Tensors in curvilinear coordinates

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.

## Contents

* 1 Vector and tensor algebra in three-dimensional curvilinear coordinates
* 1.1 Coordinate transformations
* 1.1.1 The Jacobian
* 1.2 Vectors in curvilinear coordinates
* 1.3 Second-order tensors in curvilinear coordinates
* 1.3.1 Metric tensor and relations between components
* 1.4 The alternating tensor
* 1.5 Vector operations
* 1.5.1 Identity map
* 1.5.2 Scalar (dot) product
* 1.5.3 Vector (cross) product
* 1.6 Tensor operations
* 1.6.1 Identity map
* 1.6.2 Action of a second-order tensor on a vector
* 1.6.3 Inner product of two second-order tensors
* 1.6.4 Determinant of a second-order tensor
* 1.7 Relations between curvilinear and Cartesian basis vectors
* 2 Vector and tensor calculus in three-dimensional curvilinear coordinates
* 2.1 Basic definitions
* 2.2 Tangent vector to coordinate curves
* 2.3 Gradient
* 2.3.1 Scalar field
* 2.3.2 Vector field
* 2.3.3 Christoffel symbols of the first kind
* 2.3.4 Christoffel symbols of the second kind
* 2.3.5 Explicit expression for the gradient of a vector field
* 2.3.6 Representing a physical vector field
* 2.4 Second-order tensor field
* 2.4.1 Explicit expressions for the gradient
* 2.4.2 Representing a physical second-order tensor field
* 2.5 Divergence
* 2.5.1 Vector field
* 2.5.2 Second-order tensor field
* 2.6 Laplacian
* 2.6.1 Scalar field
* 2.7 Curl of a vector field
* 3 Orthogonal curvilinear coordinates
* 3.1 Metric tensor in orthogonal curvilinear coordinates
* 3.1.1 Example: Polar coordinates
* 3.2 Line and surface integrals
* 3.2.1 Line integrals
* 3.2.2 Surface integrals
* 3.3 Grad, curl, div, Laplacian
* 4 Example: Cylindrical polar coordinates
* 4.1 Representing a physical vector field
* 4.2 Gradient of a scalar field
* 4.3 Gradient of a vector field
* 4.4 Divergence of a vector field
* 4.5 Laplacian of a scalar field
* 4.6 Representing a physical second-order tensor field
* 4.7 Gradient of a second-order tensor field
* 4.8 Divergence of a second-order tensor field
* 5 See also
* 6 References
* 7 External links

## Vector and tensor algebra in three-dimensional curvilinear coordinates

Note: the Einstein summation convention of summing on repeated indices is used below.

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,[2] Naghdi,[3] Simmonds,[4] Green and Zerna,[1] Basar and Weichert,[5] and Ciarlet.[6]

### Coordinate transformations

Consider two coordinate systems with coordinate variables (Z1,Z2,Z3) and (Z1´,Z2´,Z3´), which we shall represent in short as just Zi and Zi´ 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 Zi and Zi´ 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 Zi and Zi´ are functions of each other

Zi=fi(Z1´,Z2´,Z3´) for i=1,2,3

which can be written as

Zi=Zi(Z1´,Z2´,Z3´)=Zi(Zi´) for i´,i=1,2,3

These three equations together are also called a coordinate transformation from Zi´ to Zi. Let us denote this transformation by T. We will therefore represent the transformation from the coordinate system with coordinate variables Zi´ to the coordinate system with coordinates Zi as:

Z=T(z´)

Similarly we can represent Zi´ as a function of Zi as follows:

Zi´=gi´(Z1,Z2,Z3) for i´=1,2,3

and we can write the free equations more compactly as

Zi´=Zi´(Z1,Z2,Z3)=Zi´(Zi) for i´,i=1,2,3

These three equations together are also called a coordinate transformation from Zi to Zi´. Let us denote this transformation by S. We will represent the transformation from the coordinate system with coordinate variables Zi to the coordinate system with coordinates Zi´ as:

z´=S(z)

If the transformation T is bijective then we call the image of the transformation, namely Zi, a set of admissible coordinates for Zi´. If T is linear the coordinate system Zi will be called an affine coordinate system, otherwise Zi is called a curvilinear coordinate system.

#### The Jacobian

As we now see that the Coordinates Zi and Zi´ are functions of each other, we can take the derivative of the coordinate variable Zi with respect to the coordinate variable Zi´.

