume enclosed by the surface as the volume approaches zero. Taking this limiting ratio operation as the definition of the divergence we have

· V ( q 1 , q 2 , q 3 ) = lim d τ 0 δ V · d σ d τ
and the differential of the surface integral is
δ V ( q 1 , q 2 , q 3 , ) · d σ = [ ( V 1 h 2 h 3 ) q 1 + ( V 2 h 3 h 1 ) q 2 + ( V 3 h 1 h 2 ) q 3 ] d q 1 d q 3 d q 3
and taking ∫dτ in the limit as ds 1 ds 2 ds 3 = h 1 dq 1 h 2 dq 2 h 3 dq 3 we have for the limiting ratio

(p.239)

(C.26)
· V ( q 1 , q 2 , q 3 ) = 1 h 1 h 2 h 3 [ q 1 ( V 1 h 2 h 3 ) + q 2 ( V 2 h 3 h 1 ) + q 3 ( V 3 h 1 h 2 ) ]

Another useful general expression is the scalar Laplacian operator, ∇ · ∇ which we can get from combining the grad and div operations,

(C.27)
· ψ = 1 h 1 h 2 h 3 [ q 1 ( h 2 h 3 h 1 ψ q 1 ) + q 2 ( h 3 h 1 h 2 ψ q 2 ) + q 3 ( h 1 h 2 h 3 ψ q 3 ) ]

C.3 The Curl in Curvilinear Coordinates

The familiar differential expression for the curl operation in Cartesian coordinates is

× V = ( V z y V y z ) x ^ + ( V x z V z x ) y ^ + ( V y x V x y ) z ^
and the easiest way to remember it is by writing the expression as determinant
× V = | x ^ y ^ z ^ x y z V x V y V z |

As with the gradient and divergence operations, the curl can also be written as the limiting integral operation

× V = lim d τ 0 d σ × V d τ

It can be shown that by using this integral limiting form and applying Stokes’ theorem the curl operation can be written in curvilinear coordinates as

(C.28)
× V = 1 h 1 h 2 h 3 | q ^ 1 h 1 q ^ 2 h 2 q ^ 3 h 3 q 1 q 2 q 3 h 1 V 1 h 2 V 2 h 3 V 3 |

C.4 Expressions for Grad, Div, Curl in Cylindrical and Polar Coordinates

Circular Cylindrical Coordinates. Circular Cylindrical Coordinates consist of three independent variables, ρ, φ, z and their associated unit vectors, ρ̂, φ̂ ẑ. The limits on these variables are,

0 ρ 0 φ 2 π z +

Figure C.1 shows the relation between the Cartesian coordinates x, y, z and the cylindrical coordinates, ρ, φ, z. The z coordinate is common to both systems. From Fig. C.1 it is evident that

(p.240)

Appendix C Gradient, Divergence, and Curl in Cylindrical and Polar Coordinates

Fig. C.1 Cylindrical coordinates ρ, φ, z and unit vectors ρ̂, φ̂ ẑ and their relations to Cartesian coordinates and unit vectors, x, y, z, x̂, ŷ, ẑ.

ρ = ( x 2 + y 2 ) x = ρ cos φ y = ρ sin φ z = z

The differential volume element dV is

d V = d s ρ d s φ d s z = ρ d ρ ρ d φ d z

From which we identify

h 1 = h ρ = 1 h 2 = h φ = ρ h 3 = h z = 1

Then from the general expressions for grad, div, curl, Eqs C.24, C.26, C.28, we write these operators in cylindrical coordinates as

(C.29)
grad ψ ( ρ , φ , z ) = ψ ρ ρ ^ + 1 ρ ψ φ φ ^ + ψ z z ^
(C.30)
div · V = 1 ρ ( ρ V ρ ) ρ + 1 ρ V φ φ + V z z
(C.31)
curl × V = 1 ρ | ρ ^ ρ φ ^ z ^ ρ φ z V ρ ρ V φ V z |

(p.241) We can also find the scalar Laplacian in cylindrical coordinates by applying the general expression Eq. C.27

(C.32)
Laplacian 2 ψ = 1 ρ ρ ( ρ ψ ρ ) + 1 ρ 2 2 ψ φ 2 + 2 ψ z 2

Polar Coordinates. Polar coordinates consist of three independent variables, r, θ φ and their associated unit vectors, r̂, θ̂, φ̂. The limits on these variables are,

0 r 0 θ π 0 φ 2 π

Figure C.2 shows the relation between the Cartesian coordinates x, y, z and the polar coordinates, r, θ, φ. From Fig. C.2 we see that

r = x 2 + y 2 + z 2 x = r sin θ cos φ y = r sin θ cos φ z = rcos θ

From the differential volume dV and inspection of Fig. C.2, we can identify the scale factors

d V = d s r d s θ d s φ = d r r d θ r sin θ d φ
h r = 1 h θ = r h φ = r sin θ

(p.242)

Appendix C Gradient, Divergence, and Curl in Cylindrical and Polar Coordinates

Fig. C.2 Polar coordinates r, θ, φ and unit vectors r̂, θ̂, φ̂ and their relations to Cartesian coordinates and unit vectors, x, y, z, x̂, ŷ, ẑ.

Then from Eqs C.24, C.26, C.28 we write expressions for the grad, div and curl operators in polar coordinates
(C.33)
grad ψ = ψ r r ^ + 1 r ψ θ θ ^ + 1 r sin θ ψ φ φ ^
(C.34)
div · V = 1 r 2 sin θ [ sin θ r 2 V r r + r sin θ V θ θ + r V φ φ ]
(C.35)
curl × V = 1 r 2 sin θ | r ^ r θ ^ rsin θ φ ^ r θ φ V r r V θ rsin θ V φ |

Finally, again by applying Eq. C.27 we can obtain the Laplacian operator in polar coordinates.

(C.36)
Laplacian 2 ψ = 1 r 2 sin θ [ sin θ r ( r 2 ψ r ) + θ ( sin θ ψ θ ) + 1 sin θ 2 ψ φ 2 ]

Or, expanding the radial terms,

(C.37)
Laplacian 2 ψ = 2 ψ r 2 + 2 r ψ r + 1 r 2 sin θ [ θ ( sin θ ψ θ ) + 1 sin θ 2 ψ φ 2 ]