Tensors in curvilinear coordinates explained

Curvilinear coordinates can be formulated in tensor calculus, with important applications in physics and engineering, particularly for describing transportation of physical quantities and deformation of matter in fluid mechanics and continuum mechanics.

Vector and tensor algebra in three-dimensional curvilinear coordinates

Elementary vector and tensor algebra in curvilinear coordinates is used in some of the older scientific literature in mechanics and physics and can be indispensable to understanding work from the early and mid 1900s, for example the text by Green and Zerna.^[1] Some useful relations in the algebra of vectors and second-order tensors in curvilinear coordinates are given in this section. The notation and contents are primarily from Ogden, Naghdi,^[2] Simmonds,^[3] Green and Zerna,^[1] Basar and Weichert,^[4] and Ciarlet.^[5]

Coordinate transformations

Consider two coordinate systems with coordinate variables

(Z ^{1} , Z ^{2} ,Z ^{3} ) and (Z ^{\acute{1}} , Z ^{\acute{2}} ,Z ^{\acute{3}} ) , which we shall represent in short as just Z ^{i} and Z ^{\acute{i}} respectively and always assume our index i runs from 1 through 3. We shall assume that these coordinates systems are embedded in the three-dimensional euclidean space. Coordinates Z ^{i} and Z ^{\acute{i}} may be used to explain each other, because as we move along the coordinate line in one coordinate system we can use the other to describe our position. In this way Coordinates Z ^{i} and Z ^{\acute{i}} are functions of each other

$$Z^i=\; f^(Z^\backslash acute,\; Z^\backslash acute,Z^\backslash acute)$$ for

i =1 , 2 , 3

which can be written as

$$Z^i=\; Z^(Z^\backslash acute,\; Z^\backslash acute,Z^\backslash acute)=\; Z^(Z^\backslash acute)$$ for

\acute{i},i =1 , 2 , 3

These three equations together are also called a coordinate transformation from

Z ^{\acute{i}} to Z ^{i} .Let us denote this transformation by T . We will therefore represent the transformation from the coordinate system with coordinate variables Z ^{\acute{i}} to the coordinate system with coordinates Z ^{i} as:

$$Z=\; T(\backslash acute)$$

Similarly we can represent

Z ^{\acute{i}} as a function of Z ^{i} as follows:

$$Z^\backslash acute=\; g^(Z^,\; Z^,Z^)$$ for

\acute{i} =1 , 2 , 3

similarly we can write the free equations more compactly as

$$Z^\backslash acute=\; Z^\backslash acute(Z^,\; Z^,Z^)=\; Z^\backslash acute(Z^)$$ for

\acute{i},i =1 , 2 , 3

These three equations together are also called a coordinate transformation from

Z ^{i} to Z ^{\acute{i}} . Let us denote this transformation by S . We will represent the transformation from the coordinate system with coordinate variables Z ^{i} to the coordinate system with coordinates Z ^{\acute{i}} as:

$$\backslash acute=\; S(z)$$

If the transformation

T is bijective then we call the image of the transformation,namely Z ^{i} , a set of admissible coordinates for Z ^{\acute{i}} . If T is linear the coordinate system Z ^{i} will be called an affine coordinate system,otherwise Z ^{i} is called a curvilinear coordinate system

The Jacobian

As we now see that the Coordinates

Z ^{i} and Z ^{\acute{i}} are functions of each other, we can take the derivative of the coordinate variable Z ^{i} with respect to the coordinate variable Z ^{\acute{i}}

consider

$$\backslash frac\; \backslash ;\; \backslash overset\; \backslash ;\; J\_\backslash acute^i$$ for

\acute{i},i = 1 , 2 , 3 , these derivatives can be arranged in a matrix, say J ,in which J ^{i} _{\acute{i}} is the element in the i -th row and \acute{i} -th column

The resultant matrix is called the Jacobian matrix.

Vectors in curvilinear coordinates

Let (b₁, b₂, b₃) be an arbitrary basis for three-dimensional Euclidean space. In general, the basis vectors are neither unit vectors nor mutually orthogonal. However, they are required to be linearly independent. Then a vector v can be expressed as^[3] $$\backslash mathbf\; =\; v^k\backslash ,\backslash mathbf\_k$$The components v^k are the contravariant components of the vector v.

The reciprocal basis (b¹, b², b³) is defined by the relation ^[3] $$\backslash mathbf^i\backslash cdot\backslash mathbf\_j\; =\; \backslash delta^i\_j$$where δⁱ _j is the Kronecker delta.

The vector v can also be expressed in terms of the reciprocal basis:$$\backslash mathbf\; =\; v\_k~\backslash mathbf^k$$The components v_k are the covariant components of the vector

v .

Second-order tensors in curvilinear coordinates

A second-order tensor can be expressed as$$\backslash boldsymbol\; =\; S^~\backslash mathbf\_i\backslash otimes\backslash mathbf\_j\; =\; S^\_~\backslash mathbf\_i\backslash otimes\backslash mathbf^j\; =\; S\_^~\backslash mathbf^i\backslash otimes\backslash mathbf\_j\; =\; S\_~\backslash mathbf^i\backslash otimes\backslash mathbf^j$$The components S^ij are called the contravariant components, Sⁱ _j the mixed right-covariant components, S_i ^j the mixed left-covariant components, and S_ij the covariant components of the second-order tensor.

Metric tensor and relations between components

The quantities g_ij, g^ij are defined as^[3]

$$g\_\; =\; \backslash mathbf\_i\; \backslash cdot\; \backslash mathbf\_j\; =\; g\_\; ~;~~\; g^\; =\; \backslash mathbf^i\; \backslash cdot\; \backslash mathbf^j\; =\; g^$$From the above equations we have$$v^i\; =\; g^~v\_k\; ~;~~\; v\_i\; =\; g\_~v^k\; ~;~~\; \backslash mathbf^i\; =\; g^~\backslash mathbf\_j\; ~;~~\; \backslash mathbf\_i\; =\; g\_~\backslash mathbf^j$$

The components of a vector are related by^[3] $$\backslash mathbf\backslash cdot\backslash mathbf^i\; =\; v^k~\backslash mathbf\_k\backslash cdot\backslash mathbf^i\; =\; v^k~\backslash delta^i\_k\; =\; v^i$$$$\backslash mathbf\backslash cdot\backslash mathbf\_i\; =\; v\_k~\backslash mathbf^k\backslash cdot\backslash mathbf\_i\; =\; v\_k~\backslash delta\_i^k\; =\; v\_i$$Also,$$\backslash mathbf\backslash cdot\backslash mathbf\_i\; =\; v^k~\backslash mathbf\_k\backslash cdot\backslash mathbf\_i\; =\; g\_~v^k$$$$\backslash mathbf\backslash cdot\backslash mathbf^i\; =\; v\_k~\backslash mathbf^k\backslash cdot\backslash mathbf^i\; =\; g^~v\_k$$