Consider

∂Zi∂Zi´=defJi´i for i´,i=1,2,3, these derivatives can be arranged in a matrix, say J, in which Ji´i is the element in the i-th row and i´-th column

J=(J1´1J2´1J3´1J1´2J2´2J3´2J1´3J2´3J3´3)=(∂Z1∂Z1´∂Z1∂Z2´∂Z1∂Z3´∂Z2∂Z1´∂Z2∂Z2´∂Z2∂Z3´∂Z3∂Z1´∂Z3∂Z2´∂Z3∂Z3´)

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[4]: 27 𝐯=vk𝐛k The components vk are the contravariant components of the vector 𝐯.

The reciprocal basis (𝐛1,𝐛2,𝐛3) is defined by the relation [4]: 28–29 𝐛i⋅𝐛j=δji where δji is the Kronecker delta.

The vector 𝐯 can also be expressed in terms of the reciprocal basis: 𝐯=vk𝐛k The components vk are the covariant components of the vector 𝐯.

### Second-order tensors in curvilinear coordinates

A second-order tensor can be expressed as S=Sij𝐛i⊗𝐛j=Sji𝐛i⊗𝐛j=Sij𝐛i⊗𝐛j=Sij𝐛i⊗𝐛j The components Sij are called the contravariant components, Sji the mixed right-covariant components, Sij the mixed left-covariant components, and Sij the covariant components of the second-order tensor.

#### Metric tensor and relations between components

The quantities gij, gij are defined as[4]: 39

gij=𝐛i⋅𝐛j=gji;gij=𝐛i⋅𝐛j=gji From the above equations we have vi=gikvk;vi=gikvk;𝐛i=gij𝐛j;𝐛i=gij𝐛j

The components of a vector are related by[4]: 30–32 𝐯⋅𝐛i=vk𝐛k⋅𝐛i=vkδki=vi 𝐯⋅𝐛i=vk𝐛k⋅𝐛i=vkδik=vi Also, 𝐯⋅𝐛i=vk𝐛k⋅𝐛i=gkivk 𝐯⋅𝐛i=vk𝐛k⋅𝐛i=gkivk

The components of the second-order tensor are related by Sij=gikSkj=gjkSki=gikgjlSkl

### The alternating tensor

In an orthonormal right-handed basis, the third-order alternating tensor is defined as ℰ=εijk𝐞i⊗𝐞j⊗𝐞k In a general curvilinear basis the same tensor may be expressed as ℰ=ℰijk𝐛i⊗𝐛j⊗𝐛k=ℰijk𝐛i⊗𝐛j⊗𝐛k It can be shown that ℰijk=[𝐛i,𝐛j,𝐛k]=(𝐛i×𝐛j)⋅𝐛k;ℰijk=[𝐛i,𝐛j,𝐛k] Now, 𝐛i×𝐛j=Jεijp𝐛p=gεijp𝐛p Hence, ℰijk=Jεijk=gεijk Similarly, we can show that ℰijk=1Jεijk=1gεijk

### Vector operations

#### Identity map

The identity map 𝐈 defined by 𝐈⋅𝐯=𝐯 can be shown to be:[4]: 39

𝐈=gij𝐛i⊗𝐛j=gij𝐛i⊗𝐛j=𝐛i⊗𝐛i=𝐛i⊗𝐛i

#### Scalar (dot) product

The scalar product of two vectors in curvilinear coordinates is[4]: 32

𝐮⋅𝐯=uivi=uivi=gijuivj=gijuivj

#### Vector (cross) product

The cross product of two vectors is given by:[4]: 32–34

𝐮×𝐯=εijkujvk𝐞i

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

𝐮×𝐯=[(𝐛m×𝐛n)⋅𝐛s]umvn𝐛s=ℰsmnumvn𝐛s

where ℰijk is the third-order alternating tensor. The cross product of two vectors is given by:

𝐮×𝐯=εijku^jv^k𝐞i

where εijk is the permutation symbol and 𝐞i is a Cartesian basis vector. Therefore,

𝐞p×𝐞q=εipq𝐞i

and

𝐛m×𝐛n=∂𝐱∂qm×∂𝐱∂qn=∂(xp𝐞p)∂qm×∂(xq𝐞q)∂qn=∂xp∂qm∂xq∂qn𝐞p×𝐞q=εipq∂xp∂qm∂xq∂qn𝐞i.

