A Small Compendium on Vector and Tensor Algebra and CalculusCT/tutorials/tensors/... · 2017-01-30 · Chapter 1 Vectors and tensors algebra Algebra is concerned with operations de

Ruhr-University Bochum

Faculty of Civil and Environmental Engineering

Institute of Mechanics

A Small Compendium on Vectorand Tensor Algebra and Calculus

Klaus Hackl

Mehdi Goodarzi

2010

ii

Contents

Foreword v

1 Vectors and tensors algebra 1

1.1 Scalars and vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Geometrical operations on vectors . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Fundamental properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4 Vector decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.5 Dot product of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.6 Index notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.7 Tensors of second order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.8 Dyadic product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.9 Tensors of higher order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.10 Coordinate transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.11 Eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.12 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.13 Singular value decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.14 Cross product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.15 Triple product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.16 Dot and cross product of vectors: miscellaneous formulas . . . . . . . . . . 17

2 Tensor calculus 19

2.1 Tensor functions & tensor elds . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Derivative of tensor functions . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Derivatives of tensor elds, Gradient, Nabla operator . . . . . . . . . . . . . 20

2.4 Integrals of tensor functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5 Line integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.6 Path independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.7 Surface integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.8 Divergence and curl operators . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.9 Laplace's operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.10 Gauss' and Stoke's Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.11 Miscellaneous formulas and theorems involving nabla . . . . . . . . . . . . . 27

2.12 Orthogonal curvilinear coordinate systems . . . . . . . . . . . . . . . . . . . 28

iii

iv CONTENTS

2.13 Dierentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.14 Dierential transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.15 Dierential operators in curvilinear coordinates . . . . . . . . . . . . . . . . 31

Foreword

A quick review of vector and tensor algebra, geometry and analysis is presented. The readeris supposed to have sucient familiarity with the subject and the material is included asan entry point as well as a reference for the subsequence.

v

vi FOREWORD

Chapter 1

Vectors and tensors algebra

Algebra is concerned with operations dened in sets with certain properties. Tensor and vec-tor algebra deals with properties and operations in the set of tensors and vectors. Through-out this section together with algebraic aspects, we also consider geometry of tensors toobtain further insight.

1.1 Scalars and vectors

There are physical quantities that can be specied by a single real number, like temperatureand speed. These are called scalars. Other quantities whose specication requires mag-nitude (a positive real number) and direction, for instance velocity and force, are calledvectors. A typical illustration of a vector is an arrow which is a directed line segment.Given an appropriate unit, length of the arrow reects magnitude of the vector. In thepresent context the precise terminology would be Euclidean vector or geometric vector,to acknowledge that in mathematics, vector has a broader meaning with which we are notconcerned here. It is instructive to have a closer look at the meaning of direction of a vector.

v

eLL

θ

‖v‖ cos θ

Fig. 1.1: Projection.

Given a vector v and a directed line L (Fig. 1.1), projectionof the vector onto the direction is a scalar vL = ‖v‖ cos θ inwhich θ is the angle between v and L . Therefore, a vectorcan be thought as a function that assigns a scalar to eachdirection v(L ) = vL . This notion can be naturally extendedto understand tensors. A tensor (of second order) is a functionthat assigns vectors to directions T (L ) = TL in the sense ofprojection. In other words the projection of tensor T on direction L is a vector like TL .This looks rather abstract but its meaning is going to be clear in the sequel when we explainthe Cauchy's formula in which the dot product of stress (tensor) and area (vector) yieldstraction force (vector).

Notation 1. Vectors and tensors are denoted by bold-face letters like A and v, and mag-nitude of a vector by ‖v‖ or simply by its normal-face v. In handwriting an underbar isused to denote tensors and vectors like v.

1

2 CHAPTER 1. VECTORS AND TENSORS ALGEBRA

1.2 Geometrical operations on vectors

Denition 2 (Equality of vectors). Two vectors are equal if they have the same magnitudeand direction.

Denition 3 (Summation of vectors). The sum of every two vectors v and w is a vectoru = v +w formed by placing initial point of w on terminal point of v and joining initialpoint of v and terminal point of w (Fig. 1.2(a,b)). This is equivalent to parallelogramlaw for vector addition as shown in Fig. 1.2(c). Subtraction of vectors v −w is dened asv + (−w).

v

w

(a)

v w

u = v +w

(b)

v

w

u = v +w

(c)

Fig. 1.2: Vector summation.

Denition 4 (Multiplication of a vector by a scalar). For any real number (scalar) α andvector v, their multiplication is a vector αv whose magnitude is |α| times the magnitudeof v and whose direction is the same as v if α > 0 and opposite to v if α < 0. If α = 0then αv = 0.

1.3 Fundamental properties

The set of geometric vectors V together with the just dened summation and multiplicationoperations is subject to the following properties

Property 5 (Vector summation).1. v + 0 = 0 + v = v Existence of additive identity2. v + (−v) = (−v) + v = 0 Existence of additive inverse3. v +w = w + v Commutative law4. u+ (v +w) = (u+ v) +w Associative law

Property 6 (Multiplication with scalars).

1. α (v +w) = αv + αw Distributive law2. (α+ β)v = αv + βv Distributive law3. α (βv) = (αβ)v Associative law4. 1v = v Multiplication with scalar identity.

A vector can be rigorously dened as an element of a vector space.

Denition 7 (Real vector space). The mathematical system (V,+,R, ∗) is called a realvector space or simply a vector space, where V is the set of geometrical vectors, + is vector

1.4. VECTOR DECOMPOSITION 3

summation with the four properties mentioned, R is the set of real numbers furnished withsummation and multiplication of numbers, and ∗ stands for multiplication of a vector witha real number (scalar) having the above mentioned properties.

The reader may wonder why this recent denition is needed. Remember that a quantitywhose expression requires magnitude and direction is not necessarily a vector(!). A wellknown example is nite rotation which has both magnitude and direction but is not a vectorbecause it does not follow the properties of vector summation, as dened before. Therefore,having magnitude and direction is not sucient for a quantity to be identied as a vector,but it must also obey the rules of summation and multiplication with scalars.

Denition 8 (Unit vector). A unit vector is a vector of unit magnitude. For any vector vits unit vector is referred to by ev or v which is equal to v = v/‖v‖. One would say thatthe unit vector carries the information about direction. Therefore magnitude and directionas constituents of a vector are multiplicatively decomposed as v = vv.

1.4 Vector decomposition

Writing a vector v as the sum of its components is called decomposition. Components of avector are its projections onto the coordinate axes (Fig. 1.3)

v = v1e1 + v2e2 + v3e3 , (1.1)

where e1, e2, e3 are unit basis vectors of coordinate system, v1, v2, v3 are components

of v and v1e1, v2e2, v3e3 are component vectors of v.

X1

X2

X3

v

v1e1

v2e2

v3e3

Fig. 1.3: Components.

Since a vector is specied by its components relativeto a given coordinate system, it can be alternatively de-noted by array notation

v ≡

v1

v2

v3

= (v1 v2 v3)T . (1.2)

Using Pythagorean theorem the norm (magnitude) of avector v in three-dimensional space can be written as

‖v‖ =√v2

1 + v22 + v2

3 . (1.3)

Remark 9. Representation of a geometrical vector by its equivalent array (or tuple) isthe underlying idea of analytic geometry which basically paves the way for an algebraictreatment of geometry.

1.5 Dot product of vectors

Dot product (also called inner or scalar product) of two vectors v and w is dened as

v ·w = v w cos θ , (1.4)

where θ is the angle between v and w.


Property 10 (Dot product).

1. v ·w = w · v Commutativity2. u · (αv + βw) = αu · v + βu ·w Linearity3. v · v ≥ 0 and v · v = 0⇔ v = 0 Positivity.

