An introduction to Cartesian Vector and Tensors

Dr Karl Travis

Immobilisation Science Laboratory,Department of Engineering Materials,

University of Sheffield, UKk.travis@sheffield.ac.uk

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Analytic definition of vectors and tensors

a a1

1a

2

2a

3

3 a

ii

i

a b ai

i

i b

jj

j

aib

j

i

j j

i

aib

j

ijj

i

ai

i

bia

1b

1a

2b

2a

3b

3

Let a be a vector expanded in an orthogonal basis:

where i are unit vectors along the 3 Cartesian axes.

scalar product of 2 vectors

where the third line follows from the orthogonality relations between the unit vectors

where ijk is defined by

ijk1, for a cyclic permutation of ijk

1, for an odd permutation of ijk

0, otherwise

i.e. {123, 312, 231} = +1, {213, 132, 321} = -1

Cross (vector) product of 2 vectors

ab ai

i

i b

jj

j

aib

j

i

j j

i

aib

jk

ijk

j

i

k

aib

j

ijk

kk

j

i

Other useful relations when dealing with vector cross products are

ijk

hjk2

ihk

j

ijk

mnk

imk

jn

in

jm

ijk1

2i j j k k i

Example: Proove that

u vw v uw w u v

u vw v uw w u v LHS u

ii

i v

jw

k

jkll

k

j

l

uiv

jw

km

jkl

l

k

j

ilm

mi

uiv

jw

km

jkl

l

k

j

mil

mi

but, jkl

mil

jml

ki

ji

km

uiv

jw

km

jkl

l

k

j

mil

mi

uiv

jw

k

jm

ki

mm

k

j

i

uiv

jw

k

ji

km

mm

k

j

i

uiv

mw

im

i

m u

iv

iw

mm

i

m

uiw

ii

vm

mm

uiv

ii

wm

mm

uw v u v w RHS

r ii

ix

i y

j z

k

Differential operators

The (‘del’) operator is also a vector, and is defined as

Forming the scalar product of with another vector is called the divergence, or simply ‘div’. The divergence of a vector a, say, is

a r i

i a

jj

i

j

r i

aj

j

i

ij

r i

aia

x

xi

a

y

ya

z

z

which is a scalar quantity

The Laplacian operator or ‘del squared’ is just the scalar product of ‘del’ with itself, and is a scalar.

2 r ii

i

r jj

j 2

ri

2i

The curl of a vector is formed from the vector cross product of ‘del’ with the vector, and is itself a vector:

a r ii

i a

jj

j

ri

aj

k

j

i

ijk

k

a3

r2

a2

r3

1

a1

r3

a3

r1

2

a2

r1

a1

r2

3

Cartesian tensors.

Some physical quantities require both a magnitude and at least two directions to define them: eg Inertia tensor, Pressure tensor. The number of directions required defines the rank of the tensor. Pressure and inertia are examples of 2nd rank tensors (vectors and scalars are tensors of rank 1 and 0 respectively).

A ( 2 ) j

Aij

i

i

j

Let A be a 2nd rank tensor expanded in an orthogonal basis:

where ij is called the unit dyad. It is sometimes easier to think of the components of a tensor as a rectangular array of numbers i.e. a matrix. So for A,

A11

A12

A13

A21

A22

A23

A31

A32

A33

Operations for the unit dyads

i

j

k

l

j

k i

l jk

il

i

j

k

i

jk

i

j

k

ij

k

i

j

k

l

i

j

k l

jk

i

l

i

j

k i

j

k jkl

l

i

l

i

j

k i

j k

ijll

l

k

The transpose of a tensor

AT i

jA

jij

i

The magnitude of a tensor

A A 12

A : AT

12

Aij

2

j

i

Invariants of a tensor

3 independent scalars can be formed from a tensor by taking the trace of A, A2 and A3. These scalars are invariants since they do do change value upon a change of the coordinate system.