The components of the second-order tensor are related by$$S^\; =\; g^~S\_k^\; =\; g^~S^i\_\; =\; g^~g^~S\_$$

The alternating tensor

In an orthonormal right-handed basis, the third-order alternating tensor is defined as$$\backslash boldsymbol\; =\; \backslash varepsilon\_~\backslash mathbf^i\backslash otimes\backslash mathbf^j\backslash otimes\backslash mathbf^k$$In a general curvilinear basis the same tensor may be expressed as$$\backslash boldsymbol\; =\; \backslash mathcal\_~\backslash mathbf^i\backslash otimes\backslash mathbf^j\backslash otimes\backslash mathbf^k\; =\; \backslash mathcal^~\backslash mathbf\_i\backslash otimes\backslash mathbf\_j\backslash otimes\backslash mathbf\_k$$It can be shown that$$\backslash mathcal\_\; =\; \backslash left[\backslash mathbf\{b\}\_i,\backslash mathbf\{b\}\_j,\backslash mathbf\{b\}\_k\backslash right]\; =(\backslash mathbf\_i\backslash times\backslash mathbf\_j)\backslash cdot\backslash mathbf\_k\; ~;~~\; \backslash mathcal^\; =\; \backslash left[\backslash mathbf\{b\}^i,\backslash mathbf\{b\}^j,\backslash mathbf\{b\}^k\backslash right]$$Now,$$\backslash mathbf\_i\backslash times\backslash mathbf\_j\; =\; J~\backslash varepsilon\_~\backslash mathbf^p\; =\; \backslash sqrt~\backslash varepsilon\_~\backslash mathbf^p$$Hence,$$\backslash mathcal\_\; =\; J~\backslash varepsilon\_\; =\; \backslash sqrt~\backslash varepsilon\_$$Similarly, we can show that$$\backslash mathcal^\; =\; \backslash cfrac~\backslash varepsilon^\; =\; \backslash cfrac~\backslash varepsilon^$$

Vector operations

Identity map

The identity map I defined by

I ⋅ v = v can be shown to be:^[3]

$$\backslash mathbf\; =\; g^\backslash mathbf\_i\backslash otimes\backslash mathbf\_j\; =\; g\_\backslash mathbf^i\backslash otimes\backslash mathbf^j\; =\; \backslash mathbf\_i\backslash otimes\backslash mathbf^i\; =\; \backslash mathbf^i\backslash otimes\backslash mathbf\_i$$

Scalar (dot) product

The scalar product of two vectors in curvilinear coordinates is^[3]

$$\backslash mathbf\backslash cdot\backslash mathbf\; =\; u^i\; v\_i\; =\; u\_i\; v^i\; =\; g\_\; u^i\; v^j\; =\; g^\; u\_i\; v\_j$$

Vector (cross) product

The cross product of two vectors is given by:^[3]

$$\backslash mathbf\backslash times\backslash mathbf\; =\; \backslash varepsilon\_u\_jv\_k\backslash mathbf\_i$$

where ε_ijk is the permutation symbol and e_i is a Cartesian basis vector. In curvilinear coordinates, the equivalent expression is:

$$\backslash mathbf\backslash times\backslash mathbf\; =[(\backslash mathbf\{b\}\_m\backslash times\backslash mathbf\{b\}\_n)\backslash cdot\backslash mathbf\{b\}\_s]\; u^m\; v^n\; \backslash mathbf^s\; =\backslash mathcal\_\; u^mv^n\backslash mathbf^s$$

where

l{E} _{ijk} is the third-order alternating tensor. The cross product of two vectors is given by:

$$\backslash mathbf\backslash times\backslash mathbf\; =\; \backslash varepsilon\_\backslash hat\_j\backslash hat\_k\backslash mathbf\_i$$

where ε_ijk is the permutation symbol and

e _{i} is a Cartesian basis vector. Therefore,

$$\backslash mathbf\_p\backslash times\backslash mathbf\_q\; =\; \backslash varepsilon\_\backslash mathbf\_i$$

and

$$\backslash mathbf\_m\backslash times\backslash mathbf\_n\; =\; \backslash frac\backslash times\backslash frac\; =\; \backslash frac\backslash times\backslash frac\; =\; \backslash frac\; \backslash frac\; \backslash mathbf\_p\backslash times\backslash mathbf\_q\; =\; \backslash varepsilon\_\backslash frac\backslash frac\backslash mathbf\_i.$$

Hence,

$$(\backslash mathbf\_m\backslash times\backslash mathbf\_n)\backslash cdot\backslash mathbf\_s\; =\backslash varepsilon\_\backslash frac\backslash frac\backslash frac$$

Returning to the vector product and using the relations:

$$\backslash hat\_j\; =\; \backslash fracu^m,\; \backslash quad\; \backslash hat\_k\; =\; \backslash fracv^n,\; \backslash quad\; \backslash mathbf\_i\; =\; \backslash frac\backslash mathbf^s,$$

gives us:

$$\backslash mathbf\backslash times\backslash mathbf=\backslash varepsilon\_\; \backslash hat\_j\; \backslash hat\_k\; \backslash mathbf\_i\; =\; \backslash varepsilon\_\; \backslash frac\; \backslash frac\backslash frac\; u^m\; v^n\backslash mathbf^s\; =\; [(\backslash mathbf\{b\}\_m\; \backslash times\; \backslash mathbf\{b\}\_n)\; \backslash cdot\backslash mathbf\{b\}\_s]\; u^m\; v^n\; \backslash mathbf^s=\; \backslash mathcal\_\; u^m\; v^n\backslash mathbf^s$$

Tensor operations

Identity map

I defined by I ⋅ v = v can be shown to be^[3]

$$\backslash mathsf\; =\; g^\backslash mathbf\_i\backslash otimes\backslash mathbf\_j\; =\; g\_\backslash mathbf^i\backslash otimes\backslash mathbf^j\; =\; \backslash mathbf\_i\backslash otimes\backslash mathbf^i\; =\; \backslash mathbf^i\; \backslash otimes\; \backslash mathbf\_i$$

