TENSOR ALGEBRA

Continuum Mechanics Course (MMC) - ETSECCPB - UPC Introduction to Tensors

Tensor Algebra

2 Introduction

SCALAR , , ...

v VECTOR vf, , ...

MATRIX σε,,...

? C,...

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.

a  Scalar: zero dimension   3.14 1.2   v  0.3 a , a Vector: 1 dimension i     0.8 0.1 0 1.3  2nd order: 2 dimensions   A, A  E  02.40.5 ij   rd   A , A 3 order: 3 dimensions 1.3 0.5 5.8 A , A  4th order …

5 Cartesian Coordinate System

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

eeˆˆ12,, ee ˆˆ 23 ee ˆˆ 31 Where:

eeeˆˆˆ1231, 1, 1

 Note that 1 if ij eeˆˆi  j ij 0 if ij

6 Cylindrical Coordinate System

x3

xr1  cos  x(,rz , ) xr2 sin  xz3 

eeeˆˆˆr  cosθθ 12 sin eeeˆˆˆsinθθ cos x2  12

eeˆˆz  3 x1

7 Spherical Coordinate System

x3

xr1  sin cos  xrxr, , 2 sin sin  xr3  cos

ˆˆˆˆ x2 eeeer sinθφ sin 123sin θ cos φ cos θ

eeeˆˆˆ cosφφ 12sin x1 eeeeˆˆˆˆφ cosθφ sin 123cos θ cos φ sin θ

8 Indicial or (Index) Notation

Tensor Algebra

9 Tensor Bases – VECTOR

 A vector v can be written as a unique linear combination of the three vector basis eˆ for i   1, 2, 3  . i v

ve vv11ˆˆˆ 2 e 2 v 33 e v3  In matrix notation: v1  v  v   2  v1 v3    v2  In index notation:

ˆ tensor as a physical entity ve  vii i v  v component i of the tensor in the i i given basis i 1, 2, 3

10 Tensor Bases – 2nd ORDER TENSOR

 A 2nd order tensor A can be written as a unique linear combination of the nine dyads for . eeeeˆˆˆˆijij  ij,1,2,3 

AeeeeeeAA11ˆˆ11 12 ˆˆ 12 A 13  ˆˆ 13

AA21eeˆˆ21 22 ee ˆˆ 22 A 23  ee ˆˆ 23  AA31eeˆˆ31 32 ee ˆˆ 32 A 33 ee ˆˆ 33

Alternatively, this could have been written as:

AeeeeeeAA11ˆˆ11 12 ˆˆ 12 A 13 ˆˆ 13

AA21eeˆˆ21 22 ee ˆˆ 22  A 23 ee ˆˆ 23

AA31eeˆˆ31 32 ee ˆˆ 32  A 33 ee ˆˆ 33

11 Tensor Bases – 2nd ORDER TENSOR

Aeeeeee AA11ˆˆ11 12 ˆˆ 12   A 13  ˆˆ 13 

 AA21eeˆˆ21 22 ee ˆˆ 22  A 23  ee ˆˆ 23   AA31eeˆˆ31 32 ee ˆˆ 32 A 33 ee ˆˆ 33  In matrix notation:

 A11AA 12 13  A  A AA  21 22 23   A31AA 32 33   In index notation: AeeA ˆˆ tensor as a  ij i j  physical entity ij A  A component ij of the tensor  ij ij in the given basis ij,1,2,3 

12 Tensor Bases – 3rd ORDER TENSOR

 A 3rd order tensor A can be written as a unique linear combination

of the 27 tryads eeˆˆ ijkijk   eeee ˆˆˆˆ for ijk ,,   1,2,3  .