Hence,

(𝐛m×𝐛n)⋅𝐛s=εipq∂xp∂qm∂xq∂qn∂xi∂qs

Returning to the vector product and using the relations:

u^j=∂xj∂qmum,v^k=∂xk∂qnvn,𝐞i=∂xi∂qs𝐛s,

gives us:

𝐮×𝐯=εijku^jv^k𝐞i=εijk∂xj∂qm∂xk∂qn∂xi∂qsumvn𝐛s=[(𝐛m×𝐛n)⋅𝐛s]umvn𝐛s=ℰsmnumvn𝐛s

### Tensor operations

#### Identity map

The identity map I defined by I⋅𝐯=𝐯 can be shown to be[4]: 39

I=gij𝐛i⊗𝐛j=gij𝐛i⊗𝐛j=𝐛i⊗𝐛i=𝐛i⊗𝐛i

#### Action of a second-order tensor on a vector

The action 𝐯=S𝐮 can be expressed in curvilinear coordinates as

vi𝐛i=Sijuj𝐛i=Sjiuj𝐛i;vi𝐛i=Sijuj𝐛i=Sijuj𝐛i

#### Inner product of two second-order tensors

The inner product of two second-order tensors U=S⋅T can be expressed in curvilinear coordinates as

Uij𝐛i⊗𝐛j=SikT.jk𝐛i⊗𝐛j=Si.kTkj𝐛i⊗𝐛j

Alternatively,

U=SijT.nmgjm𝐛i⊗𝐛n=S.miT.nm𝐛i⊗𝐛n=SijTjn𝐛i⊗𝐛n

#### Determinant of a second-order tensor

If S is a second-order tensor, then the determinant is defined by the relation

[S𝐮,S𝐯,S𝐰]=det⁡S[𝐮,𝐯,𝐰]

where 𝐮,𝐯,𝐰 are arbitrary vectors and

[𝐮,𝐯,𝐰]:=𝐮⋅(𝐯×𝐰).

### 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 𝐛i=F𝐞i where F is a second-order transformation tensor that maps 𝐞i to 𝐛i. Then, 𝐛i⊗𝐞i=(F𝐞i)⊗𝐞i=F(𝐞i⊗𝐞i)=F. From this relation we can show that 𝐛i=F−T𝐞i;gij=[F−1F−T]ij;gij=[gij]−1=[FTF]ij Let J:=det⁡F be the Jacobian of the transformation. Then, from the definition of the determinant, [𝐛1,𝐛2,𝐛3]=det⁡F[𝐞1,𝐞2,𝐞3]. Since [𝐞1,𝐞2,𝐞3]=1 we have J=det⁡F=[𝐛1,𝐛2,𝐛3]=𝐛1⋅(𝐛2×𝐛3) A number of interesting results can be derived using the above relations.

First, consider g:=det⁡[gij] Then g=det⁡[FT]⋅det⁡[F]=J⋅J=J2 Similarly, we can show that det⁡[gij]=1J2 Therefore, using the fact that [gij]=[gij]−1, ∂g∂gij=2J∂J∂gij=ggij

Another interesting relation is derived below. Recall that 𝐛i⋅𝐛j=δji⇒𝐛1⋅𝐛1=1,𝐛1⋅𝐛2=𝐛1⋅𝐛3=0⇒𝐛1=A(𝐛2×𝐛3) where A is a, yet undetermined, constant. Then 𝐛1⋅𝐛1=A𝐛1⋅(𝐛2×𝐛3)=AJ=1⇒A=1J This observation leads to the relations 𝐛1=1J(𝐛2×𝐛3);𝐛2=1J(𝐛3×𝐛1);𝐛3=1J(𝐛1×𝐛2) In index notation, εijk𝐛k=1J(𝐛i×𝐛j)=1g(𝐛i×𝐛j) where εijk 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 𝐛i=∂𝐱∂qi=∂𝐱∂xj∂xj∂qi=𝐞j∂xj∂qi Similarly, 𝐞i=𝐛j∂qj∂xi From these results we have 𝐞k⋅𝐛i=∂xk∂qi⇒∂xk∂qi𝐛i=𝐞k⋅(𝐛i⊗𝐛i)=𝐞k and 𝐛k=∂qk∂xi𝐞i

## Vector and tensor calculus in three-dimensional curvilinear coordinates

Note: the Einstein summation convention of summing on repeated indices is used below.

Simmonds,[4] in his book on tensor analysis, quotes Albert Einstein saying[7]

> The magic of this theory will hardly fail to impose itself on anybody who has truly understood it; it represents a genuine triumph of the method of absolute differential calculus, founded by Gauss, Riemann, Ricci, and Levi-Civita.

Vector and tensor calculus in general curvilinear coordinates is used in tensor analysis on four-dimensional curvilinear manifolds in general relativity,[8] in the mechanics of curved shells,[6] in examining the invariance properties of Maxwell's equations which has been of interest in metamaterials[9][10] 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,[2] Simmonds,[4] Green and Zerna,[1] Basar and Weichert,[5] and Ciarlet.[6]

### Basic definitions

Let the position of a point in space be characterized by three coordinate variables (q1,q2,q3).