Action of a second-order tensor on a vector

The action

v = \boldsymbol{S} u can be expressed in curvilinear coordinates as

$$v^i\backslash mathbf\_i\; =\; S^u\_j\backslash mathbf\_i\; =\; S^i\_u^j\backslash mathbf\_i;\backslash qquad\; v\_i\backslash mathbf^i\; =\; S\_u^i\backslash mathbf^i\; =\; S\_^u\_j\; \backslash mathbf^i$$

Inner product of two second-order tensors

The inner product of two second-order tensors

\boldsymbol{U} = \boldsymbol{S} ⋅ \boldsymbol{T} can be expressed in curvilinear coordinates as

$$U\_\backslash mathbf^i\backslash otimes\backslash mathbf^j\; =\; S\_T^k\_\; \backslash mathbf^i\backslash otimes\backslash mathbf^j=\; S\_i^T\_\backslash mathbf^i\backslash otimes\backslash mathbf^j$$

Alternatively,

$$\backslash boldsymbol\; =\; S^T^m\_g\_\backslash mathbf\_i\backslash otimes\backslash mathbf^n\; =\; S^i\_T^m\_\backslash mathbf\_i\backslash otimes\backslash mathbf^n=\; S^T\_\; \backslash mathbf\_i\; \backslash otimes\backslash mathbf^n$$

Determinant of a second-order tensor

If

\boldsymbol{S} is a second-order tensor, then the determinant is defined by the relation

$$\backslash left[\backslash boldsymbol\{S\}\backslash mathbf\{u\},\; \backslash boldsymbol\{S\}\backslash mathbf\{v\},\; \backslash boldsymbol\{S\}\backslash mathbf\{w\}\backslash right]\; =\; \backslash det\backslash boldsymbol\backslash left[\backslash mathbf\{u\},\; \backslash mathbf\{v\},\; \backslash mathbf\{w\}\backslash right]$$

where

u , v , w are arbitrary vectors and

$$\backslash left[\backslash mathbf\{u\},\backslash mathbf\{v\},\backslash mathbf\{w\}\backslash right]\; :=\; \backslash mathbf\backslash cdot(\backslash mathbf\backslash times\backslash mathbf).$$

Relations between curvilinear and Cartesian basis vectors

Let (e₁, e₂, e₃) be the usual Cartesian basis vectors for the Euclidean space of interest and let$$\backslash mathbf\_i\; =\; \backslash boldsymbol\backslash mathbf\_i$$where F_i is a second-order transformation tensor that maps ei to bi. Then,$$\backslash mathbf\_i\backslash otimes\backslash mathbf\_i\; =\; (\backslash boldsymbol\backslash mathbf\_i)\backslash otimes\backslash mathbf\_i\; =\; \backslash boldsymbol(\backslash mathbf\_i\backslash otimes\backslash mathbf\_i)\; =\; \backslash boldsymbol~.$$From this relation we can show that$$\backslash mathbf^i\; =\; \backslash boldsymbol^\backslash mathbf^i\; ~;~~\; g^\; =\; [\backslash boldsymbol\{F\}^\{-\backslash rm\{1\}\}\backslash boldsymbol\{F\}^\{-\backslash rm\{T\}\}]\_\; ~;~~\; g\_\; =\; [g^\{ij\}]^\; =\; [\backslash boldsymbol\{F\}^\{\backslash rm\{T\}\}\backslash boldsymbol\{F\}]\_$$Let

J : = \det\boldsymbol{F} be the Jacobian of the transformation. Then, from the definition of the determinant,$$\backslash left[\backslash mathbf\{b\}\_1,\backslash mathbf\{b\}\_2,\backslash mathbf\{b\}\_3\backslash right]\; =\; \backslash det\backslash boldsymbol\backslash left[\backslash mathbf\{e\}\_1,\backslash mathbf\{e\}\_2,\backslash mathbf\{e\}\_3\backslash right]\; ~.$$Since$$\backslash left[\backslash mathbf\{e\}\_1,\backslash mathbf\{e\}\_2,\backslash mathbf\{e\}\_3\backslash right]\; =\; 1$$we have$$J\; =\; \backslash det\backslash boldsymbol\; =\; \backslash left[\backslash mathbf\{b\}\_1,\backslash mathbf\{b\}\_2,\backslash mathbf\{b\}\_3\backslash right]\; =\; \backslash mathbf\_1\backslash cdot(\backslash mathbf\_2\backslash times\backslash mathbf\_3)$$A number of interesting results can be derived using the above relations.

First, consider$$g\; :=\; \backslash det[g\_\{ij\}]$$Then$$g\; =\; \backslash det[\backslash boldsymbol\{F\}^\{\backslash rm\{T\}\}]\backslash cdot\backslash det[\backslash boldsymbol\{F\}]\; =\; J\backslash cdot\; J\; =\; J^2$$Similarly, we can show that$$\backslash det[g^\{ij\}]\; =\; \backslash cfrac$$Therefore, using the fact that

[g ^{ij} ] = [g _{ij} ] ^{-1} ,$$\backslash cfrac\; =\; 2~J~\backslash cfrac\; =\; g~g^$$

Another interesting relation is derived below. Recall that$$\backslash mathbf^i\backslash cdot\backslash mathbf\_j\; =\; \backslash delta^i\_j\; \backslash quad\; \backslash Rightarrow\; \backslash quad\; \backslash mathbf^1\backslash cdot\backslash mathbf\_1\; =\; 1,~\backslash mathbf^1\backslash cdot\backslash mathbf\_2=\backslash mathbf^1\backslash cdot\backslash mathbf\_3=0\; \backslash quad\; \backslash Rightarrow\; \backslash quad\; \backslash mathbf^1\; =\; A~(\backslash mathbf\_2\backslash times\backslash mathbf\_3)$$where A is a, yet undetermined, constant. Then$$\backslash mathbf^1\backslash cdot\backslash mathbf\_1\; =\; A~\backslash mathbf\_1\backslash cdot(\backslash mathbf\_2\backslash times\backslash mathbf\_3)\; =\; AJ\; =\; 1\; \backslash quad\; \backslash Rightarrow\; \backslash quad\; A\; =\; \backslash cfrac$$This observation leads to the relations$$\backslash mathbf^1\; =\; \backslash cfrac(\backslash mathbf\_2\backslash times\backslash mathbf\_3)\; ~;~~\; \backslash mathbf^2\; =\; \backslash cfrac(\backslash mathbf\_3\backslash times\backslash mathbf\_1)\; ~;~~\; \backslash mathbf^3\; =\; \backslash cfrac(\backslash mathbf\_1\backslash times\backslash mathbf\_2)$$In index notation,$$\backslash varepsilon\_~\backslash mathbf^k\; =\; \backslash cfrac(\backslash mathbf\_i\backslash times\backslash mathbf\_j)\; =\; \backslash cfrac(\backslash mathbf\_i\backslash times\backslash mathbf\_j)$$where

\varepsilon _{ijk} is the usual permutation symbol.