Geometrical interpretation of dot product

Denition of dot product (1.4) can be restated as

v ·w = (vv) · (ww) = v (v ·w) = w (v · w) = vw cos θ , (1.5)

which says the dot product of vectors v and w equals projection of w on v directiontimes magnitude of v (see Fig. 1.1), or the other way around. Also, dividing leftmost andrightmost terms of (1.5) by vw gives

v · w = cos θ , (1.6)

which says the dot product of unit vectors (standing for directions) determines the anglebetween them. A simple and yet interesting result is obtained for components of a vectorv as

v1 = v · e1 , v2 = v · e2 , v3 = v · e3. (1.7)

Important results

1. In any orthonormal coordinate system including Cartesian coordinates it holds that

e1 · e1 = 1 , e1 · e2 = 0 , e1 · e3 = 0

e2 · e1 = 0 , e2 · e2 = 1 , e2 · e3 = 0 (1.8)

e3 · e1 = 0 , e3 · e2 = 0 , e3 · e3 = 1 ,

2. therefore ∗

v ·w = v1w1 + v2w2 + v3w3 . (1.9)

3. For any two nonzero vectors a and b, a · b = 0 i a is normal to b.

4. Norm of a vector v can be obtained by

‖v‖ = (v · v)1/2 . (1.10)

∗ Note that v ·w = (v1e1 + v2e2 + v3e3) · (w1e1 + w2e2 + w3e3) = v1w1e1 · e1 + v1w2e1 · e2 + · · · .

1.6. INDEX NOTATION 5

1.6 Index notation

A typical tensorial calculation demands component-wise arithmetic derivations which isusually a tedious task. As an example let us try to expand the left-hand-side of equation(1.9)

v ·w = (v1e1 + v2e2 + v3e3) · (w1e1 + w2e2 + w3e3)

= v1w1e1 · e1 + v1w2e1 · e2 + v1w3e1 · e3

+ v2w1e2 · e1 + v2w2e2 · e2 + v2w3e2 · e3

+ v3w1e3 · e1 + v3w2e3 · e2 + v3w3e3 · e3 , (1.11)

which together with equation (1.8) yields the right-hand-side of (1.9). As can be seenfor a very basic expansion considerable eort is required. The so called index notation isdeveloped for simplication.

Index notation uses parametric index instead of explicit index. If for example the letteri is used as index in vi it addresses any of the possible values i = 1, 2, 3 in three dimensionalspace, therefore representing any of the vector v's components. Then vi is equivalent tothe set of all vi's, namely v1, v2, v3.

Notation 11. When an index appears twice in one term then that term is summed upover all possible values of the index. This is called Einstein's convention.

To clarify this, let's have a look at trace of a 3 × 3 matrix tr(A) =∑3

i=1Aii = A11 +A22 +A33. Based on Einstein's convention one could write tr(A) = Aii, because the indexi has appeared twice and shall be summed up over all possible values of i = 1, 2, 3 thenAii = A11 +A22 +A33. As another example a vector v can be written as v = viei becausethe index i has appeared twice and we can write viei = v1e1 + v2e2 + v3e3.

Remark 12. A repeating index (which is summed up over) can be freely renamed. Forexample viwiAmn can be rewritten as vkwkAmn because in fact viwiAmn = Bmn and indexi does not appear as one of the free indices which can vary among 1, 2, 3. Note thatrenaming i to m or n would not be possible as it changes the meaning.

1.7 Tensors of second order

Based on the denition of inner product, length and angle which are the building blocksof Euclidean geometry become completely general concepts (Eqs. (1.10) and (1.6)). Thatis why an n-dimensional vector space (Def. 7) together with an inner product operation(Prop. 10) is called a Euclidean space denoted by En. Since we are basically unable to drawgeometrical scenes in higher than three dimensions (or maybe two in fact!), it may seem justtoo abstract to generalize classical geometry to higher dimensions. If you remember thatthe physical space-time can be best expressed in a four-dimensional geometrical framework,then you might agree that this limitation of dimensions is more likely to be a restriction ofour perception rather than a part of physical reality.


Another generalization would concern projection. Projection can be seen as geometricaltransformation that can be dened based on inner product. Up to three dimensions vectorprojection is easily understood. Now if we allow vectors to be entities belonging to anarbitrary vector space endowed with an inner product, one can think of inner product asthe means of projection of a vector in the space onto another vector or direction. Notethat we have put no restrictions on the vector space such as its dimension. By the wayone would ask what do we need all these cumbersome generalizations for? The answer isto understand more complex constructs based on the basic understanding we already haveof components of a geometrical vector.

So far we have seen that a scalar can be expressed by a single real number. This canbe seen as: scalars have no components on the three coordinate axes so require 30 realnumbers to be specied. On the other hand a vector v has three components which are itsprojections onto the three coordinate axes Xi's each obtained by vi = v · ei. Therefore, vrequires 31 scalars to be specied.

There are more complex constructs which are called second order tensors. The threeprojections of a second order tensor A onto coordinate axes are obtained by inner productof A with basis vectors as Ai = A · ei. These projections are vectors (not scalars!), andsince each vector is determined by three scalars its components a second order tensorrequires 3× 3 = 32 real numbers to be specied. In array form the three components of atensor A are vectors denoted by

A =(A1 A2 A3

)(1.12)

where each vector Ai has three components

Ai =

A1i

A2i

A3i

(1.13)

therefore A is written in the matrix form

A =

A11 A12 A13

A21 A22 A23

A31 A32 A33

(1.14)

where each vector Ai is written component-wise in one column of the matrix.

In analogy to the notation

v = viei (1.15)

for vectors (based on Einstein's summation convention), a tensor can be written as

A = Aijeiej or A = Aijei ⊗ ej (1.16)

with ⊗ being dyadic product. Note that we usually use the rst notation for brevity.

1.8. DYADIC PRODUCT 7

1.8 Dyadic product

Dyadic product of two vectors v and w is a tensor A such that

Aij = viwj . (1.17)

Property 13. Dyadic product has the following properties

1. v ⊗w 6= w ⊗ v2. u⊗ (αv + βw) = αu⊗ v + βu⊗w

(αu+ βv)⊗w = αu⊗w + βv ⊗w3. (u⊗ v) ·w = (v ·w)u

u · (v ⊗w) = (u · v)w

From equation (1.16) considering the above properties we obtain

Aij = ei ·A · ej , (1.18)

which is the analog of equation (1.7). In equation (1.16) the dyads eiej are basis tensorswhich are given in matrix notation as

e1e1 =

1 0 00 0 00 0 0

e1e2 =

0 1 00 0 00 0 0

e1e3 =

0 0 10 0 00 0 0

(1.19)

e2e1 =

0 0 01 0 00 0 0

e2e2 =

0 0 00 1 00 0 0

e2e3 =

0 0 00 0 10 0 0

(1.20)

e3e1 =

0 0 00 0 01 0 0

e3e2 =

0 0 00 0 00 1 0

e3e3 =

0 0 00 0 00 0 1

(1.21)

Note that not every second order tensor can be expressed as dyadic product of two vectorsin general, however every second order tensor can be written as linear combination of dyadicproducts of vectors, as in its most common form given by the decomposition onto a givencoordinate system as A = Aijeiej .

1.9 Tensors of higher order

Dyadic product provides a convenient passage to denition of higher order tensors. Weknow that dyadic product of two vectors (rst-order tensor) yields a second-order tensor.Generalization of the idea is immediate for a set of n given vectors v1, . . . ,vn as

T = v1v2 · · ·vn (1.22)

which is a tensor of the nth order. To make sense of it, let's take a look at inner product ofthe above tensor with vector w

T ·w = (v1v2 · · ·vn) ·w = (vn ·w) (v1v2 · · ·vn−1) (1.23)


where (vn ·w) is of course a scalar and v1 · · ·vn−1 a tensor of (n− 1)th order. This meansthe tensor T maps a rst-order tensor (vector) to a tensor of the (n− 1)th order. Wereemphasize that not all tensors can be written as dyadic product of vectors, howevertensors can be decomposed on the basis of a coordinate system. Decomposition of thenth-order tensor T is written as

T = Ti1i2···in ei1ei2 · · · ein . (1.24)

where i1 · · · in ∈ 1, 2, 3 and ei1ei2 · · · ein is the basis tensor of the nth order. It shouldbe clear that, visualization of e.g. third-order tensors will require three-dimensional arraysand so on.