A  AA111eeeˆˆˆ111  121 eee ˆˆˆ 121   A 131  eee ˆˆˆ 131  

AAA211eeeˆˆˆ211  221 eee ˆˆˆ 221   231  eee ˆˆˆ 231    AA311eeeˆˆˆ311  321 eee ˆˆˆ 321   A 331 eee ˆˆˆ 331  

AA112eeeˆˆˆ112 122 eee ˆˆˆ 12 2... Alternatively, this could have been written as:

A AA111eeeˆˆˆ111 121 eee ˆˆˆ 121  A 131 eee ˆˆˆ 131

AAA211eeeˆˆˆ211 221 eee ˆˆˆ 221 231 eee ˆˆˆ 231

AA311eeeˆˆˆ311 321 eee ˆˆˆ 321  A 331 eee ˆˆˆ 331

AA112eeeˆˆˆ112 122 eee ˆˆˆ 12 2 ...

13 Tensor Bases – 3rd ORDER TENSOR

A  AA111eeeˆˆˆ111  121 eee ˆˆˆ 121   A 131  eee ˆˆˆ 131  

AAA211eeeˆˆˆ211  221 eee ˆˆˆ 221   231  eee ˆˆˆ 231    AA311eeeˆˆˆ311  321 eee ˆˆˆ 321   A 331 eee ˆˆˆ 331  

AA112eeeˆˆˆ112 122 eee ˆˆˆ 12 2...  In matrix notation:  AAA  113 123 133 AAA112 122 132  AAA111 121 131  AAA  213 223 233 AAA212 222 232 A   AAA211 221 231      AAA313 323 333  AAA312 322 332  AAA311 321 331  

14 Tensor Bases – 3rd ORDER TENSOR

A AA111eeeˆˆˆ111 121 eee ˆˆˆ 121 A 131  eee ˆˆˆ 131

AAA211eeeˆˆˆ211  221 eee ˆˆˆ 221   231  eee ˆˆˆ 231    AA311eeeˆˆˆ311  321 eee ˆˆˆ 321   A 331 eee ˆˆˆ 331  

AA112eeeˆˆˆ112 122 eee ˆˆˆ 12 2...  In index notation: ˆˆˆ A Aijkeee i j k  ijk tensor as a AAijkieeˆˆj e ˆkijki eee ˆˆˆj k physical entity

component ijk of the tensor A  Aijk ijk in the given basis ijk,, 1,2,3

15 Higher Order Tensors

 A tensor of order n is expressed as:

Aeeee A ˆˆˆ... ˆ ii12, ... inn i 1 i 2 i 3 i

where ii12, ... in  1,2,3

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

16 Repeated-index (or Einstein’s) Notation

 The Einstein Summation Convention: repeated Roman indices are summed over. 3 i is a mute index abii ab ii ab11 ab 2 2 ab 33 i1 i is a talking 3 index and j is a A bAbAbAbAb mute index ij j ij j i11 i 2 2 i 33 j1  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”). REMARK An index can only appear up to two times in a monomial.

17 Repeated-index (or Einstein’s) Notation

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 . uu3 22uu3  A 3 A u iiu iiA ij ij ii,  ijj,  2 ij, j  xiii1 x x jjxxj1 j x jjj1 x 5. The number of talking indices indicates the order of the tensor result

18 Kronecker Delta δ

 The Kronecker delta δij is defined as: 1 if ij ij   0 if ij

 Both i and j may take on any value in 1, 2, 3   Only for the three possible cases where i = j is δij non-zero.

11if ij11 22 33  ij   0if ij 12 13 21 ...  0 REMARK  Following Einsten’s notation:   3 ij  ji ii 11 22 33 Kronecker delta serves as a replacement operator:

ijuu j i,  ij A jk A ik

19 Levi-Civita Epsilon (permutation) ϵ

 The Levi-Civita epsilon eijk is defined as:

 0 if there is a repeated index  eijk  1ifijk  123, 231 or 312   1ifijk 213,132 or 321

 3 indices 27 possible combinations.  eeijkik j

REMARK The Levi-Civita symbol is also named permutation or alternating symbol.