The coordinate curve q1 represents a curve on which q2 and 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 [2]: 55 𝐱=φ(q1,q2,q3);qi=ψi(𝐱)=[φ−1(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 𝐱i(α)=φ(α,qj,qk),i≠j≠k with qj, qk fixed.

### Tangent vector to coordinate curves

The tangent vector to the curve 𝐱i at the point 𝐱i(α) (or to the coordinate curve qi at the point 𝐱) is d𝐱idα≡∂𝐱∂qi

### Gradient

#### Scalar field

Let f(𝐱) be a scalar field in space. Then f(𝐱)=f[φ(q1,q2,q3)]=fφ(q1,q2,q3) The gradient of the field f is defined by [∇f(𝐱)]⋅𝐜=ddαf(𝐱+α𝐜)|α=0 where 𝐜 is an arbitrary constant vector. If we define the components ci of 𝐜 are such that qi+αci=ψi(𝐱+α𝐜) then [∇f(𝐱)]⋅𝐜=ddαfφ(q1+αc1,q2+αc2,q3+αc3)|α=0=∂fφ∂qici=∂f∂qici

If we set f(𝐱)=ψi(𝐱), then since qi=ψi(𝐱), we have [∇ψi(𝐱)]⋅𝐜=∂ψi∂qjcj=ci which provides a means of extracting the contravariant component of a vector 𝐜.

If 𝐛i is the covariant (or natural) basis at a point, and if 𝐛i is the contravariant (or reciprocal) basis at that point, then [∇f(𝐱)]⋅𝐜=∂f∂qici=(∂f∂qi𝐛i)(ci𝐛i)⇒∇f(𝐱)=∂f∂qi𝐛i 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 [∇𝐟(𝐱)]⋅𝐜=∂𝐟∂qici If we consider the gradient of the position vector field 𝐫(𝐱)=𝐱, then we can show that 𝐜=∂𝐱∂qici=𝐛i(𝐱)ci;𝐛i(𝐱):=∂𝐱∂qi The vector field 𝐛i 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 ∇𝐟(𝐱)=∂𝐟∂qi⊗𝐛i

Note that the contravariant basis vector 𝐛i is perpendicular to the surface of constant ψi and is given by 𝐛i=∇ψi

#### Christoffel symbols of the first kind

The Christoffel symbols of the first kind are defined as 𝐛i,j=∂𝐛i∂qj:=Γijk𝐛k⇒𝐛i,j⋅𝐛l=Γijl To express Γijk in terms of gij we note that gij,k=(𝐛i⋅𝐛j),k=𝐛i,k⋅𝐛j+𝐛i⋅𝐛j,k=Γikj+Γjkigik,j=(𝐛i⋅𝐛k),j=𝐛i,j⋅𝐛k+𝐛i⋅𝐛k,j=Γijk+Γkjigjk,i=(𝐛j⋅𝐛k),i=𝐛j,i⋅𝐛k+𝐛j⋅𝐛k,i=Γjik+Γkij Since 𝐛i,j=𝐛j,i we have Γijk=Γjik. Using these to rearrange the above relations gives Γijk=12(gik,j+gjk,i−gij,k)=12[(𝐛i⋅𝐛k),j+(𝐛j⋅𝐛k),i−(𝐛i⋅𝐛j),k]

#### Christoffel symbols of the second kind

The Christoffel symbols of the second kind are defined as Γijk=Γjik in which

∂𝐛i∂qj=Γijk𝐛k

This implies that Γijk=∂𝐛i∂qj⋅𝐛k=−𝐛i⋅∂𝐛k∂qj Other relations that follow are ∂𝐛i∂qj=−Γjki𝐛k;∇𝐛i=Γijk𝐛k⊗𝐛j;∇𝐛i=−Γjki𝐛k⊗𝐛j

Another particularly useful relation, which shows that the Christoffel symbol depends only on the metric tensor and its derivatives, is Γijk=gkm2(∂gmi∂qj+∂gmj∂qi−∂gij∂qm)

#### 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. ∇𝐯=[∂vi∂qk+Γlkivl]𝐛i⊗𝐛k=[∂vi∂qk−Γkilvl]𝐛i⊗𝐛k

#### Representing a physical vector field

The vector field 𝐯 can be represented as 𝐯=vi𝐛i=v^i𝐛^i where vi are the covariant components of the field, v^i are the physical components, and (no summation) 𝐛^i=𝐛igii is the normalized contravariant basis vector.