Remark 14. A vector is a rst-order tensor and a scalar is a zeroth-order tensor. Order ofa tensor is equal to number of its indices in index notation. Furthermore, a tensor of ordern requires 3n scalars to be specied. We usually address a second order tensor by simplycalling it a tensor and the meaning should be clear from the context.

If an mth order tensor A is multiplied with an nth order tensor B we get an (m+ n)th

order tensor C such that

C = A⊗B = (Ai1···im ei1 · · · eim)⊗ (Bj1···jn ej1 · · · ejn)

= Ai1···imBj1···jnei1 · · · eimej1 · · · ejn= Ck1···km+n ek1 · · · ekm+n . (1.25)

Algebra and properties of higher order tensors

On the set of nth-order tensors summation operation can be dened as

A+B = (Ai1···in +Bi1···in) ei1 · · · ein , (1.26)

and multiplication with a scalar as

αA = (αAi1···in) ei1 · · · ein , (1.27)

having the properties similar to 5 and 6. In fact tensors of order n are elements of a vectorspace having the dimension 3n.

Exercise 1. Show that dimension of a vector space consisting of all nth-order tensors is3n. (Hint: nd 3n independent tensors that can span all elements of the space.)

Generalization of inner product

Inner/Dot product of two tensors A = Ai1···im ei1 · · · eim and B = Bj1···jn ej1 · · · ejn is givenas

A ·B = Ai1···im ei1 · · · eim · Bj1···jn ej1 · · · ejn= Ai1···im−1kBkj2···jnei1ei2 · · · eim−1ej2ej3 · · · ejn (1.28)

which is a tensor of (m+ n− 2)th order. Note that the innermost index k is common andsummed up. Generalization of this idea follows

1.9. TENSORS OF HIGHER ORDER 9

Denition 15 (contraction product). For any two tensors A of the mth order and B of

the nth order, their r-contraction product for r ≤ minm,n yields a tensor C = Ar B

of order (m+ n− 2r) such that

C = Ai1···im−rk1···kr Bk1···krjr+1···jn ei1 · · · eim−rejr · · · ejn . (1.29)

Note that the r innermost indices are common. The frequently used case of contraction

product is the so called double-contraction which is denoted by A : B ≡ A2 B.

Exercise 2. Write the identity σ = C : ε in index notation, where C is a fourth-ordertensor and σ and ε are second-order tensors. Then, expand the components σ12 and σ11

for the case of two-dimensional Cartesian coordinates.

Remark 16. Contraction product of two tensors is not symmetric in general, i.e. Ar B 6=

Br A, except when r = 1 and A and B are rst-order tensors (vectors), or A and B

have certain symmetry properties.

Kronecker delta

The second order unity tensor or identity tensor denoted by I is dened by the followingfundamental property.

Property 17. For every vector v it holds that

I · v = v · I = v . (1.30)

Representation of unity tensor by index notation as

δij =

1 , if i = j0 , if i 6= j .

(1.31)

is called Kronecker's delta. Its expansion for i, j ∈ 1, 2, 3 gives in matrix form

(δij) =

1 0 00 1 00 0 1

. (1.32)

Some important properties of Kronecker delta are

1. Orthonormality of the basis vectors ei in equation (1.8), can be abbreviated as

ei · ej = δij . (1.33)

2. The so called index exchange rule is given as

uiδij = uj or in general Aij···m···zδmn = Aij···n···z . (1.34)

3. Trace of Kronecker deltaδii = 3 . (1.35)

Exercise 3. Proofs are left to the reader as a practice of index notation.


Totally antisymmetric tensor

The so called totally antisymmetric tensor or permutation symbol, also named after Italianmathematician Tullio Levi-Civita as Levi-Civita symbol, is a third-order tensor dened as

εijk =

+1 , if (i, j, k) is an even permutation of (1, 2, 3)−1 , if (i, j, k) is an odd permutation of (1, 2, 3)0 , otherwise, i.e. if i = j or j = k or k = i.

(1.36)

There are important properties and relations on permutation symbol as follows

1. It is immediately known from the denition that

εijk = −εjik = −εikj = −εkji = εkij = εjki . (1.37)

2. Determinant of a 3× 3 matrix A can be expressed by

det(A) = εijkA1iA2jA3j . (1.38)

3. The general relationship between Kronecker delta and permutation symbol reads

εijkεlmn = det

δil δim δinδjl δjm δjnδkl δkm δkn

(1.39)

and as special cases

εjklεjmn = δkmδln − δknδlm , (1.40)

εijkεijm = 2δkm . (1.41)

Exercise 4. Verify the identities in equations (1.40) and (1.41) based on (1.39).

1.10 Coordinate transformation

Either seen as an empirical fact or a mathematical postulate, it is well accepted that

Denition 18 (Principle of frame indierence). A mathematical model of any physicalphenomenon must be formulated without reference to any particular coordinate system.

This is a merit of vectors and tensors that provide mathematical models with such agenerality. However in solving real world problems everything shall be expressed in termsof scalars nally for computation, and therefore introduction of a coordinate system isinevitable. The immediate question would be how to transform vectorial and tensorialquantities form one coordinate system to another. Substitution from (1.42) into the latterequation gives

v′1 = (v1e1 + v2e2 + v3e3) · e′1 =(e′1 · e1

)v1 +

(e′1 · e2

)v2 +

(e′1 · e3

)v3

v′2 = (v1e1 + v2e2 + v3e3) · e′2 =(e′2 · e1

)v1 +

(e′2 · e2

)v2 +

(e′2 · e3

)v3

v′3 = (v1e1 + v2e2 + v3e3) · e′3 =(e′3 · e1

)v1 +

(e′3 · e2

)v2 +

(e′3 · e3

)v3 . (1.44)

1.10. COORDINATE TRANSFORMATION 11

Suppose that vector v is decomposed in Cartesian coor-dinates X1X2X3 with bases e1, e2, e3 (Fig. 1.4)

v = v1e1 + v2e2 + v3e3 . (1.42)

Since a vector is completely determined by its compo-nents, we should be able to nd the decomposition ofvector v on a second coordinate system X ′1X

′2X′3 with

bases e′1, e′2, e′3 in terms of v1, v2, v3. Using equa-tion (1.7) we have

v′1 = v · e′1 , v′2 = v · e′2 , v′3 = v · e′3 . (1.43)

X1

X2

X3

v

X ′1

X ′2

X ′3

Fig. 1.4: Transformation of coor-

dinates.

Restatement in array notation reads v′1v′2v′3

=

e′1 · e1 e′1 · e2 e

′1 · e3

e′2 · e1 e′2 · e2 e

′2 · e3

e′3 · e1 e′3 · e2 e

′3 · e3

v1

v2

v3

, (1.45)

and in index notationv′j =

(e′j · ei

)vi . (1.46)

Then again, introducing transformation tensor Q =(e′j · ei

)we can rewrite the transfor-

mation rule asv′ = Qv . (1.47)

Property 19 (Orthonormal tensor). Transformation Q is geometrically a rotation mapwith the property

QTQ = I or Q−1 = QT (1.48)

which is called orthonormality.

Transformation of tensors

Having the decomposition of an nth-order tensor A in X1X2X3 coordinates as

A = Ai1···in ei1 · · · ein , (1.49)

we look for its components in another coordinates X ′1X′2X′3. Following the same approach

as for vectors, using equation (1.29)

A′j1···jn = An(e′j1 · · · e

′jn

)= Ai1···in ei1 · · · ein

n(e′j1 · · · e

′jn

)=(ei1 · e′j1

)· · ·(ein · e′jn

)Ai1···in , (1.50)

For the special case when A is a second order tensor we can write in matrix notation

A′ = QAQT , Q =(e′j · ei

). (1.51)


Remark 20. The above transformation rules are substantial. They are so fundamental that,as an alternative, we could have dened vectors and tensors based on their transformationproperties instead of the geometric and algebraic approach.

