## Mauro Fabrizio and Angelo Morro

Print publication date: 2003

Print ISBN-13: 9780198527008

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198527008.001.0001

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# D Differential operators in curvilinear coordinates

Source:
Electromagnetism of Continuous Media
Publisher:
Oxford University Press

# D.1 Differentiation

Denote by z = (z 1, z 2, z 3) the rectangular coordinates of a point in the three-dimensional space and by x = (x 1, x 2, x 3) three coordinates so that a coordinate transformation holds, namely,

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(p.645) The position vector x is expressed in rectangular coordinates as x = zkik. We can then view x as a function of x in the form x(x) = zk(x)ik. The vectors

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are tangential to the coordinate curves. They are taken to be linearly independent and hence they are a possible basis. The inner products
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determine the arc length through dx · dx = gkl dxk dx1. The curvilinear coordinates are said orthogonal if and only if gkl = 0 everywhere when kl. The reciprocal base vectors {gk} are defined uniquely by
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Since the vectors {gk} are linearly independent, the matrix [ghk] is nonsingular. Let g = det[ghk] ≠ 0. Any vector ν, at any point x, may be expressed through the two bases as

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The components νk are called contravariant, the components νk covariant. It follows that

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Letting ghk = gh · gk, we have

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Accordingly, the matrix [ghk] is the inverse of [ghk]. Since

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it follows that
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In correspondence with curvilinear coordinates, it is convenient to consider physical components, namely, the components relative to an orthonormal basis. Since gkk > 0, let hk = (gkk)1/2, k not summed, and define ek to be the unit vectors

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(p.646) The differentiation in curvilinear coordinates is based on the definition of covariant derivative, νh;k, such that

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By definition,

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Since the bases {gk} and {gk} are reciprocal of each other,

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Hence, letting

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we have
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and
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Likewise,

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The bracketed terms are known as the Christoffel symbols of the second kind. They are related to the Christoffel symbols of the first kind, [kl, m] = gm · ∂g k/∂x l, by

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Also,

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For second-order tensors T, expressed, for example, as T=T klg kg l, the covariant derivative is defined such that

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(p.647) Hence, because

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we find that
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Setting m = l provides the divergence of a second-order tensor.

# D.2 Gradient, divergence, curl and the Laplacian

A scalar function ψ : ℝ3 ⊃ Ω → ℝ is said to be differentiable at x 0 if a vector w exists such that

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If such is the case then w is said to be the gradient of ψ, ▿ψ, at x 0. In rectangular coordinates, we have

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Since

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it follows that
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In curvilinear coordinates, letting ψ(z) = ψ (x(z)), we obtain

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Consequently, we have

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This allows ▿ to be viewed as the operator

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Hence, we let the gradient of a vector, the divergence and the curl be defined by

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(p.648) Also,

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Moreover, the definitions give

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and
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where
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# D.3 Orthogonal curvilinear coordinates

Let g 1, g 2, g 3 be orthogonal to one another. Hence,

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As a consequence,

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and the covariant, contravariant and physical components, respectively, νj, νj and ν^j, of a vector ν are related by
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Accordingly, we can write

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Also, a direct application of the definition shows that

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(p.649) where equal indices are not summed and j, k, l are unequal. Consequently, we have
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Explicit expressions are now given for cylindrical and spherical coordinates. Cylindrical coordinates. (Fig. D.1). Let x 1 = r, x 2 = θ, x 3 = z; r ∈ ℝ+, θ ∈ [0, 2π), z ∈ ℝ. Hence we have

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whence
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Fig. D.1 Cylindrical polar coordinates (left) and spherical polar coordinates (right).

(p.650) Denote the orthonormal basis by er, e θ, ez and the associated (physical) components of ν by νr, νθ, νz. Substitution gives

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Spherical coordinates. Let x 1 = r, x 2 = θ, x 3 = φ; r ∈ ℝ+, θ ∈ [0, π], φ ∈ [0, 2π). Hence, we have

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whence
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Denote the orthonormal basis by er, e θ, e φ and the associated (physical) components of ν by νr, νθ, νφ. Substitution gives

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