We have not identified an explicit expression for the transformation tensor F because an alternative form of the mapping between curvilinear and Cartesian bases is more useful. Assuming a sufficient degree of smoothness in the mapping (and a bit of abuse of notation), we have$$\backslash mathbf\_i\; =\; \backslash cfrac\; =\; \backslash cfrac~\backslash cfrac\; =\; \backslash mathbf\_j~\backslash cfrac$$Similarly,$$\backslash mathbf\_i\; =\; \backslash mathbf\_j~\backslash cfrac$$From these results we have$$\backslash mathbf^k\backslash cdot\backslash mathbf\_i\; =\; \backslash frac\; \backslash quad\; \backslash Rightarrow\; \backslash quad\; \backslash frac~\backslash mathbf^i\; =\; \backslash mathbf^k\backslash cdot(\backslash mathbf\_i\backslash otimes\backslash mathbf^i)\; =\; \backslash mathbf^k$$and$$\backslash mathbf^k\; =\; \backslash frac~\backslash mathbf^i$$

Vector and tensor calculus in three-dimensional curvilinear coordinates

Simmonds,^[3] in his book on tensor analysis, quotes Albert Einstein saying^[6]

Vector and tensor calculus in general curvilinear coordinates is used in tensor analysis on four-dimensional curvilinear manifolds in general relativity,^[7] in the mechanics of curved shells,^[5] in examining the invariance properties of Maxwell's equations which has been of interest in metamaterials^{[8] [9]} and in many other fields.

Some useful relations in the calculus of vectors and second-order tensors in curvilinear coordinates are given in this section. The notation and contents are primarily from Ogden, Simmonds, Green and Zerna,^[1] Basar and Weichert,^[4] and Ciarlet.^[5]

Basic definitions

Let the position of a point in space be characterized by three coordinate variables

(q ^{1,} q ^{2,} q ^{3)} .

The coordinate curve q¹ represents a curve on which q², q³ are constant. Let x be the position vector of the point relative to some origin. Then, assuming that such a mapping and its inverse exist and are continuous, we can write ^[10] $$\backslash mathbf\; =\; \backslash boldsymbol(q^1,\; q^2,\; q^3)\; ~;~~\; q^i\; =\; \backslash psi^i(\backslash mathbf)\; =\; [\backslash boldsymbol\{\backslash varphi\}^\{-1\}(\backslash mathbf\{x\})]^i$$The fields ψⁱ(x) are called the curvilinear coordinate functions of the curvilinear coordinate system ψ(x) = φ⁻¹(x).

The qⁱ coordinate curves are defined by the one-parameter family of functions given by$$\backslash mathbf\_i(\backslash alpha)\; =\; \backslash boldsymbol(\backslash alpha,\; q^j,\; q^k)\; ~,~~\; i\backslash ne\; j\; \backslash ne\; k$$with q^j, q^k fixed.

Tangent vector to coordinate curves

The tangent vector to the curve x_i at the point x_i(α) (or to the coordinate curve q_i at the point x) is$$\backslash cfrac\; \backslash equiv\; \backslash cfrac$$

Gradient

Scalar field

Let f(x) be a scalar field in space. Then$$f(\backslash mathbf)\; =\; f[\backslash boldsymbol\{\backslash varphi\}(q^1,q^2,q^3)]\; =\; f\_\backslash varphi(q^1,q^2,q^3)$$The gradient of the field f is defined by$$[\backslash boldsymbol\{\backslash nabla\}f(\backslash mathbf\{x\})]\backslash cdot\backslash mathbf\; =\; \backslash cfrac\; f(\backslash mathbf+\backslash alpha\backslash mathbf)\backslash biggr|\_$$where c is an arbitrary constant vector. If we define the components cⁱ of c are such that$$q^i\; +\; \backslash alpha~c^i\; =\; \backslash psi^i(\backslash mathbf\; +\; \backslash alpha~\backslash mathbf)$$then$$[\backslash boldsymbol\{\backslash nabla\}f(\backslash mathbf\{x\})]\backslash cdot\backslash mathbf\; =\; \backslash cfrac\; f\_\backslash varphi(q^1\; +\; \backslash alpha~c^1,\; q^2\; +\; \backslash alpha~c^2,\; q^3\; +\; \backslash alpha~c^3)\backslash biggr|\_\; =\; \backslash cfrac~c^i\; =\; \backslash cfrac~c^i$$

If we set

f( x ) = \psi ^{i(x)} , then since q ^{i} = \psi ^{i(x)} , we have$$[\backslash boldsymbol\{\backslash nabla\}\backslash psi^i(\backslash mathbf\{x\})]\backslash cdot\backslash mathbf\; =\; \backslash cfrac~c^j\; =\; c^i$$which provides a means of extracting the contravariant component of a vector c.

If b_i is the covariant (or natural) basis at a point, and if bⁱ is the contravariant (or reciprocal) basis at that point, then$$[\backslash boldsymbol\{\backslash nabla\}f(\backslash mathbf\{x\})]\backslash cdot\backslash mathbf\; =\; \backslash cfrac~c^i\; =\; \backslash left(\backslash cfrac~\backslash mathbf^i\backslash right)\; \backslash left(c^i~\backslash mathbf\_i\backslash right)\; \backslash quad\; \backslash Rightarrow\; \backslash quad\; \backslash boldsymbolf(\backslash mathbf)\; =\; \backslash cfrac~\backslash mathbf^i$$A brief rationale for this choice of basis is given in the next section.

Vector field

A similar process can be used to arrive at the gradient of a vector field f(x). The gradient is given by$$[\backslash boldsymbol\{\backslash nabla\}\backslash mathbf\{f\}(\backslash mathbf\{x\})]\backslash cdot\backslash mathbf\; =\; \backslash cfrac~c^i$$If we consider the gradient of the position vector field r(x) = x, then we can show that$$\backslash mathbf\; =\; \backslash cfrac~c^i\; =\; \backslash mathbf\_i(\backslash mathbf)~c^i\; ~;~~\; \backslash mathbf\_i(\backslash mathbf)\; :=\; \backslash cfrac$$The vector field b_i is tangent to the qⁱ coordinate curve and forms a natural basis at each point on the curve. This basis, as discussed at the beginning of this article, is also called the covariant curvilinear basis. We can also define a reciprocal basis, or contravariant curvilinear basis, bⁱ. All the algebraic relations between the basis vectors, as discussed in the section on tensor algebra, apply for the natural basis and its reciprocal at each point x.