I Tr (A) Aij

i

II Tr (A 2 ) AijA

jij

i

III Tr (A 3 ) AijA

jkA

kik

j

i

Operations between vectors and tensors

Dyadic product of two vectors, a and b, is written as ab and is a 2nd rank tensor which is defined as

ab j

aib

ji

i

j

Scalar product between a vector and a 2nd rank tensor forms a new vector defined by

a B ( 2 ) ai

i

i

k

Bjk

j

j

k

ai

k

j

i

Bjk

i

j

k

aiB

ikk

i

k

where we have used the orthogonality relation

i

j

k

ij

k

Double contraction of two 2nd rank tensors gives a scalar

A ( 2 ) : B ( 2 ) Aij

i

j:

j

i

l

Bkl

k

k

l

l

Aij

k

j

i

Bkl

i

j:

k

l

Aij

l

k

Bkl

j

i

il

jk

AikB

kik

i

A11B

11A

12B

21A

13B

31

Einstein notation for tensors

Repeated index means sum over that index. The double contraction above becomes in Einstein notation.

AijB

ji

i

j:

k

l

il

jkWhere we have made use of

Other products:

A ( 2 ) B ( 2 ) AijB

jk

a C ( 3 ) aiC

ijk

D ( 4 ) : B ( 2 ) Dijkl

Blk

The non-repeated indices give the tensor character. The rank of the product is the sum of the ranks of the two quantities less 2 for each dot appearing in the operator.

common tensors

2nd rank isotropic tensor, 1 = ijij.

3rd rank Levi-Cevita tensor, = ijkijk which is also referred to as the alternating tensor.

Parity: polar and pseudo vectors

Vectors which change sign under a mirror inversion of the coordinate axes are called pseudovectors.

Vectors which are invariant to a mirror inversion of the coordinate system are called polar vectors.

The spin angular momentum is an example of a pseudovector since it is defined by a vector cross product: There are also polar and pseudo scalars and tensors.

s r i

i

pi

Decomposition of tensors

A cartesian tensor can be decomposed into a symmetric and antisymmetric part:

A A s A a

A s 12

AAT , A a 12

A AT

Where a superscript ‘T’ denotes the transpose.

A 2nd rank antisymmetric tensor has the form:

0 A12

A13

A12

0 A23

A13

A23

0

And hence has only 3 independent components. These components transform like a vector, so antisymmetric 2nd rank tensors are often represented as a pseudovector dual.

If Aa is an antisymmetric tensor, its pseudovector dual, ad is given by

a d 1A a

which involves the alternating tensor.

Symmetric tensors can be further split into an isotropic component and a traceless symmetric component:

A s A os 13

Tr(A)1

where Tr(A) = Aii in Einstein notation.

Note that when a tensor is formed from a dyadic product of 2 vectors, say C(2) = ab, the trace is given as the scalar product of the two vectors: Tr(C) = a•b

Pressure and strain rate tensors

= P - p1

The viscous pressure tensor, , is defined as

Where p is the equilibrium scalar pressure.

Now we decompose to give

os a 1

The strain rate tensor can be decomposed into

u u os u a 13 u 1

Fourier transforms of quantities involving vectors and tensors.

Define the Fourier transform pair by:

f (k) dr eikr f (r)

f (r) 12 3 dk e ikr f (k)

(i) Fourier transform of the divergence of a tensor quantity.

f (r)A(r)

f (k) dr eikrA(r) dr eikr

j A

ji(r)

ij

i

dr i

i

eikr j A

ji(r)

j

i

i

dr j A

ji(r)

j

eikr

Let

then

Now we can integrate by parts:

u u(x),vv(x)

vdudx

dx uv udvdx

dx

f (k) i

i

dz dy dxddx

Axi(r) d

dyA

yi(r) d

dzA

zi(r)

eikr

i

i

dz eik zz dy eikyy dx eikx x ddx

Axi(r)

Writing out the multiple integrals explicitly,

dx eikx x ddx

Axi(r)eikx x ik

xdx A

ji(r) ik

xA

xi(k

x, y, z)

Where we have used the fact that the boundary term is zero.

i i

i

j

kjA

ji(k)

ik A(k)