20 Relation between δ and ε

  ii123 i ip iq ir    e  det   eeijk pqr det  jp jq jr ijk j123 j j     kk123 k  kp kq kr 

eeijk pqk ip jq iq jp

eeijk pjk 2 pi

eeijk ijk  6

21 Example

 Prove the following expression is true:

eeijk ijk  6

22 Example - Solution

k 1 k  2 k  3 eeijk ijk  e111 e 111 e 112 e 112 e 113 e 113 j 1 1 ee121 121 ee 122 122 ee 123 123 j  2 i 1  1 ee131 131 ee 132 132 ee 133 133 j  3  1 ee211 211 ee 212 212 ee 213 213

i  2 ee221 221 ee 222 222 ee 223 223 1 ee231 231 ee 232 232 ee 233 233 1 ee311 311 ee 312 312 ee 313 313 1 i  3 ee321 321 ee 322 322 ee 323 323

ee331 331 ee 332 332 ee 333 333  6

23 Vector Operations

Tensor Algebra

24 Vector Operations

 Sum and Subtraction. Parallelogram law.

ab  bac cabiii  ab d dabiii 

 Scalar multiplication

abaa11 eˆˆˆ  2 e 2   a 33 e baii 

25 Vector Operations

 Scalar or dot product yields a scalar where  is the angle uv uvcos between the vectors u and v

 In index notation:

i3 T ˆˆ ˆˆ uvuuuuuii e vv jj e  i ji e  e j  i vv jij  ii  ii v  u v i1 u v  0(i j ) ij  1(j i )  Norm of a vector  12 12 u uu uuii 2 u uuuii eˆˆ  u j e j  uu i jij  uu ii

26 Vector Operations

 Some properties of the scalar or dot product

uv  vu u00

uvw    uvuw  Linear operator uu0 u  0 uu0 u  0 uv 0, u 0 , v 0 u  v

27 Vector Operations

 Vector product (or cross product ) yields another vector cab  ba cab sin where  is the angle between the vectors a and b 0    In index notation:

cecabcabiˆˆ e  {1,2,3} iiiee ijk j k i ijk j k

i 1 i  2 i  3 eeeˆˆˆ symb 123 ceeab abˆˆ  ab ab  ab ab e ˆ det aaa 23 32 1 31 13 2 12 21 3  123 eeab ab eeab ab bbb ee123ab23 132 ab 32 231 3 1 213 1 3 312 1 2 321 2 1  123  1111 11

28 Vector Operations

 Some properties of the vector or cross product

uv vu uv 0,, u  0v0  uv ||

uvwuvuwab  a b Linear operator

29 Vector Operations

 Tensor product (or open or dyadic product) of two vectors: Auvuv 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}:

AuvuuAiijjijijijij eˆ vv e ˆ  ee ˆˆ   ee ˆˆ  {1,2,3} Auvij Auijij  ij  iv, j   In matrix notation:

u1 uuu1vvv 1 1 2 1 3 AAA 11 12 13  uv  uvT u vvv uuuvvv AAA  2123 2 1 2 2 2 3 21 22 23  u3 uuu3vvv 1 3 2 3 3 AAA 31 32 33 

30 Vector Operations

 Some properties of the open product:

uv  vu

uvwu  vwuvw    vwu

uvwuvuw   Linear operator uvwx    uxvw 

u v w  uv w uvw  wuv 

31 Example

 Prove the following property of the tensor product is true: uv  w  uv w

32 Example - Solution

uvwuvw    c vector 2nd order scalar vector tensor (matrix) st 1st order tensor 1 order tensor (vector) (vector)

cuuuv w  u v  w vw  vw k-component kiikiik   i   kik of vector c

cuuuv wvvw  w  k-component kiiiik   k k of vector c

ceuvweuvweu vwˆˆˆ    ijkkkkkk

33 Example – Solution

uv  w  uv w

uv w uuuii eˆˆ v j e j  w kk e ˆ  ii e ˆ  vw j k e ˆˆ j  e k  i vw j k ee ˆˆˆ i  j e k

uv wuuuii eˆˆ vw j e j  kk e ˆ  i v ji e ˆˆ  e j  w kk e ˆ  i vw j k e ˆˆˆ i  e j e k