Exercise 5. Find the transformation matrix for a rotation by angle θ around X1 in three-dimensional Cartesian coordinates X1X2X3.

Exercise 6. For a symmetric second order tensor A derive the transformation formulawith a transformation given as

(Qij) =

cosϕ sinϕ 0− sinϕ cosϕ 0

0 0 1

.

1.11 Eigenvalues and eigenvectors

The question of transformation from one coordinate system to another is answered. Inpractical situations we face a second question which is the choice of coordinate system inorder to simplify the representation of the formulation at hand. A very basic example isexplained here.

Suppose that we want to write a vector v in components. With an arbitrary choice

of reference the answer is v =(v1 v2 v3

)T, with v1, v2, v3 ∈ R. However with the choice

of reference coordinates such that e.g. X1 axis coincides with vector v we come up with

v =(v1 0 0

)T. This second representation of the vector is simplied to one component

only which is in fact a scalar, and therefore being one-dimensional. Of course in morecomplicated situations an appropriate choice of coordinate system can make a much greaterdierence. We will see that the consequences of our rather practical question turns out tobe theoretically subtle. The next step in complexity is how to simplify tensors with a properchoice of coordinate system.

Denition 21. Given a second order tensor A, every nonzero vector like ψ is called aneigenvector of A if there exists a real number λ such that

A ·ψ = λψ (1.52)

where λ is the corresponding eigenvalue of ψ.

In general, a tensor multiplied by a vector gives another vector with a dierent mag-nitude and direction, interestingly however, there are vectors ψ on which the tensor actslike a scalar λ as laid down by equation (1.52). Now suppose that a coordinate system ischosen so that X1 axis coincides with ψ. It is clear that

A · e1 = A11e1 +A21e2 +A31e3

and also back to (1.52)A · e1 = λe1 (1.53)

1.11. EIGENVALUES AND EIGENVECTORS 13

comparing both equation gives

A11 = λ , A21 = 0 , A31 = 0 . (1.54)

It is possible to show that for non-degenerate symmetric tensors in R3 there are three or-thonormal eigenvectors which establish an orthonormal coordinate system which is calledprincipal coordinates. Following equation (1.54) a symmetric tensor σ in principal coordi-nates takes the diagonal form

σ =

σ11 0 00 σ22 00 0 σ33

, (1.55)

where its diagonal elements are eigenvalues corresponding to principal directions which areparallel to eigenvectors. The following theorem generalizes these ideas to n dimensions.

Theorem 22 (Spectral decomposition). Let A be a tensor of second order in n-dimensional

space with ψ1, · · · ,ψn being its n linearly independent eigenvectors and their corresponding

eigenvalues λ1, · · · , λn. Then A can be written as

A = ΨΛΨ−1 (1.56)

where Ψ is the n × n matrix having ψ's as its columns Ψ = (ψ1, · · · ,ψn), and Λ is an

n× n diagonal matrix with ith diagonal element as λi.

Remark 23. In the case of symmetric tensors equation (1.56) takes the form of

A = ΨΛΨT (1.57)

because for orthonormal eigenvectors, the matrix Ψ is an orthonormal matrix which is infact a rotation transformation.

Remark 24. The Λ is the representation of A in principal coordinates. Since it is diagonal,one can write

Λ =

n∑i=1

λinini . (1.58)

Suppose that we want to derive some relationship or develop a model based on tensorialexpressions. If the task is accomplished in principal coordinates it takes much less eort andat the same time, the results are completely general i.e. they hold for any other coordinatesystem. This is often done in material modeling and in other branches of science as well.

As the nal step we would like to solve the equation (1.52). It can be recast in the form

(A− λI)ψ = 0 . (1.59)

This equation has nonzero solutions if

det(A− λI) = 0 (1.60)


which is called the characteristic equation of tensor A. Let us have a look at its expansion∣∣∣∣∣∣A11 − λ A12 A13

A21 A22 − λ A23

A31 A32 A33 − λ

∣∣∣∣∣∣ = 0 , (1.61)

which gives

λ3−(A11 +A22 +A33)λ2+

(∣∣∣∣A22 A23

A32 A33

∣∣∣∣+

∣∣∣∣A11 A13

A31 A33

∣∣∣∣+

∣∣∣∣A11 A12

A21 A22

∣∣∣∣)λ−∣∣∣∣∣∣A11 A12 A13

A21 A22 A23

A31 A32 A33

∣∣∣∣∣∣ = 0

(1.62)which is a cubic equation with at most three distinct roots.

1.12 Invariants

Components of a vector v change from one coordinate system to another, however itslength remains the same in all coordinates because length is a scalar. This is interestingbecause one can write the length of a vector in terms of its components in any coordinatesystem as ‖v‖ = (vivi)

1/2, which means there is a combination of components that does notchange by coordinate transformation while the components themselves do. Any function ofcomponents f(v1, v2, v3) that does not change under coordinate transformation is called aninvariant. This independence of coordinate system makes invariants physically important,and the reason should become clear soon.

Tensors of higher order have also invariants. For a second order tensor A, looking backon equation (1.62), the values λ, λ2 and λ3 are scalars and stay unchanged under coordinatetransformation, therefore coecients of the equation must remain unchanged as well, andthey are all invariants of tensor A denoted by

IA = A11 +A22 +A33 (1.63)

IIA =

∣∣∣∣A22 A23

A32 A33

∣∣∣∣+

∣∣∣∣A11 A13

A31 A33

∣∣∣∣+

∣∣∣∣A11 A12

A21 A22

∣∣∣∣ (1.64)

IIIA =

∣∣∣∣∣∣A11 A12 A13

A21 A22 A23

A31 A32 A33

∣∣∣∣∣∣ (1.65)

where I, II and III are called rst, second and third principal invariants of A. The aboveformulas look familiar. The rst invariant is trace, the second is sum of principal minorsand the third is determinant. Of course, we could express invariants in terms of eigenvaluesof the tensor

IA = tr(A) = λ1 + λ2 + λ3 (1.66)

IIA =1

2

((tr(A))2 − tr

(A2))

= λ1λ2 + λ2λ3 + λ3λ1 (1.67)

IIIA = det(A) = λ1λ2λ3 . (1.68)

1.13. SINGULAR VALUE DECOMPOSITION 15

In material modeling we often look for a scalar valued function of a tensor in the formof φ(A). Since the function value φ is a scalar it must be invariant under transformationof coordinates. To fulll this requirement, the function is formulated in terms of invariantsof A, then it will be invariant itself. Therefore the most natural form of such a formulationwould be

φ = φ(IA, IIA, IIIA) . (1.69)

This will be used in the sequel when hyper-elastic material models are explained.

1.13 Singular value decomposition

The Euclidean norm of a vector is a non-negative scalar that represents its magnitude. It isalso possible to talk about norm of a tensor. Avoiding the details, we only mention that thequestion of norm of a tensor A comes down to nding the maximum eigenvalue of ATA.Since the matrix ATA is positive semidenite its eigenvalues are non-negative.

Denition 25. If λ2i is an eigenvalue of ATA, then λi > 0 is called a singular value of A.

When A is a symmetric tensor singular values of A equal the absolute values of itseigenvalues.

1.14 Cross product

v w

u = v ×w

θ

Fig. 1.5: Cross product.

Cross product of two vectors v and w is a vector u = v ×wsuch that u is perpendicular to v and w, and in the directionthat the triple (v,w,u) be right-handed (Fig. 1.5). Magnitudeof u is given by

u = vw sin θ 0 ≤ θ ≤ π . (1.70)

The cross product of vectors has the following properties.

Property 26 (Cross product).

1. v ×w = −w × v Anti-symmetry2. u× (αv + βw) = αu× v + βu×w Linearity.

