TENSOR ALGEBRAContinuum Mechanics Course (MMC) - ETSECCPB - UPC

Tensor Algebra

Introduction to Tensors

2

Introduction

SCALAR

VECTOR

MATRIX

?

, , ...σ ε

, , ...v f

, , ...

, ...C

v

3

Concept of Tensor

A TENSOR is an algebraic entity with various components which generalizes the concepts of scalar, vector and matrix.

Many physical quantities are mathematically represented as tensors.

Tensors are independent of any reference system but, by need, are commonly represented in one by means of their “component matrices”.

The components of a tensor will depend on the reference system chosen and will vary with it.

4

Order of a Tensor

The order of a tensor is given by the number of indexes needed to specify without ambiguity a component of a tensor.

Scalar: zero dimension

Vector: 1 dimension

2nd order: 2 dimensions

3rd order: 3 dimensions

4th order …

a,a a

,A A

, AA, AA

3.14 1.20.30.8

vi

0.1 0 1.30 2.4 0.5

1.3 0.5 5.8ijE

5

Cartesian Coordinate System

Given an orthonormal basis formed by three mutually perpendicular unit vectors:

Where:

Note that

1 2 2 3 3 1ˆ ˆ ˆ ˆ ˆ ˆ, , e e e e e e

1 2 3ˆ ˆ ˆ1 , 1 , 1 e e e

1ˆ ˆ

0i j ij

i ji j

e e

if

if

6

Cylindrical Coordinate System

1 2

1 2

3

ˆ ˆ ˆcos sin ˆ ˆ ˆsin cos ˆ ˆ

r

z

θ θθ θ

e e ee e ee e

1

2

3

cos ( , , ) sin

x rr z x r

x z

x

1x

2x

3x

7

Spherical Coordinate System

1 2 3

1 2

1 2 3

ˆ ˆ ˆ ˆsin sin sin cos cos ˆ ˆ ˆcos sin ˆ ˆ ˆ ˆcos sin cos cos sin

r

φ

θ φ θ φ θφ φθ φ θ φ θ

e e e ee e ee e e e

1

2

3

sin cos, , sin sin

cos

x rr x r

x r

x

3x

1x

2x

8

Tensor Algebra

Indicial or (Index) Notation

9

Tensor Bases – VECTOR

A vector can be written as a unique linear combination of the three vector basis for .

In matrix notation:

In index notation:

ˆ ie

1 1 2 2 3 3ˆ ˆ ˆv v v v e e ev

v

1

2

3

vvv

v

ˆvi ii

v e

viiv

tensor as a physical entity

component i of the tensor in the given basis

1v2v

3v

1,2,3i

1,2,3i

10

Tensor Bases – 2nd ORDER TENSOR

A 2nd order tensor can be written as a unique linear combination of the nine dyads for .

Alternatively, this could have been written as:

, 1,2,3i jˆ ˆ ˆ ˆi j i j e e e eA

1 1 1 2 1 3

2 1 2 2 2 3

3 1 3 2 3 3

ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ

11 12 13

21 22 23

31 32 33

A A AA A AA A A

A e e e e e ee e e e e ee e e e e e

1 1 1 2 1 3

2 1 2 2 2 3

3 1 3 2 3 3

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ

11 12 13

21 22 23

31 32 33

A A A

A A A

A A A

A e e e e e e

e e e e e e

e e e e e e

11

Tensor Bases – 2nd ORDER TENSOR

In matrix notation:

In index notation:

11 12 13

21 22 23

31 32 33

A A AA A AA A A

A

ˆ ˆAij i jij

A e e

ijijAA

tensor as a physical entity

component ij of the tensor in the given basis

1 1 1 2 1 3

2 1 2 2 2 3

3 1 3 2 3 3

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ

11 12 13

21 22 23

31 32 33

A A A

A A A

A A A

A e e e e e e

e e e e e e

e e e e e e

, 1,2,3i j

12

Tensor Bases – 3rd ORDER TENSOR

A 3rd order tensor can be written as a unique linear combination of the 27 tryads for .