34 Vector Operations

 Triple scalar or box product

V abc   cab  bca abccos  abccos sin height base area

 In index notation:

aaa123   Vabcabc  eijk i j k Vbbbabc  det  123 ccc123

35 Vector Operations

 Triple vector product

uvwuwvuvw     

 In index notation:

uvw  uujj eˆˆ eee klmlmkvw e  ijkj klmlm vw  e ˆ i 

 eeijk lmkuu jvw l meeˆˆ i il  jm  im jl j vw l m i 

uumivw mieeˆˆ ll vw ii

REMARK

eeijkpqk  ipjq iqjp

36 Summary

 Scalar or dot product yields a scalar T uv u  v  uvcos uv  uiiv

 Vector or cross product yields another vector cab  ba abcabe  i i ijk j k

 Triple scalar or box product yields a scalar

V abc   acos bc sin abc  eijkabc i j k

 Triple vector product yields another vector

uvwuwvuvw     uvw  uuwv  vw i kki kki

37 Tensor Operations

Tensor Algebra

38 Tensor Operations

 Summation (only for equal order tensors)

ABBAC CABij ij ij

 Scalar multiplication (scalar times tensor)

AC CAij  ij

39 Tensor Operations

 Dot product (.) or single index contraction product

Ab  c cAbii j j Index “j” disappears (index 2nd 1st 1st contraction) order order order

 bC Cbij ijkk Index “k” disappears (index A A contraction) 3rd 1st 2nd order order order

ABC CABik ij j k Index “j” disappears (index 2nd 2nd 2nd contraction) order order order REMARK AB  BA AA  A2

40 Tensor Operations

 Some properties:

Abc  AbAc  Linear operator

 2nd order unit (or identity) tensor

1u  u1 u 100 1eeeeijjiii  1  010 [1]     ij ij 001

41 2nd Order Tensor Operations

 Some properties:

1A A A1 ABC  ABAC ABC ABCABC  AB BA

42 Example

 When does the relation nT   Tn hold true ?

43 Example - Solution

nTTn  c vector 2nd order tensor vector

cnT  cnTkiik  cckk if TTik ki   cTn  cTnnTkkiiiki

cenTecnTˆˆˆ e kkk k iikk ceTnee cnTˆˆˆ kkk k ikik

44 Example - Solution

nT  Tn COMPACT NOTATION

nTiik T kii n k 1, 2, 3 INDEX NOTATION

T T nT Tn MATRIX NOTATION cT c c

45 Example - Solution

T T nT Tn MATRIX NOTATION cT c c

T TTT11TTT11 12 12 13 13 TTT TTT 11 11 12 12 13 13 n n 1 1  nnnTTnnnTT,,,, T T T T T T T T n n 1232122231 2 32122232122232 2122232 TTT TTT n TTT3131 32 32 33 33 TTT 31 31 32 32 33 33 n 3 3

c1c1  T cccc c c c c 1212 3 3 2 2 c3c3 

46 2nd Order Tensor Operations

 Transpose

 A AA AAA T T 11 12 13 11 21 31 AA   T  A A21AA 22 23A AAA 12 22 32 T   AB  BTT A  A AA AAA   31 32 33 13 23 33 T uv  vu T    A T TT A ij ji AB A B  Trace yields a scalar TrA  A() A A A  ii 11 22 33 Trab  Tr abij ab ii ab  Some properties: T TrAA Tr TrAB Tr A  Tr B TrAA Tr TrAB  Tr  BA

47 2nd Order Tensor Operations

 Double index contraction or double (vertical) dot product (:) Indices “i,j” disappear (double index AB:  c cAB ij ij contraction) 2nd 2nd zero order order order (scalar)

: Bc cii jkjkB Indices “j,k” disappear (double index A A contraction) 3rd 2nd 1st order order order : BC A CBij ijkl kl Indices “k,l” disappear (double index 4th 2nd 2nd contraction) order order order  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).