### Second-order tensor field

The gradient of a second order tensor field can similarly be expressed as ∇S=∂S∂qi⊗𝐛i

#### Explicit expressions for the gradient

If we consider the expression for the tensor in terms of a contravariant basis, then ∇S=∂∂qk[Sij𝐛i⊗𝐛j]⊗𝐛k=[∂Sij∂qk−ΓkilSlj−ΓkjlSil]𝐛i⊗𝐛j⊗𝐛k We may also write ∇S=[∂Sij∂qk+ΓkliSlj+ΓkljSil]𝐛i⊗𝐛j⊗𝐛k=[∂Sji∂qk+ΓkliSjl−ΓkjlSli]𝐛i⊗𝐛j⊗𝐛k=[∂Sij∂qk−ΓiklSlj+ΓkljSil]𝐛i⊗𝐛j⊗𝐛k

#### 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., S=Sij𝐛i⊗𝐛j=S^ij𝐛^i⊗𝐛^j where the hatted basis vectors have been normalized. This implies that (again no summation)

S^ij=Sijgiigjj

### Divergence

#### Vector field

The divergence of a vector field 𝐯 is defined as div⁡𝐯=∇⋅𝐯=tr(∇𝐯) In terms of components with respect to a curvilinear basis ∇⋅𝐯=∂vi∂qi+Γℓiivℓ=[∂vi∂qj−Γjiℓvℓ]gij

An alternative equation for the divergence of a vector field is frequently used. To derive this relation recall that ∇⋅𝐯=∂vi∂qi+Γℓiivℓ Now, Γℓii=Γiℓi=gmi2[∂gim∂qℓ+∂gℓm∂qi−∂gil∂qm] Noting that, due to the symmetry of g, gmi∂gℓm∂qi=gmi∂giℓ∂qm we have ∇⋅𝐯=∂vi∂qi+gmi2∂gim∂qℓvℓ Recall that if [gij] is the matrix whose components are gij, then the inverse of the matrix is [gij]−1=[gij]. The inverse of the matrix is given by [gij]=[gij]−1=Aijg;g:=det⁡([gij])=det⁡g where Aij is the cofactor matrix of the components gij. From matrix algebra we have g=det⁡([gij])=∑igijAij⇒∂g∂gij=Aij Hence, [gij]=1g∂g∂gij Plugging this relation into the expression for the divergence gives ∇⋅𝐯=∂vi∂qi+12g∂g∂gmi∂gim∂qℓvℓ=∂vi∂qi+12g∂g∂qℓvℓ A little manipulation leads to the more compact form ∇⋅𝐯=1g∂∂qi(vig)

#### Second-order tensor field

The divergence of a second-order tensor field is defined using (∇⋅S)⋅𝐚=∇⋅(S𝐚) where 𝐚 is an arbitrary constant vector. [11] In curvilinear coordinates, ∇⋅S=[∂Sij∂qk−ΓkilSlj−ΓkjlSil]gik𝐛j=[∂Sij∂qi+ΓiliSlj+ΓiljSil]𝐛j=[∂Sji∂qi+ΓiliSjl−ΓijlSli]𝐛j=[∂Sij∂qk−ΓiklSlj+ΓkljSil]gik𝐛j

### Laplacian

#### Scalar field

The Laplacian of a scalar field φ(𝐱) is defined as ∇2φ:=∇⋅(∇φ) Using the alternative expression for the divergence of a vector field gives us ∇2φ=1g∂∂qi([∇φ]ig) Now ∇φ=∂φ∂ql𝐛l=gli∂φ∂ql𝐛i⇒[∇φ]i=gli∂φ∂ql Therefore, ∇2φ=1g∂∂qi(gli∂φ∂qlg)

### Curl of a vector field

The curl of a vector field 𝐯 in covariant curvilinear coordinates can be written as ∇×𝐯=ℰrstvs|r𝐛t where vs|r=vs,r−Γsrivi

## Orthogonal curvilinear coordinates

Assume, for the purposes of this section, that the curvilinear coordinate system is orthogonal, i.e., 𝐛i⋅𝐛j={giiif i=j0if i≠j, or equivalently, 𝐛i⋅𝐛j={giiif i=j0if i≠j, where gii=gii−1. As before, 𝐛i,𝐛j are covariant basis vectors and 𝐛i, 𝐛j 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

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., 𝐱=∑i=13xi𝐞i

Using the chain rule, we can then express d𝐱 in terms of three-dimensional orthogonal curvilinear coordinates (q1,q2,q3) as d𝐱=∑i=13∑j=13(∂xi∂qj𝐞i)dqj Therefore, the metric is given by d𝐱⋅d𝐱=∑i=13∑j=13∑k=13∂xi∂qj∂xi∂qkdqjdqk

The symmetric quantity gij(qi,qj)=∑k=13∂xk∂qi∂xk∂qj=𝐛i⋅𝐛j is called the fundamental (or metric) tensor of the Euclidean space in curvilinear coordinates.