Important results

1. In any orthonormal coordinate system including Cartesian coordinates it holds that

e1 × e1 = 0 , e1 × e2 = e3 , e1 × e3 = −e2

e2 × e1 = −e3 , e2 × e2 = 0 , e2 × e3 = e1

e3 × e1 = e2 , e3 × e2 = −e1 , e3 × e3 = 0 ,(1.71)


2. therefore

v ×w =

∣∣∣∣∣∣e1 e2 e3

v1 v2 v3

w1 w2 w3

∣∣∣∣∣∣ , (1.72)

3. which in index notation reads

v ×w = εijkvjwkei or [v ×w]i = εijkvjwk . (1.73)

Exercise 7. Using equation (1.71) prove the identity (1.72).

Geometrical interpretation of cross product

v

w

wsi

nθ

θ

Fig. 1.6: Area of triangle.

Norm of the vector v×w is equal to the area of parallelogramhaving sides v and w, and since it is normal to plane of theparallelogram we consider it to be the area vector. Then it isclear the the area A of a triangle with sides v and w is givenby

A =1

2v ×w . (1.74)

Cross product for tensors

Cross product can be naturally extended to the case of a tensor A and a vector v as

A× v = εjmnAimvneiej and v ×A = εimnvmAnjeiej . (1.75)

1.15 Triple product

For every three vectors u, v and w, combination of dot and cross products gives

u · (v ×w) , (1.76)

u

v

w

Fig. 1.7: Parallelepiped.

the so called triple product. Based on equation (1.72) it canbe written in component form

u · (v ×w) =

∣∣∣∣∣∣u1 u2 u3

v1 v2 v3

w1 w2 w3

∣∣∣∣∣∣ , (1.77)

or in index notation

u · (v ×w) = εijkuivjwk . (1.78)

Geometrically, triple product of the three vectors equals the volume of parallelepiped withsides u, v and w (Fig. 1.7).

1.16. DOT AND CROSS PRODUCT OF VECTORS: MISCELLANEOUS FORMULAS17

1.16 Dot and cross product of vectors: miscellaneous formu-

las

u× (v ×w) = (u ·w)v − (u · v)w (1.79)

(u× v)×w = (u ·w)v − (v ·w)u (1.80)

(u× v) · (w × x) = (u ·w) (v · x)− (u · x) (v ·w) (1.81)

(u× v)× (w × x) = (u · (v × x))w − (u · (v ×w))x

= (u · (w × x))v − (v · (w × x))u (1.82)

Exercise 8. Using index notation verify the above identities.

Exercise 9. For a tensor given as T = I + vw with v and w being orthogonal vectors i.e.v ·w = 0, calculate T 2,T 3, · · · ,T n and nally eT (Hint: eT =

∑∞n=0

1n!T

n).


Chapter 2

Tensor calculus

So far, vectors and tensors are considered based on operations by which we can combinethem. In this section we study vector and tensor valued functions in three dimensionalspace, together with their derivatives and integrals. It is important to keep in mind thatwe address tensors and vectors both by the term tensor, and the reader should alreadyknow that a vector is a rst-order tensor and a scalar a zeroth-order tensor.

2.1 Tensor functions & tensor elds

A tensor function A(u) assigns a tensor A to a real number u. In formal mathematicalnotation

A : R −→ Tu 7−→ A(u)

(2.1)

where T is the space of tensors of the same order, for instance second order tensors T =R3×3 or vectors T = R3 or maybe fourth order tensors T = R3×3×3×3.

O X1

X2

x(t)

Fig. 2.1: Position vector.

A typical example is the position vector of a moving particlegiven as a function of time x(t) where t is a real number andx is a vector with initial point at origin and terminal point atthe moving particle (Fig. 2.1).

On the other hand a tensor led A(x) assigns a tensor Ato a point x in space, say three dimensional. Again one couldformally write

A : R3 −→ Tx 7−→ A(x)

. (2.2)

Here, R3 is the usual three dimensional Euclidean space. Forinstance, we can think of temperature distribution in a given domain T (x) which is a scalareld that assigns scalars (temperatures) to points x in the domain. Another example couldbe strain led ε(x) in a solid body under deformation, which assigns second-order tensorsε to points x in the body.

19

20 CHAPTER 2. TENSOR CALCULUS

2.2 Derivative of tensor functions

Derivative of a dierentiable tensor function A(u) is dened as

dA

du= lim

∆u→0

A(u+ ∆u)−A(u)

∆u. (2.3)

We assume that all derivatives exist unless otherwise stated. For the case of a vectorfunction A(u) = Ai(u) ei in Cartesian coordinates the above denition becomes

dA

du=dAiduei =

dA1

due1 +

dA2

due2 +

dA3

due3 . (2.4)

Because ei's remain unchanged in Cartesian coordinates their derivatives are zero.

Exercise 10. For a given second-order tensor function A(u) = Aij(u) eiej in Cartesiancoordinates write down the derivatives.

Property 27. For tensor functions A(u) ,B(u) ,C(u), and vector function a(u) we have

1.d

du(A ·B) = A · dB

du+dA

du·B (2.5)

2.d

du(A×B) = A× dB

du+dA

du×B (2.6)

3.d

du[A · (B ×C)] =

dA

du· (B ×C) +A ·

(dB

du×C

)+A ·

(B × dC

du

)(2.7)

4. a · dadu

= ada

du(2.8)

5. a · dadu

= 0 i ‖a‖ = const (2.9)

Exercise 11. Verify the 4th and the 5th properties above.

2.3 Derivatives of tensor elds, Gradient, Nabla operator

Since a tensor led A(x) is a multi-variable function its dierential dA depends on thedirection of dierential of independent variable dx which is a vector dx = dxiei. When dx isparallel to one of coordinate axes, partial derivatives of the tensor eldA(x) = A(x1, x2, x3)are obtained

∂A

∂x1= lim

∆x1→0

A(x1 + ∆x1, x2, x3)−A(x1, x2, x3)

∆x1

∂A

∂x2= lim

∆x2→0

A(x1, x2 + ∆x2, x3)−A(x1, x2, x3)

∆x2(2.10)

∂A

∂x3= lim

∆x3→0

A(x1, x2, x3 + ∆x3)−A(x1, x2, x3)

∆x3

2.4. INTEGRALS OF TENSOR FUNCTIONS 21

or in compact form∂A

∂xi= lim

∆xi→0

A(x+ ∆xiei)−A(x)

∆xi. (2.11)

In general, dx is not parallel to coordinate axes. Therefore based on the chain rule ofdierentiation

dA(x1, x2, x3) =∂A

∂xidxi =

∂A

∂x1dx1 +

∂A

∂x2dx3 +

∂A

∂x3dx3 , (2.12)

Remembering from calculus of multi-variable functions, this can also be written as

dA = grad(A) · dx with grad(A) =∂A

∂xiei . (2.13)

In the latter formula gradient of tensor eld grad(A) appears which is a tensor of one orderhigher than A itself. To highlight this fact, we rewrite the above equation as

grad(A) =∂A

∂xi⊗ ei . (2.14)

A proper choice of notation in mathematics is substantial and introduction of operatorsnotation in calculus is no exception. Since linear operators follow algebraic rules somewhatsimilar to that of arithmetics of real numbers, writing relations in operators notation isinvaluable. Here, we introduce the so called nabla operator as

∇ ≡ ei∂

∂xi= e1

∂

∂x1+ e2

∂

∂x2+ e3

∂

∂x3. (2.15)

Then gradient of the tensor led in equation (2.13) can be written as∗

grad(A) = A∇ = A⊗∇ . (2.16)

2.4 Integrals of tensor functions

If A(u) and B(u) are tensor functions such that A(u) = d/duB(u), the indenite integralof A is ∫

A(u) du = B(u) +C (2.17)

with C = const, and the denite integral of A over the interval [a, b] is∫ b

aA(u) du = B(b)−B(a) . (2.18)

Decomposition of equation (2.17) for (say) a second-order tensor gives∫Aij(u) du = Bij(u) + Cij , (2.19)

where Aij is a real valued function of real numbers. Therefore, integration of a tensorfunction comes down to integration of its components.

∗In recent equations the dyadic ⊗ is written explicitly sometimes for pedagogical reasons.


2.5 Line integrals

X1

X2

X3

C

P1

P2

xn

Fig. 2.2: Space curve.