48 2nd Order Tensor Operations

 Some properties

AB::Tr ATT B Tr B A Tr AB T Tr  BA  T BA

1A::Tr A A1 ABC::: BACTT   ACB Au: v uAv uv: wx uwvx

REMARK AB:: CB A C

49 2nd Order Tensor Operations

 Double index contraction or double (horizontal) dot product (··) Indices “i,j” disappear (double index ABc cAB ij ji contraction) 2nd 2nd zero order order order (scalar)

Bc cii jkkjB Indices “j,k” disappear (double index A A contraction) 3rd 2nd 1st order order order BC A CBij ijkl lk Indices “k,l” disappear (double index 4th 2nd 2nd contraction) order order order  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).

50 Tensor Operations

 Norm of a tensor is a non-negative real number defined by 12 12 AAA:0AAij ij 

AB Tr AB   Tr  BA   BA 

1A Tr A  A1  REMARK AB:  AB Unless one of the two tensors is symmetric.

51 Example

 Prove that:

AB: Tr  AT B

ABTr  AB 

52 Example - Solution

cTrABTT  AB A T B AB AB kk ki ik ik ik ij ij k  j

cTrAB AB A B AB  AB    kk  ki  ik ki ik ijji i  j k  i

53 2nd Order Tensor Operations

 Determinant yields a scalar

AAA11 12 13  1 detAA det det A21AA 22 23 eeeijk AAA 1 i 2 j 3 k  ijk pqr AAA pi qj rk  6 AAA  Some properties: 31 32 33 REMARK detAB det A  det B The tensor A is SINGULAR if and detAAT  det only if det A = 0. detAA 3 det A is NONSINGULAR if det A ≠ 0.  Inverse There exists a unique inverse A-1 of A when A is nonsingular, which satisfies the reciprocal relation:

11 AA 1 A A  11 AAik kj A ik A kj ij ijk,, {1, 2, 3} 54 2nd Order Tensor Operations

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

1 AB B11 A 1 AA1 

1 1 AA 1

T 1 AAA1 TT AAA211  detAA1  det 1 1 det A

55 Example

 Prove that det A  e ijk A 12 i AA j 3. k

56 Example - Solution

AAA11 12 13 detA  det AAA 21 22 23 AAA31 32 33

A11AA 22 33 AAA 21 32 13 AAA 31 12 23 AAA 13 22 31 AAA 23 32 11 AAA 33 12 21

57 Example - Solution

k 1 k  2 k  3 eijkAA12 i j A 3 k  eee111AAA 11 21 31 112 AAA 11 21 32 113 AAA 11 21 33 j 1 1 eee121AAA 11 22 31 122 AAA 11 22 32 123 AAA 11 22 33 j  2 i 1  1 eee131AAA 11 23 31 132 AAA 11 23 32 133 AAA 11 23 33 j  3 1

211AAA 12 21 31ee 212 AAA 12 21 32  213 AAA 12 21 33

i  2 ee221AAA 12 22 31 222 AAA 12 22 32 e223AAA 12 22 33  1 eee231AAA 12 23 31 232 AAA 12 23 32 233 AAA 12 23 33 1 eee311AAA 13 21 31 312 AAA 13 21 32 313 AAA 13 21 33 1 eeeAAA AAA AAA i  3 321 13 22 31 322 13 22 32 323 13 22 33

eee331AAA 13 23 31 332 AAA 13 23 32 333 AAA 13 23 33 

A11AA 22 33 AAA 12 23 31 AAA 13 21 32 AAA 13 22 31 AAA 12 21 33 AAA 11 23 32

58 Summary - Tensor Operations

 Dot product – contraction of one index:

cu uviiv      CABij ij AikBC kj  ij DBAij ij BikAD kj  ij

cuAjj   uAiij  c j     C ijkl AB ijkl ABijm mkl  ijkl    dAu A ud     iiij j i D ijklBA ijkl BAijm mkl C ijkl D     CBijkA ijk AimBC mjk  ijk    BD DBijkA ijk ijmA mk  ijk