Since c is arbitrary, we can write$$\backslash boldsymbol\backslash mathbf(\backslash mathbf)\; =\; \backslash cfrac\backslash otimes\backslash mathbf^i$$

Note that the contravariant basis vector bⁱ is perpendicular to the surface of constant ψⁱ and is given by$$\backslash mathbf^i\; =\; \backslash boldsymbol\backslash psi^i$$

Christoffel symbols of the first kind

The Christoffel symbols of the first kind are defined as$$\backslash mathbf\_\; =\; \backslash frac\; :=\; \backslash Gamma\_~\backslash mathbf^k\; \backslash quad\; \backslash Rightarrow\; \backslash quad\; \backslash mathbf\_\; \backslash cdot\; \backslash mathbf\_l\; =\; \backslash Gamma\_$$To express Γ_ijk in terms of g_ij we note thatSince b_i,j = b_j,i we have Γ_ijk = Γ_jik. Using these to rearrange the above relations gives$$\backslash Gamma\_\; =\; \backslash frac(g\_\; +\; g\_\; -\; g\_)\; =\; \backslash frac[(\backslash mathbf\{b\}\_i\backslash cdot\backslash mathbf\{b\}\_k)\_\{,j\}\; +\; (\backslash mathbf\{b\}\_j\backslash cdot\backslash mathbf\{b\}\_k)\_\{,i\}\; -\; (\backslash mathbf\{b\}\_i\backslash cdot\backslash mathbf\{b\}\_j)\_\{,k\}]$$

Christoffel symbols of the second kind

The Christoffel symbols of the second kind are defined as$$\backslash Gamma\_^k\; =\; \backslash Gamma\_^k$$in which

$$\backslash cfrac\; =\; \backslash Gamma\_^k~\backslash mathbf\_k$$

This implies that$$\backslash Gamma\_^k\; =\; \backslash cfrac\backslash cdot\backslash mathbf^k\; =\; -\backslash mathbf\_i\backslash cdot\backslash cfrac$$Other relations that follow are$$\backslash cfrac\; =\; -\backslash Gamma^i\_~\backslash mathbf^k\; ~;~~\; \backslash boldsymbol\backslash mathbf\_i\; =\; \backslash Gamma\_^k~\backslash mathbf\_k\backslash otimes\backslash mathbf^j\; ~;~~\; \backslash boldsymbol\backslash mathbf^i\; =\; -\backslash Gamma\_^i~\backslash mathbf^k\backslash otimes\backslash mathbf^j$$

Another particularly useful relation, which shows that the Christoffel symbol depends only on the metric tensor and its derivatives, is$$\backslash Gamma^k\_\; =\; \backslash frac\backslash left(\backslash frac\; +\; \backslash frac\; -\; \backslash frac\; \backslash right)$$

Explicit expression for the gradient of a vector field

The following expressions for the gradient of a vector field in curvilinear coordinates are quite useful.

Representing a physical vector field

The vector field v can be represented as$$\backslash mathbf\; =\; v\_i~\backslash mathbf^i\; =\; \backslash hat\_i~\backslash hat^i$$where

v _{i} are the covariant components of the field, \hat{v} _{i} are the physical components, and (no summation)$$\backslash hat^i\; =\; \backslash cfrac$$is the normalized contravariant basis vector.

Second-order tensor field

The gradient of a second order tensor field can similarly be expressed as$$\backslash boldsymbol\backslash boldsymbol\; =\; \backslash frac\backslash otimes\backslash mathbf^i$$

Explicit expressions for the gradient

If we consider the expression for the tensor in terms of a contravariant basis, then$$\backslash boldsymbol\backslash boldsymbol\; =\; \backslash frac\; [S\_\{ij\}~\backslash mathbf\{b\}^i\backslash otimes\backslash mathbf\{b\}^j]\backslash otimes\backslash mathbf^k\; =\; \backslash left[\backslash frac\{\backslash partial\; S\_\{ij\}\}\{\backslash partial\; q^k\}\; -\; \backslash Gamma^l\_\{ki\}~S\_\{lj\}\; -\; \backslash Gamma^l\_\{kj\}\; ~S\_\{il\}\backslash right]\; ~\backslash mathbf^i\; \backslash otimes\backslash mathbf^j\; \backslash otimes\; \backslash mathbf^k$$We may also write

Representing a physical second-order tensor field

The physical components of a second-order tensor field can be obtained by using a normalized contravariant basis, i.e.,$$\backslash boldsymbol\; =\; S\_~\backslash mathbf^i\backslash otimes\backslash mathbf^j\; =\; \backslash hat\_~\backslash hat^i\backslash otimes\backslash hat^j$$where the hatted basis vectors have been normalized. This implies that (again no summation)

$$\backslash hat\_\; =\; S\_\; ~\; \backslash sqrt$$

Divergence

Vector field

The divergence of a vector field (

v )is defined as$$\backslash operatorname~\backslash mathbf\; =\; \backslash boldsymbol\backslash cdot\backslash mathbf\; =\; \backslash text(\backslash boldsymbol\backslash mathbf)$$In terms of components with respect to a curvilinear basis$$\backslash boldsymbol\backslash cdot\backslash mathbf\; =\; \backslash cfrac\; +\; \backslash Gamma^i\_~v^\backslash ell\; =\; \backslash left[\backslash cfrac\{\backslash partial\; v\_i\}\{\backslash partial\; q^j\}\; -\; \backslash Gamma^\backslash ell\_\{ji\}~v\_\backslash ell\backslash right]~g^$$

An alternative equation for the divergence of a vector field is frequently used. To derive this relation recall that$$\backslash boldsymbol\; \backslash cdot\; \backslash mathbf\; =\; \backslash frac\; +\; \backslash Gamma\_^i~v^\backslash ell$$Now,$$\backslash Gamma\_^i\; =\; \backslash Gamma\_^i\; =\; \backslash cfrac\backslash left[\backslash frac\{\backslash partial\; g\_\{im\}\}\{\backslash partial\; q^\backslash ell\}\; +\; \backslash frac\{\backslash partial\; g\_\{\backslash ell\; m\}\}\{\backslash partial\; q^i\}\; -\; \backslash frac\{\backslash partial\; g\_\{il\}\}\{\backslash partial\; q^m\}\backslash right]$$Noting that, due to the symmetry of