Alternatively, this could have been written as:

ˆ ˆ ˆ ˆ ˆ ˆi j k i j k e e e e e eA

, , 1,2,3i j k

1 1 1 1 2 1 1 3 1

2 1 1 2 2 1 2 3 1

3 1 1 3 2 1 3 3 1

1 1 2 1 2 2

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ ...

111 121 131

211 221 231

311 321 331

112 122

e e e e e e e e e

e e e e e e e e e

e e e e e e e e e

e e e e e e

A A A A

A A A

A A A

A A

1 1 1 1 2 1 1 3 1

2 1 1 2 2 1 2 3 1

3 1 1 3 2 1 3 3 1

1 1 2 1 2 2

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ...

111 121 131

211 221 231

311 321 331

112 122

e e e e e e e e ee e e e e e e e ee e e e e e e e ee e e e e e

A A A A

A A A

A A A

A A

13

Tensor Bases – 3rd ORDER TENSOR

In matrix notation:

1 1 1 1 2 1 1 3 1

2 1 1 2 2 1 2 3 1

3 1 1 3 2 1 3 3 1

1 1 2 1 2 2

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ ...

111 121 131

211 221 231

311 321 331

112 122

e e e e e e e e e

e e e e e e e e e

e e e e e e e e e

e e e e e e

A

A A A

A A A

A A A

A A

113 123 133

213 223 233

313 323 333

A A A

A A A

A A A

112 122 132

212 222 232

312 322 332

A A A

A A A

A A A

111 121 131

211 221 231

311 321 331

A A A

A A A

A A A

A

14

Tensor Bases – 3rd ORDER TENSOR

In index notation:

ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ

ijk i j kijk

ijk i j k ijk i j k

e e e

e e e e e e

A A

A A

ijkijkA A

1 1 1 1 2 1 1 3 1

2 1 1 2 2 1 2 3 1

3 1 1 3 2 1 3 3 1

1 1 2 1 2 2

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ ...

111 121 131

211 221 231

311 321 331

112 122

e e e e e e e e e

e e e e e e e e e

e e e e e e e e e

e e e e e e

A A A A

A A A

A A A

A A

tensor as a physical entity

component ijk of the tensor in the given basis , , 1,2,3i j k

15

A tensor of order n is expressed as:

The number of components in a tensor of order n is 3n.

Higher Order Tensors

1 2 1 2 3, ... ˆ ˆ ˆ ˆ...n ni i i i i i iA A e e e e

1 2, ... 1,2,3ni i i where

16

The Einstein Summation Convention: repeated Roman indices are summed over.

A “MUTE” (or DUMMY) INDEX is an index that does not appear in a monomial after the summation is carried out (it can be arbitrarily changed of “name”).

A “TALKING” INDEX is an index that is not repeated in the same monomial and is transmitted outside of it (it cannot be arbitrarily changed of “name”).

3

1 1 2 2 3 313

1 1 2 2 3 31

i i i ii

ij j ij j i i ij

a b a b a b a b a b

A b A b A b A b A b

REMARKAn index can only appear up to two times in a monomial.

Repeated-index (or Einstein’s) Notation

i is a mute index

i is a talking index and j is a

mute index

17

Rules of this notation:

1. Sum over all repeated indices.

2. Increment all unique indices fully at least once, covering all combinations.

3. Increment repeated indices first.

4. A comma indicates differentiation, with respect to coordinate xi .

5. The number of talking indices indicates the order of the tensor result

Repeated-index (or Einstein’s) Notation

3

,1

i ii i

ii i

u uux x

2 23

, 21

i ii jj

jj j j

u uux x x

3

,1

ij ijij j

jj j

A AA

x x

18

Kronecker Delta δ

The Kronecker delta δij is defined as:

Both i and j may take on any value in Only for the three possible cases where i = j is δij non-zero.