Note also that gij=∂𝐱∂qi⋅∂𝐱∂qj=(∑khki𝐞k)⋅(∑mhmj𝐞m)=∑khkihkj where hij are the Lamé coefficients.

If we define the scale factors, hi, using 𝐛i⋅𝐛i=gii=∑khki2=:hi2⇒|∂𝐱∂qi|=|𝐛i|=gii=hi 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)=(rcos⁡θ,rsin⁡θ) (r,θ) are the curvilinear coordinates, and the Jacobian determinant of the transformation (r,θ)->(rcos⁡θ,rsin⁡θ) is r.

The orthogonal basis vectors are 𝐛r=(cos⁡θ,sin⁡θ), 𝐛θ=(−rsin⁡θ,rcos⁡θ). The normalized basis vectors are 𝐞r=(cos⁡θ,sin⁡θ), 𝐞θ=(−sin⁡θ,cos⁡θ) and the scale factors are hr=1 and hθ=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 ∫Cfds=∫abf(𝐱(t))|∂𝐱∂t|dt where 𝐱(t) parametrizes C in Cartesian coordinates. In curvilinear coordinates, the term

|∂𝐱∂t|=|∑i=13∂𝐱∂qi∂qi∂t|

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

∂𝐱∂qi=∑khki𝐞k

and thus

|∂𝐱∂t|=|∑k(∑ihki∂qi∂t)𝐞k|=∑i∑j∑khkihkj∂qi∂t∂qj∂t=∑i∑jgij∂qi∂t∂qj∂t

Now, since gij=0 when i≠j, we have |∂𝐱∂t|=∑igii(∂qi∂t)2=∑ihi2(∂qi∂t)2 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: ∫SfdS=∬Tf(𝐱(s,t))|∂𝐱∂s×∂𝐱∂t|dsdt Again, in curvilinear coordinates, we have |∂𝐱∂s×∂𝐱∂t|=|(∑i∂𝐱∂qi∂qi∂s)×(∑j∂𝐱∂qj∂qj∂t)| and we make use of the definition of curvilinear coordinates again to yield ∂𝐱∂qi∂qi∂s=∑k(∑i=13hki∂qi∂s)𝐞k;∂𝐱∂qj∂qj∂t=∑m(∑j=13hmj∂qj∂t)𝐞m

Therefore, |∂𝐱∂s×∂𝐱∂t|=|∑k∑m(∑i=13hki∂qi∂s)(∑j=13hmj∂qj∂t)𝐞k×𝐞m|=|∑p∑k∑mℰkmp(∑i=13hki∂qi∂s)(∑j=13hmj∂qj∂t)𝐞p| where ℰ is the permutation symbol.

In determinant form, the cross product in terms of curvilinear coordinates will be: |𝐞1𝐞2𝐞3∑ih1i∂qi∂s∑ih2i∂qi∂s∑ih3i∂qi∂s∑jh1j∂qj∂t∑jh2j∂qj∂t∑jh3j∂qj∂t|

### Grad, curl, div, Laplacian

In orthogonal curvilinear coordinates of 3 dimensions, where 𝐛i=∑kgik𝐛k;gii=1gii=1hi2 one can express the gradient of a scalar or vector field as ∇φ=∑i∂φ∂qi𝐛i=∑i∑j∂φ∂qigij𝐛j=∑i1hi2∂f∂qi𝐛i;∇𝐯=∑i1hi2∂𝐯∂qi⊗𝐛i For an orthogonal basis g=g11g22g33=h12h22h32⇒g=h1h2h3 The divergence of a vector field can then be written as ∇⋅𝐯=1h1h2h3∂∂qi(h1h2h3vi) Also, vi=gikvk⇒v1=g11v1=v1h12;v2=g22v2=v2h22;v3=g33v3=v3h32 Therefore, ∇⋅𝐯=1h1h2h3∑i∂∂qi(h1h2h3hi2vi) We can get an expression for the Laplacian in a similar manner by noting that gli∂φ∂ql={g11∂φ∂q1,g22∂φ∂q2,g33∂φ∂q3}={1h12∂φ∂q1,1h22∂φ∂q2,1h32∂φ∂q3} Then we have ∇2φ=1h1h2h3∑i∂∂qi(h1h2h3hi2∂φ∂qi) The expressions for the gradient, divergence, and Laplacian can be directly extended to n-dimensions.

