PH6_L3 1 Einstein Summation Convention This is a method to write equation involving several summations in a uncluttered form Example: i i ij j ij i B A or j i j i where B A B A = ⎩ ⎨ ⎧ ≠ = = = 0 1 . δ δ •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.
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Einstein Summation Convention - Department of Physics - Indian
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PH6_L3 1
Einstein Summation Convention
This is a method to write equation involving several summations in a uncluttered form
Example:
ii
ijjiji
BAorjiji
whereBABA
=⎩⎨⎧
≠=
==01
. δδ
•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 2
Vector or Cross or Outer Product
)()()(
ˆsin)()()(
)()(
)(ˆsin
AssocitiveNotCBACBA
nmABBmABAmBAm
veDistributiCABACBA
eCommutativNotABnABBA
⎩⎨⎧
××≠××=×=×=×
×+×=+×
×−==×
θ
θ
Product varies under change of basis, i.e. coordinate systemDirection of the product is given by right hand screw ruleProduct gives the area of the parallelogram consisting the two vectors as its arms
PH6_L3 3
Cross Product: Graphical Representation
A
BθBs
inθ
n
PH6_L3 4
Examples:Magnetic force on a moving charge
.magF qv B= ×
Torque on a bodyr fτ = ×
r
fθ
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In the component formˆ ˆ ˆx y z
x Y Z
X Y Z
e e eA B A A A
B B B
⎡ ⎤⎢ ⎥× = ⎢ ⎥⎢ ⎥⎣ ⎦
( ) ( ) ( )x y z z y y z x x z z x y y xe A B A B e A B A B e A B A B= − + − + −
In the Einstein summation notation
( )i ijk j kA B A Bε× =
Where εijk is a Levi‐Civita Tensor
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The tensor operator and ijkε ijkε
• The tensor is defined for i,j,k=1,...,3 as
0 unless , ,and are distinct1, if rst is an even permutation of 123
-1, if rst is an odd permutation of 123ijk
i j kε
⎧⎪= +⎨⎪⎩
3 2 11 3 2
2 1 3
1 2 32 3 13 1 2
evenodd
ijkε
6
PH6_L3 7
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
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 continuous mass distribution
For axis of rotation about the scalar form can be calculated as
n
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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 11
Tensor notation
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
== =3
2
1
3,2,1)(ppp
pP iii• In tensor notation a superscript
stands for a column vector
• a subscript for a row vector (useful to specify lines)
• A matrix is written as
You know about Matrix Methods
),,( 321 lllLi =
),,( 321
3
2
1
33
32
31
23
22
21
13
12
11
jjj
i
i
ij
i
MMM
MMM
mmmmmmmmm
M
=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
=⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
=
11
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Tensor notation
• Tensor summation convention:– an index repeated as sub and superscript in a product represents summation over the range of the index.
• Example:
33
22
11 plplplPL i
i ++=
12
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Tensor notation
• Scalar product can be written as
• where the subscript has the same index as the superscript. This implicitly computes the sum.
• This is commutative
• Multiplication of a matrix and a vector
• This means a change of P from the coordinate system i to the coordinate system j (transformation).
33
22
11 plplplPL i
i ++=
iii
i LPPL =
iji
j PMP =
13
PH6_L3 14
Line equation
• In classical methods, a line is defined by the equation
• In homogenous coordinates we can write this as
• In tensor notation we can write this as
0=++ cbyax
01
),,( =⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛⋅=⋅ y
xcbaPLT
0=iiPL
14
PH6_L3 15
Determinant in tensor notation
( )jjjji mmmM 321 ,,=
1 2 3det( )j i j ki ijkM m m mε=
15
PH6_L3 16
Cross product in tensor notation
bac ×=
( ) j ki i ijkc a b a bε= × =
16
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Example
• Intersection of two lines
• L: l1x+l2y+l3=0, M: m1x+m2y+m3=0
• Intersection:
• Tensor:
• Result:
12213113,
12212332
mlmlmlmly
mlmlmlmlx
−−
=−−
=
kjijki MLEP =
12213
31132
23321
mlmlp
mlmlp
mlmlp
−=
−=
−=
17
PH6_L3 18
Translation
• Classic
• Tensor notation
• T is a transformation from the system A to B
• Homogenous coordinates
y
x
tyytxx
+=+=
12
12
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
111
1001001
122
yx
tytx
yx
1,2,3 BA, with == ABA
B PTP
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PH6_L3 19
Rotation
• Homogenous coordinates
1)cos(1)sin(1)sin(1)cos(
2
2
yaxayyaxax
+=−=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛ −=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
111
1000)cos()sin(0)sin()cos(
122
yx
aaaa
yx
• Classic
• Tensor notation1,2,3 BA, with == AB
AB PRP
19
PH6_L3 20
Scalar Triple Product
.( ) .( ) .( )x y z
x y z
x y z
A A AC A B A B C B C A B B B
C C C
⎛ ⎞⎜ ⎟
× = × = × = ⎜ ⎟⎜ ⎟⎝ ⎠
Can we take these vectors in any other sequence?
PH6_L3 21
vector triple productThe cross product of a vector with a cross product The expansion formula of the triple cross product is
This vector is in the plane spanned by the vectors and(when these are not parallel).
b c
Exercise: Prove it: Hint: use ijk ilm jl km jm klε ε δ δ δ δ= −
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 22
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.