10ij

i ji j

if

if

11 22 33

12 13 21

1 10 ... 0ij

i ji j

if

if

ij ji REMARKFollowing Einsten’s notation: Kronecker delta serves as a replacement operator:

11 22 33 3ii

,ij j i ij jk iku u A A

1,2,3

19

Levi-Civita Epsilon (permutation) ϵ

The Levi-Civita epsilon is defined as:

3 indices 27 possible combinations.

01 123, 231 3121 213,132 321

ijk ijkijk

if there is a repeated index

if or

if or

e

REMARKThe Levi-Civita symbol is also named permutation or alternating symbol.

ijk ikj e e

ijke

20

Relation between δ and ε

1 2 3

1 2 3

1 2 3

deti i i

ijk j j j

k k k

e detip iq ir

ijk pqr jp jq jr

kp kq kr

e e

ijk pqk ip jq iq jp e e

2ijk pjk pie e

6ijk ijk e e

21

Example

Prove the following expression is true:

6ijk ijk e e

22

211 211 212 212 213 213

221 221 222 222 223 223

231 231 232 232 233 233

e e e e e e

e e e e e e

e e e e e e

2i

311 311 312 312 313 313

321 321 322 322 323 323

331 331 332 332 333 333

e e e e e e

e e e e e e

e e e e e e

3i

121 121 122 122 123 123 e e e e e e 2j 131 131 132 132 133 133 e e e e e e 3j

Example - Solution

111 111 112 112 113 113ijk ijk e e e e e e e e1

1

1

1

1

1

6

1i

1j 1k 2k 3k

23

Tensor Algebra

Vector Operations

24

Sum and Subtraction. Parallelogram law.

Scalar multiplication

Vector Operations

a b b a ca b d

1 1 2 2 3 3ˆ ˆ ˆa a a a b e e e

i i i

i i i

c a bd a b

i ib a

25

Scalar or dot product yields a scalar

In index notation:

Norm of a vector

Vector Operations

cos u v u vwhere is the angle

between the vectors u and v

2 ˆ ˆi i j j i j ij i iu u u u u uu u e eu

1 2 1 2i iu uu uu

3

1

ˆ ˆ ˆ ˆv v v v vi

Ti i j j i j i j i j ij i i i i

i

u u u u u

u v e e e e u v

u v ij 0( )1 ( )

i jj i

26

Some properties of the scalar or dot product

Vector Operations

0

000, ,

u v v uu 0u v w u v u wu u u 0u u u 0u v u 0 v 0 u v

Linear operator

27

Vector product (or cross product ) yields another vector

In index notation:

Vector Operations

sin

c a b b ac a b

i iˆ ˆi ijk j k i ijk j kc a b c a b i c e ee e {1,2,3}

2 3 3 2 1 3 1 1 3 2 1 2 2 1 3ˆ ˆ ˆa b a b a b a b a b a b c e e e

where is the angle between the vectors a and b

0

1 2 3

1 2 3

1 2 3

ˆ ˆ ˆdet

symb

a a ab b b

e e e

123 1321 1

2 3 3 2a b a b

e e

1i

231 3 1 213 1 3

1 1

a b a b

e e

2i

312 1 2 321 2 11 1

a b a b

e e

3i

28

Some properties of the vector or cross product

Vector Operations

, , ||

a b a b

u v v uu v 0 u 0 v 0 u vu v w u v u w Linear operator

29

Tensor product (or open or dyadic product) of two vectors:

Also known as the dyad of the vectors u and v, which results in a 2nd

order tensor A.