The curl of a vector field is given by ∇×𝐯=1h1h2h3∑i=1n𝐞i∑jkεijkhi∂(hkvk)∂qj where εijk is the Levi-Civita symbol.

## Example: Cylindrical polar coordinates

For cylindrical coordinates we have (x1,x2,x3)=𝐱=φ(q1,q2,q3)=φ(r,θ,z)={rcos⁡θ,rsin⁡θ,z} and {ψ1(𝐱),ψ2(𝐱),ψ3(𝐱)}=(q1,q2,q3)≡(r,θ,z)={x12+x22,tan−1(x2/x1),x3} where 0<r<∞,0<θ<2π,−∞<z<∞

Then the covariant and contravariant basis vectors are 𝐛1=𝐞r=𝐛1𝐛2=r𝐞θ=r2𝐛2𝐛3=𝐞z=𝐛3 where 𝐞r,𝐞θ,𝐞z are the unit vectors in the r,θ,z directions.

Note that the components of the metric tensor are such that gij=gij=0(i≠j);g11=1,g22=1r,g33=1 which shows that the basis is orthogonal.

The non-zero components of the Christoffel symbol of the second kind are Γ122=Γ212=1r;Γ221=−r

### Representing a physical vector field

The normalized contravariant basis vectors in cylindrical polar coordinates are 𝐛^1=𝐞r;𝐛^2=𝐞θ;𝐛^3=𝐞z and the physical components of a vector 𝐯 are (v^1,v^2,v^3)=(v1,v2/r,v3)=:(vr,vθ,vz)

### 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 ∇f=∂f∂r𝐞r+1r∂f∂θ𝐞θ+∂f∂z𝐞z

### Gradient of a vector field

Similarly, the gradient of a vector field, 𝐯(𝐱), in cylindrical coordinates can be shown to be ∇𝐯=∂vr∂r𝐞r⊗𝐞r+1r(∂vr∂θ−vθ)𝐞r⊗𝐞θ+∂vr∂z𝐞r⊗𝐞z+∂vθ∂r𝐞θ⊗𝐞r+1r(∂vθ∂θ+vr)𝐞θ⊗𝐞θ+∂vθ∂z𝐞θ⊗𝐞z+∂vz∂r𝐞z⊗𝐞r+1r∂vz∂θ𝐞z⊗𝐞θ+∂vz∂z𝐞z⊗𝐞z

### 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 ∇⋅𝐯=∂vr∂r+1r(∂vθ∂θ+vr)+∂vz∂z

### Laplacian of a scalar field

The Laplacian is more easily computed by noting that ∇2f=∇⋅∇f. In cylindrical polar coordinates 𝐯=∇f=[vrvθvz]=[∂f∂r1r∂f∂θ∂f∂z] Hence, ∇⋅𝐯=∇2f=∂2f∂r2+1r(1r∂2f∂θ2+∂f∂r)+∂2f∂z2=1r[∂∂r(r∂f∂r)]+1r2∂2f∂θ2+∂2f∂z2

### 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:

S^11=S11=:Srr,S^12=S12r=:Srθ,S^13=S13=:SrzS^21=S21r=:Sθr,S^22=S22r2=:Sθθ,S^23=S23r=:SθzS^31=S31=:Szr,S^32=S32r=:Szθ,S^33=S33=:Szz

