## 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

Show Summary Details
Page of

PRINTED FROM OXFORD SCHOLARSHIP ONLINE (www.oxfordscholarship.com). (c) Copyright Oxford University Press, 2017. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a monograph in OSO for personal use (for details see http://www.oxfordscholarship.com/page/privacy-policy). Subscriber: null; date: 01 March 2017

# (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)
$Display mathematics$

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

(C.2)
$Display mathematics$
(C.3)
$Display mathematics$
(C.4)
$Display mathematics$

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

$Display mathematics$

The complete differential in x can be written

$Display mathematics$
and similarly for y and z. The differential of the position vector r in the Cartesian basis is
$Display mathematics$

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

$Display mathematics$

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

(C.5)
$Display mathematics$
and
(C.6)
$Display mathematics$

Remembering that we have supposed the curvilinear coordinates orthogfinal,

$Display mathematics$
we write Eq. C.6
$Display mathematics$
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
$Display mathematics$
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)
$Display mathematics$

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

$Display mathematics$
and
$Display mathematics$

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

(C.8)
$Display mathematics$

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

(C.9)
$Display mathematics$
(C.10)
$Display mathematics$

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

(C.11)
$Display mathematics$
(C.12)
$Display mathematics$
(C.13)
$Display mathematics$

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

(C.14)
$Display mathematics$
and the differential volume element in spherical coordinates is obtained from the product of ds 1, ds 2, ds 3 with
(C.15)
$Display mathematics$
(C.16)
$Display mathematics$
(C.17)
$Display mathematics$

The differential volume element in spherical coordinates is thus

(C.18)
$Display mathematics$

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)
$Display mathematics$
(C.20)
$Display mathematics$

# 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)
$Display mathematics$

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

(C.22)
$Display mathematics$

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)
$Display mathematics$
and also remembering that dsi = hi dqi we have
(C.24)
$Display mathematics$

# 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)
$Display mathematics$

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

$Display mathematics$
and the differential of the surface integral is
$Display mathematics$
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)
$Display mathematics$

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

(C.27)
$Display mathematics$

# C.3 The Curl in Curvilinear Coordinates

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

$Display mathematics$
and the easiest way to remember it is by writing the expression as determinant
$Display mathematics$

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

$Display mathematics$

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)
$Display mathematics$

# 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,

$Display mathematics$

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)

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.

$Display mathematics$

The differential volume element dV is

$Display mathematics$

From which we identify

$Display mathematics$

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)
$Display mathematics$
(C.30)
$Display mathematics$
(C.31)
$Display mathematics$

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

(C.32)
$Display mathematics$

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,

$Display mathematics$

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

$Display mathematics$

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

$Display mathematics$
$Display mathematics$

(p.242)

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)
$Display mathematics$
(C.34)
$Display mathematics$
(C.35)
$Display mathematics$

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

(C.36)
$Display mathematics$

Or, expanding the radial terms,

(C.37)
$Display mathematics$