59 Summary - Tensor Operations

 Double dot product – contraction of two indices:

cABAB: ij ij

C  AB: AB C cA :B A c ij ij ikm kmj ij kk  ij ijk k dA B:  AB d D BA: BA D iiijk jk i ij ij ikm kmj ij B

ijkl : ijkl     CABABijmp mpkl C ijkl CBijk A : ijk A ijlmBC lmk ijk

ijkl : ijkl  BD DBA BAijmp mpij D ijkl DBijk : A ijk ilm lmjk ijk A

60 Summary - Tensor Operations

 Transposed double dot product – contraction of two indexes:

cABAB ij ji

C  AB AB C cA B A B  c ij ij ikm mkj ij  kk  ij jik k B B D BA D dAii  ijkA kj d i ij ij BAikm mkj ij

ijkl ijkl     A CABABijmp pmkl C ijkl CBijkA ijk ijlmBC mlk  ijk    BD ijkl ijkl ijmp pmij  ijkl DB   DBA BA D ijkA ijk ilmA mljk  ijk

61 Summary - Tensor Operations

 Open product – expansion of indexes: Auv   uAv  ij ij ij ij Bvuij  ij vijuB  ij AB  AB      C ijkl ijkl ij klC ijkl D ijklBA ijkl BAij kl D ijkl

C ijkuA  ijk uA i jk C ijk D   ijkAu ijk Auij k D ijk

C ijklm A Bijklm AijB klm C ijklm  B D ijklm B A ijklm ijkA lm D ijklm

62 Differential Operators

Tensor Algebra

63 Differential Operators

 A differential operator is a mapping that transforms a field vx ,... Ax  into another field by means of partial derivatives.

 The mapping is typically understood to be linear.

 Examples:  Nabla operator  Gradient  Divergence  Rotation  …

64 Nabla Operator

 The Nabla operator  is a differential operator “symbolically” defined as: symbolic symb.    eˆi x xi

 In Cartesian coordinates, it can be used as a (symbolic) vector on its own:      x  1  symb.      x2       x3 

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  symb.            i {1, 2, 3} iii   xxii      eeˆˆ    eˆi  i ii x  xi i  Gradient of a vector field v(x):  Yields a 2nd order tensor  symb.  v vv   v,{1,2,3} j ij  ij i j j  xxii  v j v veeˆˆij vv veeeeˆˆ  j ˆˆ  x  ij ij ij i  xi 66 Gradient

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

 symb.  A AAAA,, jk ijk {1, 2, 3} ijk  ijk  i  jk jk  xxii  A AA Aeeeeeeˆˆ   ˆ jk ˆˆ   ˆ  ijk ijk ijk  xi

A jk Aeeeˆˆijk ˆ xi

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 symb.  vi vv   v  vi ii i v  xiix xi  Divergence of a 2nd order tensor A(x):  Yields a vector  symb.  A AA   A ij j {1, 2, 3}  j iij ij  xxii  A AAee  ˆˆ  ij A  j jj ij  xi Ae  ˆ j xi

68 Divergence

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

 If v 0 , the vector field vx  is said to be solenoid (or divergence-free).

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  symb.. symb   v vv eee  v k i  {1, 2, 3}  i ijkjk ijk k ijk  xxjj   vk  vk vvee  ˆˆ e ve eijkˆ i  i iijki x  x j j  Rotation of a 2nd order tensor A(x):  Yields a 2nd order tensor symb.   A  kl  A il eeijkA kl  ijk ijk , ,  {1, 2, 3}  xxjj   A  A Aeee kl ˆˆ    kl ijk i l A   Aeeil ˆˆilijkil  e ee ˆˆ  x  j  x j 70 Rotation

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

 If   v 0 , the vector field vx   is said to be irrotational (or curl-free).

71 Differential Operators - Summary

scalar field vector field 2nd order tensor Φ(x) v(x) A(x)

  v  A   i ij  ijk GRADIENT  v j A jk  v  A  i x ij ijk i xi xi

v A v  i A  ij DIVERGENCE x j i xi

 vk  A v i e kl ROTATION ijk A il eijk x j x j

72 Example

 Given the vector vvx  x1231xx eˆˆˆ  xx 122 e  x 13 e determine vvv,,.  