\boldsymbol{g} ,$$g^~\backslash frac\; =\; g^~\; \backslash frac$$we have$$\backslash boldsymbol\; \backslash cdot\; \backslash mathbf\; =\; \backslash frac\; +\; \backslash cfrac~\backslash frac~v^\backslash ell$$Recall that if [''g_ij''] is the matrix whose components are g_ij, then the inverse of the matrix is [g _{ij} ] ^{-1} = [g ^{ij} ] . The inverse of the matrix is given by$$[g^\{ij\}]\; =\; [g\_\{ij\}]^\; =\; \backslash cfrac\; ~;~~\; g\; :=\; \backslash det([g\_\{ij\}])\; =\; \backslash det\backslash boldsymbol$$where A^ij are the Cofactor matrix of the components g_ij. From matrix algebra we have$$g\; =\; \backslash det([g\_\{ij\}])\; =\; \backslash sum\_i\; g\_~A^\; \backslash quad\; \backslash Rightarrow\; \backslash quad\; \backslash frac\; =\; A^$$Hence,$$[g^\{ij\}]\; =\; \backslash cfrac~\backslash frac$$Plugging this relation into the expression for the divergence gives$$\backslash boldsymbol\; \backslash cdot\; \backslash mathbf\; =\; \backslash frac\; +\; \backslash cfrac~\backslash frac~\backslash frac~v^\backslash ell\; =\; \backslash frac\; +\; \backslash cfrac~\backslash frac~v^\backslash ell$$A little manipulation leads to the more compact form$$\backslash boldsymbol\; \backslash cdot\; \backslash mathbf\; =\; \backslash cfrac~\backslash frac(v^i~\backslash sqrt)$$

Second-order tensor field

The divergence of a second-order tensor field is defined using$$(\backslash boldsymbol\backslash cdot\backslash boldsymbol)\backslash cdot\backslash mathbf\; =\; \backslash boldsymbol\backslash cdot(\backslash boldsymbol\backslash mathbf)$$where a is an arbitrary constant vector.^[11] In curvilinear coordinates,

Laplacian

Scalar field

The Laplacian of a scalar field φ(x) is defined as$$\backslash nabla^2\; \backslash varphi\; :=\; \backslash boldsymbol\; \backslash cdot\; (\backslash boldsymbol\; \backslash varphi)$$Using the alternative expression for the divergence of a vector field gives us$$\backslash nabla^2\; \backslash varphi\; =\; \backslash cfrac~\backslash frac([\backslash boldsymbol\{\backslash nabla\}\; \backslash varphi]^i~\backslash sqrt)$$Now$$\backslash boldsymbol\; \backslash varphi\; =\; \backslash frac~\backslash mathbf^l\; =\; g^~\backslash frac~\backslash mathbf\_i\; \backslash quad\; \backslash Rightarrow\; \backslash quad\; [\backslash boldsymbol\{\backslash nabla\}\; \backslash varphi]^i\; =\; g^~\backslash frac$$Therefore,$$\backslash nabla^2\; \backslash varphi\; =\; \backslash cfrac~\backslash frac\backslash left(g^~\backslash frac~\backslash sqrt\backslash right)$$

Curl of a vector field

The curl of a vector field v in covariant curvilinear coordinates can be written as$$\backslash boldsymbol\backslash times\backslash mathbf\; =\; \backslash mathcal^\; v\_~\; \backslash mathbf\_t$$where$$v\_\; =\; v\_\; -\; \backslash Gamma^i\_~v\_i$$

Orthogonal curvilinear coordinates

Assume, for the purposes of this section, that the curvilinear coordinate system is orthogonal, i.e.,or equivalently,where

g ^{ii} = g ^{-1} _{ii} . As before, b _{i,} b _{j} are covariant basis vectors and bⁱ, b^j are contravariant basis vectors. Also, let (e¹, e², e³) be a background, fixed, Cartesian basis. A list of orthogonal curvilinear coordinates is given below.

Metric tensor in orthogonal curvilinear coordinates

See main article: article and Metric tensor.

Let r(x) be the position vector of the point x with respect to the origin of the coordinate system. The notation can be simplified by noting that x = r(x). At each point we can construct a small line element dx. The square of the length of the line element is the scalar product dx • dx and is called the metric of the space. Recall that the space of interest is assumed to be Euclidean when we talk of curvilinear coordinates. Let us express the position vector in terms of the background, fixed, Cartesian basis, i.e.,$$\backslash mathbf\; =\; \backslash sum\_^3\; x\_i~\backslash mathbf\_i$$

Using the chain rule, we can then express dx in terms of three-dimensional orthogonal curvilinear coordinates (q¹, q², q³) as$$\backslash mathrm\backslash mathbf\; =\; \backslash sum\_^3\; \backslash sum\_^3\; \backslash left(\backslash cfrac~\backslash mathbf\_i\backslash right)\backslash mathrmq^j$$Therefore, the metric is given by$$\backslash mathrm\backslash mathbf\backslash cdot\backslash mathrm\backslash mathbf\; =\; \backslash sum\_^3\; \backslash sum\_^3\; \backslash sum\_^3\; \backslash cfrac~\backslash cfrac~\backslash mathrmq^j~\backslash mathrmq^k$$

The symmetric quantity$$g\_(q^i,q^j)\; =\; \backslash sum\_^3\; \backslash cfrac~\backslash cfrac\; =\; \backslash mathbf\_i\backslash cdot\backslash mathbf\_j$$is called the fundamental (or metric) tensor of the Euclidean space in curvilinear coordinates.

Note also that$$g\_\; =\; \backslash cfrac\backslash cdot\backslash cfrac\; =\; \backslash left(\backslash sum\_\; h\_~\backslash mathbf\_k\backslash right)\backslash cdot\backslash left(\backslash sum\_\; h\_~\backslash mathbf\_m\backslash right)\; =\; \backslash sum\_\; h\_~h\_$$where h_ij are the Lamé coefficients.

If we define the scale factors, h_i, using$$\backslash mathbf\_i\backslash cdot\backslash mathbf\_i\; =\; g\_\; =\; \backslash sum\_\; h\_^2\; =:\; h\_i^2\; \backslash quad\; \backslash Rightarrow\; \backslash quad\; \backslash left|\backslash cfrac\backslash right|\; =\; \backslash left|\backslash mathbf\_i\backslash right|\; =\; \backslash sqrt\; =\; h\_i$$we get a relation between the fundamental tensor and the Lamé coefficients.