Deriving the tensor product along an orthonormal basis {êi}:

In matrix notation:

Vector Operations

A u v uv

ˆ ˆ ˆ ˆ ˆ ˆv vi i j j i j i j ij i ju u A A u v e e e e e e

v v v1

T2 1 2 3

3

uuu

u v u v

v ,ij i jij ijA u i j A u v {1,2,3}

v v vv v vv v v

1 1 1 2 1 3 11 12 13

2 1 2 2 2 3 21 22 23

3 1 3 2 3 3 31 32 33

u u u A A Au u u A A Au u u A A A

30

Some properties of the open product:

Vector Operations

u v w u v w u v w v w u

u v v u

u v w u v u w

u v w u v w u v w w u v

u v w x u x v wLinear operator

31

Example

Prove the following property of the tensor product is true:

u v w u v w

32

Example - Solution

v v wk i i i i kkkc u u u v w w

v w v wk i i k i i kik ikc u u u v w u v w

ˆ ˆ ˆv wi j k k k kk ku c e u v w e u v w e

scalar vector

1st order tensor (vector)

1st order tensor (vector)

2nd order tensor (matrix)

u v w u v wvector

c

k-component of vector c

k-component of vector c

33

Example – Solution

u v w u v w

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆv w v w v wi i j j k k i i j k j k i j k i j ku u u u v w e e e e e e e e e

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆv w v w v wi i j j k k i j i j k k i j k i j ku u u u v w e e e e e e e e e

34

Vector Operations

Triple scalar or box product

In index notation:

cos

cos sin

V

a b c

c a b b c a

a b c

a b c

a b c ijk i j kV a b c e 1 2 3

1 2 3

1 2 3

deta a a

V b b bc c c

a b c

base areaheight

35

Vector Operations

Triple vector product

In index notation:

u v w u w v u v w

ˆ ˆ ˆv w v w

ˆ ˆv w v w

ˆ ˆv w v w

u v w e e e

e e

e e

j j klm l m k ijk j klm l m i

ijk lmk j l m i il jm im jl j l m i

m i m i l l i i

u u

u u

u u

e e e

e e

ijk pqk ip jq iq jp e eREMARK

36

Scalar or dot product yields a scalar

Vector or cross product yields another vector

Triple scalar or box product yields a scalar

Triple vector product yields another vector

Summary

c a b b a a b i ijk j kic a b e

cos sinV a b c a b c a b c ijk i j ka b c e

u v w u w v u v w w v v wk k i k k iiu u u v w

cosT u v u v u v vi iu u v

37

Tensor Algebra

Tensor Operations

38

Summation (only for equal order tensors)

Scalar multiplication (scalar times tensor)

Tensor Operations

A B B A C

A C

ij ij ijC A B

ij ijC A

39

Dot product (.) or single index contraction product

Tensor Operations

REMARK2A A A A B B A

Index “j” disappears (index contraction)

i ij jc A b2nd

order1st

order

A b c1st

order

Index “k” disappears (index contraction)

ij i k kjC bA b CA3rd

order1st

order2nd

order

Index “j” disappears (index contraction)

ik i kj jC A B A B C2nd

order2nd

order2nd

order

40

Some properties:

2nd order unit (or identity) tensor

Tensor Operations

1 u u 1 u

[1]ij j i i i

ij ij

1 e e e e

1 0 0

0 1 0

0 0 1

1

A b c A b A c Linear operator

41

Some properties:

2nd Order Tensor Operations

1 A A A 1A B C A B A C

A B C A B C A B C A B B A

42

Example

When does the relation hold true ? n T T n

43

Example - Solution

n T T nvector 2nd order

tensor

vector

c

k i ikc nT

k ki i i kic T n nT

c n T

c T nk kc c ik kiT Tif

ˆ ˆ ˆk k k i ik kkc nT c e n T e e

ˆ ˆ ˆk k k i ki kkc nT c e T n e e

44

Example - Solution

n T T n COMPACT NOTATION

INDEX NOTATION

TT n T T n

1,2,3i ik ki in T T n k

MATRIX NOTATION

Tcc

c

45

Example - Solution

TT n T T n MATRIX NOTATION

Tcc

c

11 12 13 11 12 13 1

1 2 3 21 22 23 21 22 23 2

31 32 33 31 32 33 3

, ,T T T T T T n

n n n T T T T T T nT T T T T T n

1

1 2 3 2

3

cc c c c

c

11 12 13 11 12 13 1

1 2 3 21 22 23 21 22 23 2

31 32 33 31 32 33 3

, ,

TT T T T T T n

n n n T T T T T T nT T T T T T n

1

1 2 3 2

3

Tc

c c c cc

46

Transpose

Trace yields a scalar

Some properties:

2nd Order Tensor Operations

Tjiij AA

11 22 33( )iiTr A A A AA

A ATTr Tr

Tr Tr A B B A Tr Tr Tr A B A B

Tr Tr A A

11 12 13 11 21 31

21 22 23 12 22 32

31 32 33 13 23 33

T

A A A A A AA A A A A AA A A A A A

A A

i j i iTr Tr a b a b a b a b

A ATT

T T T A B B A

u v v uT

A B A BT T T

47

Double index contraction or double (vertical) dot product (:)

Indices contiguous to the double-dot (:) operator get vertically repeated (contraction) and they disappear in the resulting tensor (4 order reduction of the sum of orders).

2nd Order Tensor Operations

Indices “i,j” disappear (double index contraction)

ij ijc A B2nd

order2nd

order

: cA Bzero order

(scalar)

Indices “j,k” disappear (double index contraction)

jk jki i Bc A: B cA3rd

order2nd

order1st

order

Indices “k,l” disappear (double index contraction)

ij ijkl klC B: B CA4th

order2nd

order2nd

order

48

Some properties

2nd Order Tensor Operations

: :T T T TTr Tr Tr Tr A B A B B A A B B A B A

: :

: : :

:

:

1 A A A 1

A B C B A C A C B

A u v u A v

u v w x u w v x

T T

Tr

REMARK: :A B C B A C

49

Double index contraction or double (horizontal) dot product (··)

Indices contiguous to the double-dot (··) operator get horizontally repeated (contraction) and they disappear in the resulting tensor (4 orders reduction of the sum of orders).

2nd Order Tensor Operations

Indices “i,j” disappear (double index contraction)

ij jic A B2nd

order2nd

order

c A B

Indices “j,k” disappear (double index contraction)

jk kji i Bc A B cA3rd

order2nd

order1st

order

Indices “k,l” disappear (double index contraction)

ij ijkl lkC B B CA4th

order2nd

order2nd

order

zero order

(scalar)

50

Norm of a tensor is a non-negative real number defined by

Tensor Operations

REMARK

Unless one of the two tensors is symmetric.

:A B A B Tr 1 A A A 1

1 21 2: 0ij ijA A A A A

Tr Tr A B A B B A B A

51

Example

Prove that:

Tr A B A B

: TTr A B A B

52

A B A BT T Tik ik ij ijikkk ki

c Tr A B A B A B

Example - Solution

k j

A B A Bki ik ij jikk ki ikc Tr A B A B A B

k ii j

53

Determinant yields a scalar

Some properties:

Inverse There exists a unique inverse A-1 of A when A is nonsingular, which satisfies the reciprocal relation:

2nd Order Tensor Operations

11 12 13

21 22 23 1 2 3

31 32 33

1

6det det detA A ijk i j k ijk pqr pi qj rk

A A AA A A A A A A A AA A A

e e e

det det det A B A Bdet detA AT

REMARKThe tensor A is SINGULAR if and only if det A = 0. A is NONSINGULAR if det A ≠ 0.

1 1

1 1 , , {1,2,3}ik kj ik kj ijA A A A i j k

A A 1 A A

3det det A A

54

If A and B are invertible, the following properties apply:

2nd Order Tensor Operations

1 1 1

11

1 1

11

2 1 1

11

1

1det det det

T T T

A B B A

A A

A A

A A A

A A A

A A A

55

Example

Prove that . 1 2 3det A ijk i j kA A A e

56

Example - Solution

11 12 13

21 22 23

31 32 33

11 22 33 21 32 13 31 12 23 13 22 31 23 32 11 33 12 21

det detA A AA A AA A A

A A A A A A A A A A A A A A A A A A

A

57

Example - Solution

111 11 21 31 112 11 21 32 113 11 21 33

121 11 22 31 122 11 22 32 123 11 22 33

131 11 23 31 132 11 23 32 133 11 23 33

211 12 21 31 212 12 21 32 213 12 21 33

221 12 22 31 222 12 22 32

A A A A A A A A A

A A A A A A A A A

A A A A A A A A A

A A A A A A A A A

A A A A A A

e e e

e e e

e e e

e e

e e 223 12 22 33

231 12 23 31 232 12 23 32 233 12 23 33

311 13 21 31 312 13 21 32 313 13 21 33

321 13 22 31 322 13 22 32 323 13 22 33

331 13 23 31 332 13 23 32 333 13 23 33

A A A

A A A A A A A A A

A A A A A A A A A

A A A A A A A A A

A A A A A A A A A

e

e e e

e e e

e e e

e e e

1

1

1

1

1

1

11 22 33 12 23 31 13 21 32 13 22 31 12 21 33 11 23 32A A A A A A A A A A A A A A A A A A

1 2 3ijk i j kA A A e

1i

1j 1k 2k 3k

2i

3i

2j 3j

58

Dot product – contraction of one index:

Summary - Tensor Operations

vi i

ijij ik kj ij

c uA B C

u vC A B

i ij jj j

ij j ii i

u A c

A u d

c u A

d A u

ijk ijk im mjk ijk

ijk ijk ijm mk ijk

A

A

A

A

C B

D B

B C

B D

ijklijkl ijm mkl ijkl

ijkl ijkl ijm mkl ijkl

C

D

A B

B A

A B

B A

CD

ij ij ik kj ijB A D D B A

59

Double dot product – contraction of two indices:

Summary - Tensor Operations

: ij ijc A B A B

:

:

ikm kmj ijij ij

ij ikm kmj ijij

C

D

C

D

A B

B A

A B

B A

:

:ijkl ijkl ijmp mpkl ijkl

ijkl ijkl ijmp mpij ijkl

C A B

D B A

A B CB A D

:

:ijk ijk ijlm lmk ijk

ijk ijk ilm lmjk ijk

AA

C B

D B

A

A

B C

B D

:

:ij ijk kk k

ijk jk ii i

A c

A d

c A

d A

B

B

B

B

60

Transposed double dot product – contraction of two indexes:

Summary - Tensor Operations

ij jic A B A B

ikm mkj ijij ij

ij ikm mkj ijij

C

D

C

D

A B

B A

A B

B A

ijklijkl ijmp pmkl ijkl

ijkl ijkl ijmp pmij ijkl

C A B

D B A

A B CB A D

ijk ijk ijlm mlk ijk

ijk ijk ilm mljk ijk

A

A

C B

D B

B C

B D

AA

ij jik kk k

ijk kj ii i

A c

A d

c A

d A

B

B

B

B

61

Open product – expansion of indexes:

Summary - Tensor Operations

vi j ijij ij

ij kl ijklijkl ijkl

u A

A B

A u v

A BC C

i jk ijkijk ijk

ij k ijkijk ijk

u A

A u

u A

A u

C

D

C

D

ijklmijklm ij klm ijklm

ijklmijklm ijk lm ijklm

A

A

A

A

B

B

B

B

C

D

C

D

vi j ijij ij

ij kl ijklijkl ijkl

u B

B A

B v u

B AD D

62

Tensor Algebra

Differential Operators

63

Differential Operators

A differential operator is a mapping that transforms a field into another field by means of partial derivatives.

The mapping is typically understood to be linear.

Examples: Nabla operator Gradient Divergence Rotation …

, ...v x A x

64

Nabla Operator

The Nabla operator is a differential operator “symbolically” defined as:

In Cartesian coordinates, it can be used as a (symbolic) vector on its own:

.ˆ

symbolic symb

iix

e

x

1

.