73 Example - Solution

xxx123 vvxx xx eˆˆˆ  xx e  x e 1231 122 13 v   xx   12  x1   Divergence:

v v  i xi

vvi vv12 3 v    xx23  x 1 xxxxi 123

74 Example - Solution

 Divergence: xxx123 v    i v  xx v   12 xi  x1   In matrix notation:

T xxx symb symb123 symb T  vv   ,, xx  12 xxx123 x1 13 31 symb xxx123 xx12 x1 x123xx xx 12 x 1 xx 23 x 1 xxxxxx123123

11

75 Example - Solution

 Rotation: xxx123  v    k v  xx12 v i eijk   x j  x1 

 In index notation:

 vk v i eijk  x j

vvvv2112vv33 eeeeeeiiiiii12 13 21 23 31 32 x112233xxxxx

76 Example - Solution

 Rotation: vv21 v3 v i eeeiii12  13  21  xxx123 xxx112   v  xx  v vv  12 eee3 12 iii23 31 32  x1  x233xx  In matrix notation  v  v ee3  2 123xx 132 1 23 1  0  v  v  v ee3 1  xx 1  213xx 231 1 2  1 13 1 x213 xx vv21 ee312 321 xx12 1  1  In compact notation: ve xx12 1ˆˆ 2  x 2  xx 13 e 3

77 Example - Solution

 Rotation: v1123 xxx  Calculated directly in matrix notation: v  xx  212  v  x x ˆˆˆ 31 1 eee123 v1 symb    v  vdet2   xxxx  2123 v3   vvv  123 x3

vv33vv21  vv 21  eeˆˆ123 e ˆ xx23  xx 31  xx 12

xx121 eeˆˆ 2 x 2 xx 13 3

78 Example - Solution

 Gradient: v1123 xxx v v  xx vv    j 212 ij ij x i v31 x  In matrix notation  x 1 xx x 1 symb symb symb 23 2 T    vvv  xxx,, xx x  xx x 0 x 123 12 1 13 1  2 xx12 00  13  x3 33 31  In compact notation:

veeeeeeeeeeeexx23ˆˆ 1 1 x 21 ˆˆˆˆ 2 1 3 xx 13 ˆˆ 2 1 x 1 ˆˆ 2 2 xx 12 ˆˆ 3 1

79 Integral Theorems

Tensor Algebra

80 Divergence or Gauss Theorem

 GivenA 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:

AAdV n  dS VV

AAdV  n dS VV Where: A represents either a vector field ( v(x) ) or a tensor field ( A(x) ).

81 Generalized Divergence Theorem

 Given a field A in a volume V with closed boundary surface ∂V and unit outward normal to the boundary n , the Generalized Divergence Theorem states: AAdV n  dS VV

AAdV  n dS VV Where:   represents either the dot product ( · ), the cross product (  ) or the tensor product (  ).  A represents either a scalar field ( ϕ(x) ), a vector field ( v(x) ) or a tensor field ( A(x) ).

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:

un dS  u  d r SS

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 Example

 Use the Generalized Divergence Theorem to show that x ndSV  S i j ij nj where x i is the position vector of n j .

AAn dS dV VV

84 Example - Solution

x ndSV  x ndS xn dS S i j ij S ij S

 Applying the Generalized Divergence Theorem:

xndS x dV VV

 Applying the definition of gradient of a vector:

x j x xi ij x ij xi x j

85 Example - Solution

 The Generalized Divergence Theorem xindSVj  ij in index notation: S x x ndS i dV SVij x j

 Then,

x x ndSi dV dV V SVij  V ij ij x j

86 References

Tensor Algebra

87 References

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

 Eduardo W. V. Chaves, Mecánica del Medio Continuo,Vol. 1 Conceptos básicos, 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.

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