Example: Polar coordinates

If we consider polar coordinates for R², note that$$(x,\; y)=(r\; \backslash cos\; \backslash theta,\; r\; \backslash sin\; \backslash theta)$$(r, θ) are the curvilinear coordinates, and the Jacobian determinant of the transformation (r,θ) → (r cos θ, r sin θ) is r.

The orthogonal basis vectors are b_r = (cos θ, sin θ), b_θ = (−r sin θ, r cos θ). The normalized basis vectors are e_r = (cos θ, sin θ), e_θ = (−sin θ, cos θ) and the scale factors are h_r = 1 and h_θ= r. The fundamental tensor is g₁₁ =1, g₂₂ =r², g₁₂ = g₂₁ =0.

Line and surface integrals

If we wish to use curvilinear coordinates for vector calculus calculations, adjustments need to be made in the calculation of line, surface and volume integrals. For simplicity, we again restrict the discussion to three dimensions and orthogonal curvilinear coordinates. However, the same arguments apply for

n -dimensional problems though there are some additional terms in the expressions when the coordinate system is not orthogonal.

Line integrals

Normally in the calculation of line integrals we are interested in calculating$$\backslash int\_C\; f\; \backslash ,ds\; =\; \backslash int\_a^b\; f(\backslash mathbf(t))\backslash left|\backslash right|\backslash ;\; dt$$where x(t) parametrizes C in Cartesian coordinates.In curvilinear coordinates, the term

$$\backslash left|\backslash right|\; =\; \backslash left|\; \backslash sum\_^3\; \backslash right|$$

by the chain rule. And from the definition of the Lamé coefficients,

$$=\; \backslash sum\_\; h\_~\; \backslash mathbf\_$$

and thus

Now, since

g _{ij} = 0 when i \ne j , we have$$\backslash left|\backslash right|\; =\; \backslash sqrt\; =\; \backslash sqrt$$and we can proceed normally.

Surface integrals

Likewise, if we are interested in a surface integral, the relevant calculation, with the parameterization of the surface in Cartesian coordinates is:$$\backslash int\_S\; f\; \backslash ,dS\; =\; \backslash iint\_T\; f(\backslash mathbf(s,\; t))\; \backslash left|\backslash times\; \backslash right|\; \backslash ,\; ds\; \backslash ,\; dt$$Again, in curvilinear coordinates, we have$$\backslash left|\backslash times\; \backslash right|\; =\; \backslash left|\backslash left(\backslash sum\_i\; \backslash right)\; \backslash times\; \backslash left(\backslash sum\_j\; \backslash right)\backslash right|$$and we make use of the definition of curvilinear coordinates again to yield$$=\; \backslash sum\_k\; \backslash left(\backslash sum\_^3\; h\_~\backslash right)\; \backslash mathbf\_k\; ~;~~\; =\; \backslash sum\_m\; \backslash left(\backslash sum\_^3\; h\_~\backslash right)\; \backslash mathbf\_$$

Therefore,where

l{E} is the permutation symbol.

In determinant form, the cross product in terms of curvilinear coordinates will be:

Grad, curl, div, Laplacian

In orthogonal curvilinear coordinates of 3 dimensions, where$$\backslash mathbf^i\; =\; \backslash sum\_k\; g^~\backslash mathbf\_k\; ~;~~\; g^\; =\; \backslash cfrac\; =\; \backslash cfrac$$one can express the gradient of a scalar or vector field as$$\backslash nabla\backslash varphi\; =\; \backslash sum\_\; ~\; \backslash mathbf^i\; =\; \backslash sum\_\; \backslash sum\_j\; ~\; g^~\backslash mathbf\_j\; =\; \backslash sum\_i\; \backslash cfrac~~\backslash mathbf\_i\; ~;~~\; \backslash nabla\backslash mathbf\; =\; \backslash sum\_i\; \backslash cfrac~\backslash otimes\backslash mathbf\_i$$For an orthogonal basis$$g\; =\; g\_~g\_~g\_\; =\; h\_1^2~h\_2^2~h\_3^2\; \backslash quad\; \backslash Rightarrow\; \backslash quad\; \backslash sqrt\; =\; h\_1\; h\_2\; h\_3$$The divergence of a vector field can then be written as$$\backslash boldsymbol\; \backslash cdot\; \backslash mathbf\; =\; \backslash cfrac~\backslash frac(h\_1\; h\_2\; h\_3~v^i)$$Also,$$v^i\; =\; g^~v\_k\; \backslash quad\; \backslash Rightarrow\; v^1\; =\; g^~v\_1\; =\; \backslash cfrac\; ~;~~\; v^2\; =\; g^~v\_2\; =\; \backslash cfrac~;~~\; v^3\; =\; g^~v\_3\; =\; \backslash cfrac$$Therefore,$$\backslash boldsymbol\; \backslash cdot\; \backslash mathbf\; =\; \backslash cfrac~\backslash sum\_i\; \backslash frac\backslash left(\backslash cfrac~v\_i\backslash right)$$We can get an expression for the Laplacian in a similar manner by noting that$$g^~\backslash frac\; =\; \backslash left\backslash \; =\; \backslash left\backslash$$Then we have$$\backslash nabla^2\; \backslash varphi\; =\; \backslash cfrac~\backslash sum\_i\backslash frac\backslash left(\backslash cfrac~\backslash frac\backslash right)$$The expressions for the gradient, divergence, and Laplacian can be directly extended to n-dimensions.

The curl of a vector field is given by$$\backslash nabla\backslash times\backslash mathbf\; =\; \backslash frac\; \backslash sum\_^n\; \backslash mathbf\_i\backslash sum\_\; \backslash varepsilon\_\; h\_i\; \backslash frac$$where ε_ijk is the Levi-Civita symbol.

Example: Cylindrical polar coordinates

For cylindrical coordinates we have$$(x\_1,\; x\_2,\; x\_3)\; =\; \backslash mathbf\; =\; \backslash boldsymbol(q^1,\; q^2,\; q^3)\; =\; \backslash boldsymbol(r,\; \backslash theta,\; z)\; =\; \backslash$$and$$\backslash \; =\; (q^1,\; q^2,\; q^3)\; \backslash equiv\; (r,\; \backslash theta,\; z)\; =\; \backslash$$where

Then the covariant and contravariant basis vectors arewhere

e _{r,} e _{\theta,} e _{z} are the unit vectors in the r, \theta, z directions.