Consider a space curve C joining two points P1 (a1, a2, a3)and P2 (b1, b2, b3) as shown in Fig. 2.2. If the curve is dividedinto N parts by subdivision points x1, . . . ,xN−1, then the lineintegral of a tensor eld A(x) along C is given by

∫CA · dx =

∫ P2

P1

A · dx = limN→∞

N∑i=1

A(xi) ·∆xi (2.20)

where ∆xi = xi − xi−1, and when N → ∞ the largest ofdivisions tends to zero

maxi‖∆xi‖ → 0 .

In Cartesian coordinates the line integral can be written as∫CA · dx =

∫C

(A1 dx1 +A2 dx2 +A3 dx3) . (2.21)

Line integral has the following properties.

Property 28.

1.

∫ P2

P1

A · dx = −∫ P1

P2

A · dx (2.22)

2.

∫ P2

P1

A · dx =

∫ P3

P1

A · dx+

∫ P2

P3

A · dx P3 is between P1 and P2 (2.23)

Parameterization

A curve in space is a one-dimensional entity which can be specied by a parameterizationof the form

x(s) (2.24)

which obviously means

x1 = x1(s) , x2 = x2(s) , x3 = x3(s) .

Therefore a path integral can be parametrized in terms of s by∫ x2

x1

A · dx =

∫ s2

s1

A · dxds

ds . (2.25)

2.6. PATH INDEPENDENCE 23

2.6 Path independence

A line integral as introduced above, depends on its limit points P1 and P2, on particularchoice of the path C, and on its integrand A(x). However, there are cases where the lineintegral is independent of the path C. Then, having specied the tensor eld A(x) and theinitial point P1, the line integral is a function of the position of P2 only. If we denote theposition of P2 by x then

φ(x) =

∫ x

P1

A · dx . (2.26)

Now if we consider dierentiating the tensor eld φ(x) it yields

dφ = A · dx , (2.27)

which according denition of gradient (2.13) means

A = grad(φ) = φ∇ . (2.28)

This argument could be reversed which would lead to the following theorem.

Theorem 29. The line integral∫C A · dx is path independent if and only if there is a

function φ(x) such that A = φ∇.

Exercise 12. Prove the reverse argument of the above theorem. That is, starting fromexistence of φ(x) such that A = φ∇, prove that the line integral is path independent.

Remark 30. Note that in particular case of C being a closed path the line integral is pathindependent if and only if ∮

CA · dx = 0 , (2.29)

for arbitrary path C. The circle on integral sign shows that C is closed. This is anotherstatement of path-independence which is equivalent to the ones mentioned before.

2.7 Surface integrals

X2

X3

X1

ni∆Si

S

Fig. 2.3: Surface integral.

Consider a smooth connected surface S in space (Fig. 2.3)subdivided into N elements ∆Si, i = 1, . . . , N at positions xiwith unit normal vectors to the surface ni. Let tensor eldA(x) be dened over the domain Ω ⊇ S. Then the surfaceintegral of normal component of A over S is dened as∫

SA · n dS = lim

N→∞

N∑i=1

A(xi) · ni ∆Si (2.30)

where it holds that

N →∞ : maxi‖∆Si‖ → 0 .


The surface integral of tangential component of A over S is dened as∫SA× n dS = lim

N→∞

N∑i=1

A(xi)× ni ∆Si (2.31)

Note the surface integral (2.30) is a tensor of one order lower than A due to appearance ofA · n in the integrand, while the surface integral (2.31) is a tensor of the same order as Adue to the term A× n.

Surface integrals in terms of double-integrals

Surface integrals can be calculated in Cartesian coordinates. This is the case usually whenthe surface S does not have any symmetries, such as rotational symmetry. For this thesurface S is projected onto Cartesian plane X1X2 which gives a domain like S (Fig. 2.3).Then ∫

SA · n dS =

∫∫S

A · n dx1 dx2

n · e3. (2.32)

The transformation between the area element dS and its projection dx1dx2 is given by

dx1dx2 = (ndS) · e3 = (n · e3) dS ,

where n is the unit normal vector to dS and e3 is the unit normal vector to dx1dx2.

Exercise 13. Calculate the surface integral of the normal component of vector eld v =1/r2 er over the unit sphere centered at origin.

2.8 Divergence and curl operators

X2

X3

X1

ndS

S

Fig. 2.4: Closed surface.

In surface integrals introduced above, the case when the sur-face S is closed has a special importance. In a physical con-text, integral of the normal component of a tensor eld overa closed surface expresses the overall ux of the eld throughthe surface which is a measure of intensity of sources†. On theother hand, closed surface integral of the tangential compo-nent represents the so called rotation of the eld around thesurface.

Assume that S is a closed surface with outward unit nor-mal vector n, occupying the domain Ω with volume V . Inte-gral over closed surfaces is usually denoted by

∮. We can dene surface integrals of a given

tensor eld A(x) over S as in equations (2.30) and (2.31). As mentioned before, in the caseof closed surfaces these have special physical meanings. A question that naturally arises is:what are the average intensity of sources and rotations over the domain Ω. The answer is

D =1

V

∮SA · n dS C =

1

V

∮SA× n dS , (2.33)

†A source, as it literally suggests, is what causes the tensor eld.

2.9. LAPLACE'S OPERATOR 25

where integrals are divided by volume V . Now if the domain Ω shrinks, that is V → 0, thenthese average values reect the local intensity or concentration of sources and rotations ofthe tensor eld. This is the physical analogue of density which is the concentration of mass.These local values are called divergence and curl of the tensor eld A, denoted and formallydened by

div(A) = limV→0

1

V

∮SA · n dS (2.34)

curl(A) = limV→0

1

V

∮SA× n dS . (2.35)

It can be shown thatdiv(A) = A · ∇ (2.36)

curl(A) = A×∇ (2.37)

in operator notation. The proof is straightforward in Cartesian coordinates and can befound in most calculus books.

Exercise 14. Using a good calculus book, starting from (2.34) and (2.35) derive equations(2.36) and (2.37) in Cartesian coordinates for a vector eld u(x) as

div(u) = u · ∇ = ui∂i and curl(u) = u×∇ = εkijui∂j .

Remark 31. So far gradient, divergence and curl operators are applied to their operandform the right-hand-side i.e. grad(A) = A∇ , div(A) = A · ∇ , curl(A) = A×∇. This isnot a strict requirement by their denition. In fact the direction from which the operatoris applied depends on the context. If we exchange the order of dyadic, inner and outerproducts in equations (2.26), (2.34) and (2.35) respectively, the direction of application ofnabla will be reversed, to wit

φ =

∫dx · ∇φ lim

V→0

1

V

∮Sn ·A dS = ∇ ·A lim

V→0

1

V

∮Sn×A dS = ∇×A (2.38)

which are equally valid. However, keeping a consistence convention shall not be overlooked.

Exercise 15. Calculate ∇× (∇× v) for the vector eld v.

2.9 Laplace's operator

We remember from equation (2.28) that a tensor eld A whose line integrals are path-independent can be written as the gradient of a lower-order tensor eld as A = φ∇. Nowif we are interested in the concentration of sources that generate A then

div(A) = div(grad(φ)) = φ∇ · ∇ (2.39)

This combination of gradient and divergence operators appears so often that it is given adistinct name, the so called Laplace's operator or Laplacian which is denoted by

∆ ≡ ∇ · ∇ = ∂k∂k . (2.40)


Laplacian of a eld ∆A is sometimes denoted by ∇2A. Laplace's operator is also calledharmonic operator. That is why two consecutive application of Laplacian over a eld iscalled biharmonic operator denoted by

∇4A = ∆∆A = ∆ (∆A) (2.41)

which is expanded in Cartesian coordinates as

∆∆φ =∂4φ

∂x41

+∂4φ

∂x42

+∂4φ

∂x43

+ 2∂4φ

∂x21∂x

22

+ 2∂4φ

∂x22∂x

23

+ 2∂4φ

∂x23∂x

21

. (2.42)

2.10 Gauss' and Stoke's Theorems

Theorem 32 (Gauss theorem). Let S be a closed surface bounding a region Ω of volume

V with outward unit normal vector n, and A(x) a tensor eld (Fig. 2.4). Then we have

∫VA · ∇ dV =

∮SA · n dS . (2.43)

This result is also called Green's or divergence theorem. If we remember the physicalinterpretation of divergence, then the Gauss theorem means the ux of a tensor eld througha closed surface equals the intensity of the sources bounded by the surface, which physicallymakes perfect sense.