### 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 ∇S=∂Srr∂r𝐞r⊗𝐞r⊗𝐞r+1r[∂Srr∂θ−(Sθr+Srθ)]𝐞r⊗𝐞r⊗𝐞θ+∂Srr∂z𝐞r⊗𝐞r⊗𝐞z+∂Srθ∂r𝐞r⊗𝐞θ⊗𝐞r+1r[∂Srθ∂θ+(Srr−Sθθ)]𝐞r⊗𝐞θ⊗𝐞θ+∂Srθ∂z𝐞r⊗𝐞θ⊗𝐞z+∂Srz∂r𝐞r⊗𝐞z⊗𝐞r+1r[∂Srz∂θ−Sθz]𝐞r⊗𝐞z⊗𝐞θ+∂Srz∂z𝐞r⊗𝐞z⊗𝐞z+∂Sθr∂r𝐞θ⊗𝐞r⊗𝐞r+1r[∂Sθr∂θ+(Srr−Sθθ)]𝐞θ⊗𝐞r⊗𝐞θ+∂Sθr∂z𝐞θ⊗𝐞r⊗𝐞z+∂Sθθ∂r𝐞θ⊗𝐞θ⊗𝐞r+1r[∂Sθθ∂θ+(Srθ+Sθr)]𝐞θ⊗𝐞θ⊗𝐞θ+∂Sθθ∂z𝐞θ⊗𝐞θ⊗𝐞z+∂Sθz∂r𝐞θ⊗𝐞z⊗𝐞r+1r[∂Sθz∂θ+Srz]𝐞θ⊗𝐞z⊗𝐞θ+∂Sθz∂z𝐞θ⊗𝐞z⊗𝐞z+∂Szr∂r𝐞z⊗𝐞r⊗𝐞r+1r[∂Szr∂θ−Szθ]𝐞z⊗𝐞r⊗𝐞θ+∂Szr∂z𝐞z⊗𝐞r⊗𝐞z+∂Szθ∂r𝐞z⊗𝐞θ⊗𝐞r+1r[∂Szθ∂θ+Szr]𝐞z⊗𝐞θ⊗𝐞θ+∂Szθ∂z𝐞z⊗𝐞θ⊗𝐞z+∂Szz∂r𝐞z⊗𝐞z⊗𝐞r+1r∂Szz∂θ𝐞z⊗𝐞z⊗𝐞θ+∂Szz∂z𝐞z⊗𝐞z⊗𝐞z

### 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, ∇⋅S=∂Srr∂r𝐞r+∂Srθ∂r𝐞θ+∂Srz∂r𝐞z+1r[∂Srθ∂θ+(Srr−Sθθ)]𝐞r+1r[∂Sθθ∂θ+(Srθ+Sθr)]𝐞θ+1r[∂Sθz∂θ+Srz]𝐞z+∂Szr∂z𝐞r+∂Szθ∂z𝐞θ+∂Szz∂z𝐞z

## 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

1. ↑ 1.0 1.1 1.2 Green, A. E.; Zerna, W. (1968). Theoretical Elasticity. Oxford University Press. ISBN 0-19-853486-8.
2. ↑ 2.0 2.1 2.2 Ogden, R. W. (2000). Nonlinear elastic deformations. Dover.
3. ↑ Naghdi, P. M. (1972). "Theory of shells and plates". in S. Flügge. Handbook of Physics. VIa/2. pp. 425–640.
4. ↑ 4.00 4.01 4.02 4.03 4.04 4.05 4.06 4.07 4.08 4.09 4.10 Simmonds, J. G. (1994). A brief on tensor analysis. Springer. ISBN 0-387-90639-8.
5. ↑ 5.0 5.1 Basar, Y.; Weichert, D. (2000). Numerical continuum mechanics of solids: fundamental concepts and perspectives. Springer.
6. ↑ 6.0 6.1 6.2 Ciarlet, P. G. (2000). Theory of Shells. 1. Elsevier Science.
7. ↑ Einstein, A. (1915). "Contribution to the Theory of General Relativity". in Laczos, C.. The Einstein Decade. p. 213.
8. ↑ Misner, C. W.; Thorne, K. S.; Wheeler, J. A. (1973). Gravitation. W. H. Freeman and Co.. ISBN 0-7167-0344-0.
9. ↑ Greenleaf, A.; Lassas, M.; Uhlmann, G. (2003). "Anisotropic conductivities that cannot be detected by EIT". Physiological Measurement 24 (2): 413–419. doi:10.1088/0967-3334/24/2/353. PMID 12812426.
10. ↑ Leonhardt, U.; Philbin, T. G. (2006). "General relativity in electrical engineering". New Journal of Physics 8 (10): 247. doi:10.1088/1367-2630/8/10/247. Bibcode: 2006NJPh....8..247L.
11. ↑ "The divergence of a tensor field". Introduction to Elasticity/Tensors. Wikiversity. http://en.wikiversity.org/wiki/Introduction_to_Elasticity/Tensors#The_divergence_of_a_tensor_field. Retrieved 2010-11-26.

Further reading

* Spiegel, M. R. (1959). Vector Analysis. New York: Schaum's Outline Series. ISBN 0-07-084378-3.
* Arfken, George (1995). Mathematical Methods for Physicists. Academic Press. ISBN 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

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Orthogonal coordinate systems

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Tensors

Glossary of tensor theory

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* Physics
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* Kronecker delta
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Physics|

* moment of inertia
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Mathematicians|

* Leonhard Euler
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