Einstein Summation Convention

This is a method to write equation involving several summations in a uncluttered form Example: G G ⎧1 i = j A.B = Aiδ ij B j where δ ij = ⎨ ⎩0 i ≠ j

or = Ai Bi •Summation runs over 1 to 3 since we are 3 dimension •No indices appear more than two times in the equation •Indices which is summed over is called dummy indices appear only in one side of equation •Indices which appear on both sides of the equation is free indices. PH6_L3 1 Vector or Cross or Outer Product G G G G A× B = ABsinθ nˆ = −B× A (Not Commutative) G G G G G G G A×(B + C) = A× B + A×C (Distributive) G G G G G G ⎧m(A× B) = (mA)× B = A×(mB) = mABsinθ nˆ ⎨ G G G G G G (Not Associtive) ⎩ A×(B×C) ≠ (A× B)×C z Product varies under change of basis, i.e. coordinate system z Direction of the product is given by right hand screw rule z Product gives the area of the parallelogram consisting the two vectors as its arms

PH6_L3 2 Cross Product: Graphical Representation

nˆ G θ B

Bsin θ G A

PH6_L3 3 Examples: Magnetic force on a moving charge G G G Fqmag. = v× B

Torque on a body G G G τ = r × f

f θ r

PH6_L3 4 In the component form ⎡ eeeˆˆˆ⎤ G G xyz A×=BA⎢ AA⎥ ⎢ xYZ⎥ ⎣⎢BBXYBZ⎦⎥ GGG =−eAxy()BzzAByy+e(AzxB−AxBz)+ez(AxByy−ABx )

In the Einstein summation notation G G ()AB×=i εijk Aj Bk

Where εijk is a Levi‐Civita Tensor

PH6_L3 5 ijk The tensor operator εijk and ε

• The tensor εijk is defined for i,j,k=1,...,3 as

⎧ 0 unless ij, ,and k are distinct ⎪ εijk =+⎨ 1, if rst is an even permutation of 123 ⎪ ⎩ -1, if rst is an odd permutation of 123

odd even 3 2 1 1 2 3 1 3 2 2 3 1 2 1 3 3 1 2

PH6_L3 6 What is a Tensor?

A Tensor is a method to represent the Physical Properties in an anisotropic system

For example:

You apply a force in one direction and look for the affect in other direction (Piezo‐electricity)

Elasticity : Elastic Tensor Dielectric constant : Dielectric Tensor Conductivity : Conductivity Tensor

PH6_L3 7 This generalized notation allows an easy writing of equations of the continuum mechanics, such as the generalized Hook's law : 2nd rank tensors z Stress, strain

z Conductivity ( Jii= σ jE j) z susceptibility

z Kroneker Delta δij 3rd rank tensors z Piezoelectricity z Levi‐Civita 4th rank tensors z Elastic moduli σ → Stress on ith plane nth rank tensor has 3n ij components in 3 ‐ dimensional in jth direction space PH6_L3 8 Moment of Inertia tensor: When axis of rotation is not given, then we can generalize moment of inertia into a tensor of rank 2.

Angular momentum For discrete particles Lii= I jω j

For continuous mass distribution

For axis of rotation about nˆ the scalar form can be calculated as

PH6_L3 9 Rank of a Tensor

Rank = 0 : Scalar Only One component

Rank = 1 : Vector Three components

Rank = 2 Nine Components

Rank = 3 Twenty Seven Components

Rank = 4 Eighty One Components

Symmetry plays a very important role in evaluating these components

PH6_L3 10 Tensor notation

⎛ p1 ⎞ ⎜ ⎟ i i 2 • In tensor notation a superscript P = ( p )i=1,2,3 = ⎜ p ⎟ ⎜ 3 ⎟ stands for a column vector ⎝ p ⎠ • a subscript for a row vector (useful to specify lines) Li = (l1,l2 ,l3 )

• A matrix is written as ⎛ m1 m1 m1 ⎞ ⎛ M 1 ⎞ You know about Matrix Methods ⎜ 1 2 3 ⎟ ⎜ i ⎟ j 2 2 2 2 M i = ⎜m1 m2 m3 ⎟ = ⎜ M i ⎟ ⎜ 3 3 3 ⎟ ⎜ 3 ⎟ ⎝ m1 m2 m3 ⎠ ⎝ M i ⎠ j j j = (M1 , M 2 , M 3 )