Theorem 33 (Stoke's theorem). Let S be an open two-sided

surface bounded by a closed non-self-intersecting curve C as

in Fig. 2.5. Then∮CA · dx =

∫S

(A×∇) · n dS (2.44)

where n is directed towards counter-clockwise view of the path

direction as indicated in the gure.

X2

X3

X1

C

ndS

S

Fig. 2.5: Stoke's theorem.

Physical interpretation of Stoke's theorem is much similar to that of Gauss' theoremexcept that it involves closed line integration and enclosed surface, instead of closed surfaceintegration and enclosed volume. Namely, the overall rotation of the tensor eld along aclosed path equals the intensity of rotations enclosed by the path.

2.11. MISCELLANEOUS FORMULAS AND THEOREMS INVOLVING NABLA 27

2.11 Miscellaneous formulas and theorems involving nabla

For given vector elds v and w, tensor elds A and B, and scalar elds α and β we have

∇(A+B) = ∇A+∇B (2.45)

∇ · (A+B) = ∇ ·A+∇ ·B (2.46)

∇× (A+B) = ∇×A+∇×B (2.47)

∇ · (αA) = (∇α) ·A+ α(∇ ·A) (2.48)

∇× (αA) = (∇α)×A+ α(∇×A) (2.49)

∇× (∇×A) = ∇(∇ ·A)−∇2A (2.50)

∇ · (v ×w) = (∇× v) ·w + v · (∇×w) (2.51)

∇× (v ×w) = (w · ∇)v − (∇ · v)w − (v · ∇)w + (∇ ·w)v (2.52)

∇(v ·w) = (w · ∇)v + (v · ∇)w +w × (∇× v) + v × (∇×w) (2.53)

Theorem 34. For a tensor eld A

∇× (∇A) = 0 (A∇)×∇ = 0 (2.54)

∇ · (∇×A) = 0 (A×∇) · ∇ = 0 (2.55)

given that A is smooth enough.

Let us rephrase the above theorem as follows:

Gradient of a tensor eld is curl-free, i.e. its curl vanishes.

Curl of a tensor eld is divergence-free, i.e. its divergence vanishes.

Comparing to Gauss and Stokes's theorems, these mean∮S

(A×∇) · n dS =

∫V

(A×∇) · ∇ dV = 0 (2.56)

and ∮C

(A∇) · dx =

∫S

(A∇)×∇ · n dS = 0 . (2.57)

Exercise 16. Show that in general ∇ ·A×∇ 6= 0 and ∇×A · ∇ 6= 0.

Remark 35. The recent equation, according (2.29), is equivalent to path-independence.

Theorem 36 (Green's identities). For two scalar elds φ and ψ the rst Green's identity

is ∫V

[φ (ψ∆) + (φ∇) · (ψ∇)] dV =

∮Sφ (ψ∇) · n dS (2.58)

and the second Green's identity∫V

[φ (ψ∆)− (φ∆)ψ] dV =

∮S

[φ (ψ∇)− (φ∇)ψ] · n dS . (2.59)


Alternative forms of Gauss' and Stoke's theorems

For tensor eld A and scalar eld φ we have∫VA×∇ dV =

∮SA× n dS

∮Cφdx =

∫S

(φ∇)× n dS . (2.60)

which could be equivalently stated as∫V∇×A dV =

∮Sn×A dS

∮Cφdx =

∫Sn× (∇φ) dS . (2.61)

e2

e3

e1

u2

u3

u1 u3 = c3

u1 = c1u2

=c2

Fig. 2.6: Curvilinear coordinates.

2.12 Orthogonal curvilinear coordinate systems

A point P in space (Fig. 2.6) can be specied by its rectangular coordinates (another namefor Cartesian coordinates) (x1, x2, x3), or curvilinear coordinates (u1, u2, u3). Then theremust be transformation rules of the form

x = x(u) and u = u(x) (2.62)

where the transformation functions are smooth and on-to-one. Smoothness means beingdierentiable, and being one-to-one requires Jacobian determinant to be non-zero

det

(∂ (u1, u2, u3)

∂ (x1, x2, x3)

)=

∣∣∣∣∣∣∣∂u1∂x1

∂u1∂x2

∂u1∂x3

∂u2∂x1

∂u2∂x2

∂u2∂x3

∂u3∂x1

∂u3∂x2

∂u3∂x3

∣∣∣∣∣∣∣ 6= 0 (2.63)

Denition 37 (Jacobian). For a dierentiable function f(x) such that f : Rm → Rn, itsJacobian is an m× n matrix dened as J =

(∂fi∂xj

)with i = 1, . . . ,m and j = 1, . . . , n.

At the point P there are three surfaces described by

u1 = const , u2 = const , u3 = const (2.64)

passing through P , called coordinate surfaces. And there are also three curves which areintersections of every two coordinate surfaces. These are called coordinate curves. On each

2.13. DIFFERENTIALS 29

coordinate curve only one of the three coordinates ui varies and the other two are constant.If the position vector is denoted by r then the vector ∂r/∂ui is tangent to ui coordinatecurve, and the unit tangent vectors (Fig. 2.6) are obtained from

e1 =∂r/∂u1

‖∂r/∂u1‖, e2 =

∂r/∂u2

‖∂r/∂u2‖, e3 =

∂r/∂u3

‖∂r/∂u3‖(2.65)

Introducing scale factors hi = ‖∂r/∂ui‖ we have

∂r/∂u1 = h1e1 , ∂r/∂u2 = h2e2 , ∂r/∂u3 = h3e3 (2.66)

If e1, e2 and e3 are mutually orthogonal then we call (u1, u2, u3) an orthogonal curvilinear

coordinate system.

Remark 38. In rectangular coordinates the coordinate surfaces are at planes and coordi-nate curves are straight lines. Also, the scale factors hi's are equal to one due to straightcoordinate curves. Furthermore, the unit vectors ei's are the same for all points in space.These properties do not generally hold for curvilinear coordinate systems.

2.13 Dierentials

In three dimensional space, there are three types of geometrical objects (other than points)regarding dimensions

one-dimensional objects: lines

two-dimensional objects: surfaces

three-dimensional objects: volumes.

Each type has its own dierential element, namely line elements, surface elements andvolume elements. We assume that the reader has already dealt with the related ideas inrectangular (Cartesian) coordinates. Now we want to generalize those ideas to curvilinearcoordinates.

Line elements

An innitesimal line segment with an arbitrary direction in space can be projected ontocoordinate axes. Consider the position vector r explained in a curvilinear coordinate systemby r(u1, u2, u3). A line element is the dierential of the position vector obtained by chainrule and using equation (2.66)

dr =∂r

∂u1du1 +

∂r

∂u2du2 +

∂r

∂u3du3 = h1 du1 e1 + h2 du2 e2 + h3 du3 e3 . (2.67)

where

dr1 = h1 du1 , dr2 = h2 du2 , dr3 = h3 du3 (2.68)


are the basis line elements. If the position vector sweeps a curve (which is typical forexample in kinematics) the arc length element denoted by ds is obtained by

(ds)2 = dr · dr = h21 (du1)2 + h2

2 (du2)2 + h23 (du3)2 . (2.69)

Note that the above formulas are completely general. For instance in Cartesian coordi-nates where hi = 1 they are reduced to familiar forms

dr = dx1e1 + dx2e2 + dx3e3 and (ds)2 = (dx1)2 + (dx2)2 + (dx3)2 .