2

3

symb

x

x

x

65

Gradient

The gradient (or open product of Nabla) is a differential operator defined as: Gradient of a scalar field Φ(x):

Yields a vector

Gradient of a vector field v(x): Yields a 2nd order tensor

.

{1, 2,3}

ˆ ˆ

symb

i i ii i

i iii

ix x

x

e e

. vv , {1, 2,3}

vˆ ˆ ˆ ˆ

symbj

jij i ji i

ji j i jij

i

i jx x

x

v v

v v v e e e e

ˆ iix

e

vˆ ˆj

i jix

v e e

66

Gradient

Gradient of a 2nd order tensor field A(x): Yields a 3rd order tensor

. AA , , {1,2,3}

Aˆ ˆ ˆ ˆ ˆ ˆ

symbjk

jkijk ijk i jki i

jki j k i j kijk

i

i j kx x

x

A A A

A A A e e e e e e

Aˆ ˆ ˆjk

i j kix

A e e e

67

Divergence

The divergence (or dot product of Nabla) is a differential operator defined as : Divergence of a vector field v(x):

Yields a scalar

Divergence of a 2nd order tensor A(x): Yields a vector

vi

ix

v . vv

symbi

ii ii ix x

v v

Aˆij

jix

A e

. AA {1,2,3}

Aˆ ˆ

symbij

j iji iji i

ijj jj

i

jx x

x

A A

A A e e

68

Divergence

The divergence can only be performed on tensors of order 1 or higher.

If , the vector field is said to be solenoid (or divergence-free).

0 v v x

69

Rotation

The rotation or curl (or vector product of Nabla) is a differential operator defined as: Rotation of a vector field v(x):

Yields a vector

Rotation of a 2nd order tensor A(x): Yields a 2nd order tensor

v ˆv ekijk i

jx

e

A ˆ ˆA e eklijk i l

jx

e

. . vv {1,2,3}

vˆ ˆ

v v

v v e e

symb symbk

i ijk ijk k ijkj kj j

ki i ijk i

j

ix x

x

e e e

e

. AA , , {1, 2,3}

Aˆ ˆ ˆ ˆ

A

A A e e e e

symbkl

il ijk kl ijkj j

klil i l ijk i l

j

i j kx x

x

e e

e

70

Rotation

The rotation can only be performed on tensors of order 1 or higher.

If , the vector field is said to be irrotational (or curl-free).

0 v v x

71

Differential Operators - Summary

scalar fieldΦ(x)

vector field v(x)

2nd order tensor A(x)

GRADIENT

DIVERGENCE

ROTATION AA klil ijk

jx

e vv ki ijk

jx

e

vi

ix

v Aijj

ix

A

i

iix

vij

jij

ix

v

v

Aijk

jkijk

ix

A

A

72

Example

Given the vector determine

1 2 3 1 1 2 2 1 3ˆ ˆ ˆx x x x x x v v x e e e, , . v v v

73

Example - Solution

Divergence:

vi

ix

v

31 22 3 1

1 2 3

v vv vi

i

x x xx x x x

v

1 2 3

1 2

1

x x xx xx

v 1 2 3 1 1 2 2 1 3ˆ ˆ ˆx x x x x x v v x e e e

74

Example - Solution

Divergence:

In matrix notation:

vi

ix

v

1 2 3

1 21 2 3

1

1 2 3 1 2 11 2 3 1 2 1 2 3 1

1 2 3 1 2 3

, ,T

symb symb symbT

symb

x x xx x

x x xx

x x x x x xx x x x x x x x xx x x x x x

v v

1331

11

1 2 3

1 2

1

x x xx xx

v

75

In index notation:

Example - Solution

Rotation:

vv ki ijk

jx

e

3 32 1 1 212 13 21 23 31 32

1 1 2 2 3 3

v

v vv v v v

v ki ijk

j

i i i i i i

x

x x x x x x

e

e e e e e e

1 2 3

1 2

1

x x xx xx

v

76

3 2123 132

2 3

3 1213 231 1 2

1 32 1 3

2 1312 321

1 2

v v

0v v 1

v v

v

x x

x xx x

x x x

x x

e e

e e

e e

Example - Solution

Rotation:

In matrix notation

In compact notation:

32 112 13 21

1 1 2

3 1 223 31 32

2 3 3

vv v

v v v

v i i i i

i i i

x x x

x x x

e e e

e e e

1

1

1

1

1

1

1 2 2 2 1 3 3ˆ ˆ1x x x x x v e e

1 2 3

1 2

1

x x xx xx

v

77

Example - Solution

Rotation: Calculated directly in matrix notation:

1 1 2 3

2 1 2

3 1

vvv

x x xx xx

1 1 2 31

22 1 2 3

31 2 3

3

3 32 1 2 11 2 3

2 3 3 1 1 2

1 2 2 2 1 3 3

ˆ ˆ ˆvv detv

v v v

v vv v v vˆ ˆ ˆ

ˆ ˆ1

symb

x

x x x x

x

x x x x x x

x x x x x

e e e

v

e e e

e e

78

Example - Solution

Gradient:

In matrix notation

In compact notation:

v j

ij ijix

v v

1 1 2 3

2 1 2

3 1

vvv

x x xx xx

1

2 3 2

1 2 3 1 2 1 1 3 12

1 2

3

1, , 0

0 0

symb symb symbT

x x x xx x x x x x x x x

xx x

x

v v v

2 3 1 1 2 1 2 1 3 1 3 2 1 1 2 2 1 2 3 1ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆx x x x x x x x v e e e e e e e e e e e e

13

31

33

79

Tensor Algebra

Integral Theorems

80

Divergence or Gauss Theorem

Given a field in a volume V with closed boundary surface ∂V and unit outward normal to the boundary n , the Divergence (or Gauss) Theorem states:

Where: represents either a vector field ( v(x) ) or a tensor field ( A(x) ).A

A

V VdV dS

nA A

V VdV dS

nA A

81

Generalized Divergence Theorem

Given a field in a volume V with closed boundary surface ∂V and unit outward normal to the boundary n , the Generalized Divergence Theorem states:

Where: represents either the dot product ( · ), the cross product ( ) or the

tensor product ( ). represents either a scalar field ( ϕ(x) ), a vector field ( v(x) ) or a

tensor field ( A(x) ).

V VdV dS

nA A

V VdV dS

nA A

A

A

82

Curl or Stokes Theorem

Given a vector field u in a surface S with closed boundary surface ∂S and unit outward normal to the boundary n , the Curl (or Stokes) Theorem states:

S S

dS d

u n u r

where the curve of the line integral must have positive orientation, such that dr points counter-clockwise when the unit normal points to the viewer, following the right-hand rule.

83

jn

Example

Use the Generalized Divergence Theorem to show that

where is the position vector of .

i j ijSx n dS V

jnix

V VdS dV

nA A

84

Example - Solution

Applying the Generalized Divergence Theorem:

Applying the definition of gradient of a vector:

V VdS dV

x n x

i jSx n dS S

dS x n

jij

i

xx

x i

ijj

xx

x

i j ijSx n dS V

85

Example - Solution

The Generalized Divergence Theorem in index notation:

Then,

i j ijSx n dS V

ii jS V

j

xx n dS dVx

ii j ij ijS V V

j

xx n dS dV dV Vx

86

Tensor Algebra

References

87

José Mª Goicolea, Mecánica de Medios Continuos: Resumen de Álgebra y Cálculo Tensorial, UPM.

Eduardo W. V. Chaves, Mecánica del Medio Continuo,Vol. 1 Conceptosbásicos, Capítulo 1: Tensores de Mecánica del Medio Continuo, CIMNE, 2007.

L. E. Malvern. Introduction to the mechanics of a continuous medium. Prentice-Hall, Englewood Clis, NJ, 1969.

G. A. Holzapfel. Nonlinear solid mechanics: a continuum approach for engineering. 2000.

References

88