PH6_L3 11 Tensor notation

• Tensor summation convention: – an index repeated as sub and superscript in a product represents summation over the range of the index. • Example:

i 1 2 3 Li P = l1 p + l2 p + l3 p

PH6_L3 12 Tensor notation

• Scalar product can be written as i 1 2 3 Li P = l1 p + l2 p + l3 p • where the subscript has the same index as the superscript. This implicitly computes the sum. • This is commutative i i Li P = P Li • Multiplication of a matrix and a vector

j j i P = M i P • This means a change of P from the coordinate system i to the coordinate system j (transformation).

PH6_L3 13 Line equation

• In classical methods, a line is defined by the equation ax + by + c = 0

• In homogenous coordinates we can write this as ⎛ x ⎞ G G T ⎜ ⎟ L ⋅ P = (a,b,c)⋅⎜ y⎟ = 0 ⎜ ⎟ ⎝ 1 ⎠ • In tensor notation we can write this as L Pi = 0 i PH6_L3 14 Determinant in tensor notation

j j j j M i = (m1 ,m2 ,m3 )

j ijk det(M i ) = εijk mm12m3

PH6_L3 15 Cross product in tensor notation

G G G c = a ×b

j k cai =×()bi =εijk ab

PH6_L3 16 Example

• Intersection of two lines

• L: l1x+l2y+l3=0, M: m1x+m2y+m3=0 • Intersection: l2m3− l3m2 l3m1− l1m3 x = , y = l1m2 − l2m1 l1m2 − l2m1 • Tensor: i ijk P = E L j M k 1 • Result: p = l2m3 − l3m2 2 p = l3m1 − l1m3 3 p = l1m2 − l2m1

PH6_L3 17 Translation

• Homogenous • Classic coordinates

x2 = x1 + tx ⎛ x2⎞ ⎛1 0 tx⎞⎛ x1⎞ ⎜ ⎟ ⎜ ⎟⎜ ⎟ y = y + t ⎜ y2⎟ = ⎜0 1 ty⎟⎜ y1⎟ 2 1 y ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ 1 ⎠ ⎝0 0 1 ⎠⎝ 1 ⎠ • Tensor notation B B A P = TA P with A,B = 1,2,3 • T is a transformation from the system A to B

PH6_L3 18 Rotation

• Homogenous • Classic coordinates

x = cos(a)x1− sin(a)y1 ⎛ x2⎞ ⎛cos(a) − sin(a) 0⎞⎛ x1⎞ 2 ⎜ ⎟ ⎜ ⎟⎜ ⎟ y = sin(a)x1+ cos(a)y1 ⎜ y2⎟ = ⎜ sin(a) cos(a) 0⎟⎜ y1⎟ 2 ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ 1 ⎠ ⎝ 0 0 1⎠⎝ 1 ⎠

•Tensor notation B B A P = RA P with A,B = 1,2,3

PH6_L3 19 Scalar Triple Product

⎛⎞Ax AAyz GG GGG G GG G ⎜⎟ CA.( ×=B) A.(B×C) =B.(C×=A) ⎜⎟Bx ByzB ⎜⎟ ⎝⎠CCx yzC

Can we take these vectors in any other sequence?

PH6_L3 20 vector triple product The cross product of a vector with a cross product The expansion formula of the triple cross product is

Exercise: Prove it: Hint: use εijkεδilm = jlδkm −δjmδkl G G This vector is in the plane spanned by the vectors b and c (when these are not parallel).

Note that the use of parentheses in the triple cross products is necessary, since the cross product operation is not associative, i.e., generally we have

PH6_L3 21 Coordinate Transformations: translation

In engineering it is often necessary to express vectors in different coordinate frames. This requires the rotation and translation matrixes, which relates coordinates, i.e. basis (unit) vectors in one frame to those in another frame.

Translation of Coordinate systems eˆ z' Position coordinate ˆ G G G ez ' G A = AT− T eˆ G y' A eˆ ' x eˆy eˆx PH6_L3 22