Area elements

An area element is a vector described by its magnitude and direction

dS = dS n , (2.70)

whose decomposition is obtained by

dS = (dS · e1) e1 + (dS · e2) e2 + (dS · e3) e3 (2.71)

where dS · ei is in fact the projection of dS on the coordinate surface normal to ei.

On the other hand, having the coordinate line elements (2.68), the area elements oneach coordinate surface are obtained by

dS1e1 = dr2e2 × dr3e3 = h2 h3 du2 du3e1

dS2e2 = dr3e3 × dr1e1 = h3 h1 du3 du1e2 (2.72)

dS3e3 = dr1e1 × dr2e2 = h1 h2 du1 du2e3

which are called basis area elements. Note that an area element is a parallelogram withits side being line elements, and its area is obtained by cross product of the line elementvectors. Comparing the two above equation gives

dS = (h2 h3 du2 du3) e1 + (h3 h1 du3 du1) e2 + (h1 h2 du1 du2) e3 (2.73)

As the simplest case, in Cartesian coordinates the latter formula gives the familiar form

dS = dx2 dx3 e1 + dx3 dx1 e2 + dx1 dx2 e3 .

Volume element

A volume element dV is a parallelepiped built by the three coordinate line elements dr1e1,dr2e2 and dr3e3. Since the volume of a parallelepiped is obtained by triple product of itsside vectors as in equation (1.77), we have

dV = (dr1e1) · (dr2e2)× (dr3e3) = h1 h2 h3 du1 du2 du3 . (2.74)

2.14. DIFFERENTIAL TRANSFORMATION 31

2.14 Dierential transformation

For two given orthogonal curvilinear coordinate (u1, u2, u3) and (q1, q2, q3), we are interestedin transformation of dierentials between them. Here we adopt the index notation. Thereare two types of expressions in the aforementioned formulas to be transformed. One is thedierentials dui and the other is the factors of the form ∂r/∂ui or their norms hi. Weconsider the transformation of each separately. Starting from transformations of the formu(q), applying the chain rule we have

dui =∂ui∂qj

dqj and∂r

∂ui=

∂r

∂qj

∂qj∂ui

(2.75)

and because of orthogonality of coordinates

∥∥∥∥ ∂r∂ui∥∥∥∥ =

∑j

(∂r

∂qj

∂qj∂ui

)21/2

. (2.76)

Transformation of area element form one curvilinear coordinates to another is obtained by

dAq =

∣∣∣∣ ∂ (q1, q2, q3)

∂ (u1, u2, u3)

∣∣∣∣ ( ∂ (q1, q2, q3)

∂ (u1, u2, u3)

)−T· dAu (2.77)

And volume element transforms according

dVq =

∣∣∣∣ ∂ (q1, q2, q3)

∂ (u1, u2, u3)

∣∣∣∣ dVu (2.78)

2.15 Dierential operators in curvilinear coordinates

Here we briey address the general formulations of gradient, divergence, curl and Laplacianin orthogonal curvilinear coordinates. Formulations are given in a concrete form for a scalareld φ and a vector eld a = a1e1 + a2e2 + a3e3 however the reader should be able to usethem for tensor elds of any order.

∇φ =1

h1

∂φ

∂u1+

1

h2

∂φ

∂u2+

1

h3

∂φ

∂u3(2.79)

∇ · a =1

h1h2h3

[∂

∂u1(h2h3a1) +

∂

∂u2(h3h1a2) +

∂

∂u3(h1h2a3)

](2.80)

∇× a =1

h1h2h3

∣∣∣∣∣∣h1e1 h2e2 h3e3

∂/∂u1 ∂/∂u2 ∂/∂u3

h1a1 h2a2 h3a3

∣∣∣∣∣∣ (2.81)

∆φ =1

h1h2h3

[∂

∂u1

(h2h3

h1

∂φ

∂u1

)+

∂

∂u2

(h3h1

h2

∂φ

∂u2

)+

∂

∂u3

(h1h2

h3

∂φ

∂u3

)].(2.82)

Let us apply these formulas for the two most commonly used curvilinear coordinate systems,namely, cylindrical and spherical coordinates.


X2

X3

X1

x

rϕ

z = x3

x1

x2

Fig. 2.7: Cylindrical coordinates.

Cylindrical coordinates

In cylindrical coordinates a point is determined by (r, ϕ, z) (Fig. 2.7). Transformation torectangular coordinates are given as

x1 = r cosϕ , x2 = r sinϕ , x3 = z (2.83)

r =√x2

1 + x22 , ϕ = arctan(x2/x1) , z = x3 (2.84)

Scale factors are

hr = 1 hϕ = r hz = 1 , (2.85)

and dierential elements

dr = dr er + r dϕ eϕ + dz ez (2.86)

dS = r dϕ dz er + dr dz eϕ + r dr dϕ ez (2.87)

dV = r dr dϕ dz . (2.88)

Finally the dierential operators can be listed as

∇ ≡ er∂

∂r+ eϕ

1

r

∂

∂ϕ+ ez

∂

∂z(2.89)

∇ · a =1

r

∂

∂r(rar) +

1

r

∂aϕ∂ϕ

+∂az∂z

(2.90)

∆φ =1

r

∂

∂r

(r∂φ

∂r

)+

1

r2

∂2φ

∂ϕ2+∂2φ

∂z2=∂2φ

∂r2+

1

r

∂φ

∂r+

1

r2

∂2φ

∂ϕ2+∂2φ

∂z2(2.91)

∇× a =

[1

r

∂az∂ϕ− ∂aϕ

∂z

]er +

[∂ar∂z− ∂az

∂r

]eϕ +

1

r

[∂

∂r(raϕ)− ∂ar

∂ϕ

]ez (2.92)

2.15. DIFFERENTIAL OPERATORS IN CURVILINEAR COORDINATES 33

X2

X3

X1

r

ϕ

θ

x3

x1

x2

Fig. 2.8: Spherical coord.

Spherical coordinates

In spherical coordinates a point is determined by (r, θ, φ) (Fig. 2.8). Transformation torectangular coordinates are given as

x1 = r sin θ cosφ , x2 = r sin θ sinφ , x3 = r cos θ (2.93)

r =√x2

1 + x22 + x2

3 , θ = arccos(x3/r) , φ = arctan(x2/x1) .

Scale factors arehr = 1 hθ = r hφ = r sin θ , (2.94)

and dierential elements

dr = dr er + r dθ eθ + r sin θ dφ eφ (2.95)

dS = r2 sin θ dθ dφ er + r sin θ dr dφ eθ + r dr dθ eφ (2.96)

dV = r2 sin θ dr dθ dφ . (2.97)

Finally the dierential operators can be listed as

∇ ≡ er∂

∂r+ eθ

1

r

∂

∂θ+ eφ

1

r sin θ

∂

∂φ(2.98)

∇ · a =1

r2

∂

∂r

(r2ar

)+

1

r sin θ

∂

∂θ(sin θaθ) +

1

r sin θ

∂aφ∂φ

(2.99)

∆ξ =∂2ξ

∂r2+

2

r

∂ξ

∂r+

cos θ

r2 sin θ

∂ξ

∂θ+

1

r2

∂2ξ

∂θ2+

1

r2 sin2 θ

∂2ξ

∂φ2(2.100)

∇× a =1

r sin θ

[∂

∂θ(sin θaφ)− ∂aθ

∂φ

]er +

1

r

[1

sin θ

∂ar∂φ− ∂

∂r(raφ)

]eθ (2.101)

+1

r

[∂

∂r(raθ)−

∂ar∂θ

]eφ (2.102)


Bibliography

[1] A.I. Borisenko and I.E. Tarapov. Vector and Tensor Analysis with Applications. DoverPub. Inc, 1968.

[2] P.C. Chou and N.J. Pagano. Elasticity: Tensor, Dyadic, and Engineering Approaches.Dover Pub. Inc, 1992.

[3] R.W. Ogden. Non-Linear Elastic Deformations. Dover Pub. Inc, 1997.

[4] M.R. Spiegel. Mathematical Handbook of Formulas and Tables. McGraw-Hill (Schaum'soutline), 1979.

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