Note that the components of the metric tensor are such that$$g^\; =\; g\_\; =\; 0\; (i\; \backslash ne\; j)\; ~;~~\; \backslash sqrt\; =\; 1,~\backslash sqrt\; =\; \backslash cfrac,~\backslash sqrt=1$$which shows that the basis is orthogonal.

The non-zero components of the Christoffel symbol of the second kind are$$\backslash Gamma\_^2\; =\; \backslash Gamma\_^2\; =\; \backslash cfrac\; ~;~~\; \backslash Gamma\_^1\; =\; -r$$

Representing a physical vector field

The normalized contravariant basis vectors in cylindrical polar coordinates are$$\backslash hat^1\; =\; \backslash mathbf\_r\; ~;~~\backslash hat^2\; =\; \backslash mathbf\_\backslash theta\; ~;~~\backslash hat^3\; =\; \backslash mathbf\_z$$and the physical components of a vector v are$$(\backslash hat\_1,\; \backslash hat\_2,\; \backslash hat\_3)\; =\; (v\_1,\; v\_2/r,\; v\_3)\; =:\; (v\_r,\; v\_\backslash theta,\; v\_z)$$

Gradient of a scalar field

The gradient of a scalar field, f(x), in cylindrical coordinates can now be computed from the general expression in curvilinear coordinates and has the form$$\backslash boldsymbolf\; =\; \backslash cfrac~\backslash mathbf\_r\; +\; \backslash cfrac~\backslash cfrac~\backslash mathbf\_\backslash theta\; +\; \backslash cfrac~\backslash mathbf\_z$$

Gradient of a vector field

Similarly, the gradient of a vector field, v(x), in cylindrical coordinates can be shown to be

Divergence of a vector field

Using the equation for the divergence of a vector field in curvilinear coordinates, the divergence in cylindrical coordinates can be shown to be

Laplacian of a scalar field

The Laplacian is more easily computed by noting that

\boldsymbol{\nabla} ^{2} f = \boldsymbol{\nabla} ⋅ \boldsymbol{\nabla}f . In cylindrical polar coordinates$$\backslash mathbf\; =\; \backslash boldsymbolf\; =\; \backslash left[v\_r~~\; v\_\backslash theta~~\; v\_z\backslash right]\; =\; \backslash left[\backslash cfrac\{\backslash partial\; f\}\{\backslash partial\; r\}~~\; \backslash cfrac\{1\}\{r\}\backslash cfrac\{\backslash partial\; f\}\{\backslash partial\; \backslash theta\}~~\; \backslash cfrac\{\backslash partial\; f\}\{\backslash partial\; z\}\; \backslash right]$$Hence,$$\backslash boldsymbol\backslash cdot\backslash mathbf\; =\; \backslash boldsymbol^2\; f\; =\; \backslash cfrac\; +\; \backslash cfrac\backslash left(\backslash cfrac\backslash cfrac\; +\; \backslash cfrac\; \backslash right)\; +\; \backslash cfrac\; =\; \backslash cfrac\backslash left[\backslash cfrac\{\backslash partial\}\{\backslash partial\; r\}\backslash left(r\backslash cfrac\{\backslash partial\; f\}\{\backslash partial\; r\}\backslash right)\backslash right]\; +\; \backslash cfrac\backslash cfrac\; +\; \backslash cfrac$$

Representing a physical second-order tensor field

The physical components of a second-order tensor field are those obtained when the tensor is expressed in terms of a normalized contravariant basis. In cylindrical polar coordinates these components are:

Gradient of a second-order tensor field

Using the above definitions we can show that the gradient of a second-order tensor field in cylindrical polar coordinates can be expressed as

Divergence of a second-order tensor field

The divergence of a second-order tensor field in cylindrical polar coordinates can be obtained from the expression for the gradient by collecting terms where the scalar product of the two outer vectors in the dyadic products is nonzero. Therefore,

See also

• Covariance and contravariance
• Basic introduction to the mathematics of curved spacetime
• Orthogonal coordinates
• Frenet–Serret formulas
• Covariant derivative
• Tensor derivative (continuum mechanics)
• Curvilinear perspective
• Del in cylindrical and spherical coordinates

References

Notes

Further reading

• Book: Spiegel, M. R. . Vector Analysis . Schaum's Outline Series . New York . 1959. 0-07-084378-3 .
• Book: Arfken, George . Mathematical Methods for Physicists . Academic Press . 1995. 0-12-059877-9.

External links

• Derivation of Unit Vectors in Curvilinear Coordinates
• MathWorld's page on Curvilinear Coordinates
• Prof. R. Brannon's E-Book on Curvilinear Coordinates

Notes and References

1. Book: Green . A. E. . Zerna . W. . 1968 . Theoretical Elasticity . Oxford University Press . 0-19-853486-8 .
2. Book: P. M. . Naghdi . 1972 . Theory of shells and plates . S. Flügge . Handbook of Physics . VIa/2 . 425–640.
3. Book: Simmonds, J. G. . 1994 . A brief on tensor analysis . Springer . 0-387-90639-8.
4. Book: Basar . Y. . Weichert . D. . 2000 . Numerical continuum mechanics of solids: fundamental concepts and perspectives . Springer.
5. Book: Ciarlet, P. G. . 2000 . Theory of Shells . 1 . Elsevier Science .
6. Book: Einstein, A. . 1915 . Contribution to the Theory of General Relativity . Laczos . C. . The Einstein Decade . 213.
7. Book: Misner . C. W. . Thorne . K. S. . Wheeler . J. A. . 1973 . Gravitation . W. H. Freeman and Co. . 0-7167-0344-0.
8. 10.1088/0967-3334/24/2/353 . Greenleaf . A. . Lassas . M. . Uhlmann . G. . 2003 . Anisotropic conductivities that cannot be detected by EIT . Physiological Measurement . 24 . 2 . 413–419 . 12812426. 250813768 .
9. Leonhardt . U. . Philbin . T. G. . 2006 . General relativity in electrical engineering . New Journal of Physics . 8 . 10 . 247 . cond-mat/0607418 . 2006NJPh....8..247L . 10.1088/1367-2630/8/10/247 . 12100599 .
10. Book: Ogden, R. W. . 2000 . Nonlinear elastic deformations . Dover.
11. Web site: The divergence of a tensor field . . Introduction to Elasticity/Tensors . 2010-11-26 .