Page 55 - Intro to Tensor Calculus

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If a dyadic equals its conjugate A = A c ,then A ij = A ji and the dyadic is called symmetric. If a dyadic
equals the negative of its conjugate A = −A c ,then A ij = −A ji and the dyadic is called skew-symmetric. A
special dyadic called the identical dyadic or idemfactor is defined by

J = b e 1 b e 1 + b e 2 b e 2 + b e 3 b e 3 .
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This dyadic has the property that pre or post dot product multiplication of J with a vector V produces the
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same vector V. For example,
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V · J =(V 1 b e 1 + V 2 b e 2 + V 3 b e 3 ) · J
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= V 1 b e 1 · b e 1 b e 1 + V 2 b e 2 · b e 2 b e 2 + V 3 b e 3 · b e 3 b e 3 = V
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and J · V = J · (V 1 b e 1 + V 2 b e 2 + V 3 b e 3 )
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= V 1 b e 1 b e 1 · b e 1 + V 2 b e 2 b e 2 · b e 2 + V 3 b e 3 b e 3 · b e 3 = V
A dyadic operation often used in physics and chemistry is the double dot product A : B where A and
B are both dyadics. Here both dyadics are expanded using the distributive law of multiplication, and then
each unit dyad pair b e i b e j : b e m b e n are combined according to the rule
b e i b e j : b e m b e n =( b e i · b e m )( b e j · b e n ).

For example, if A = A ij b e i b e j and B = B ij b e i b e j , then the double dot product A : B is calculated as follows.
A : B =(A ij b e i b e j ):(B mn b e m b e n )= A ij B mn ( b e i b e j : b e m b e n )= A ij B mn ( b e i · b e m )( b e j · b e n )
= A ij B mn δ im δ jn = A mj B mj

= A 11 B 11 + A 12 B 12 + A 13 B 13
+ A 21 B 21 + A 22 B 22 + A 23 B 23

+ A 31 B 31 + A 32 B 32 + A 33 B 33
When operating with dyads, triads and polyads, there is a definite order to the way vectors and polyad
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components are represented. For example, for A = A i b e i and B = B i b e i vectors with outer product
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AB = A m B n b e m b e n = φ
there is produced the dyadic φ with components A m B n . In comparison, the outer product
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BA = B m A n b e m b e n = ψ
produces the dyadic ψ with components B m A n . That is
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φ = AB =A 1 B 1 b e 1 b e 1 + A 1 B 2 b e 1 b e 2 + A 1 B 3 b e 1 b e 3
A 2 B 1 b e 2 b e 1 + A 2 B 2 b e 2 b e 2 + A 2 B 3 b e 2 b e 3

A 3 B 1 b e 3 b e 1 + A 3 B 2 b e 3 b e 2 + A 3 B 3 b e 3 b e 3
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and ψ = BA =B 1 A 1 b e 1 b e 1 + B 1 A 2 b e 1 b e 2 + B 1 A 3 b e 1 b e 3
B 2 A 1 b e 2 b e 1 + B 2 A 2 b e 2 b e 2 + B 2 A 3 b e 2 b e 3

B 3 A 1 b e 3 b e 1 + B 3 A 2 b e 3 b e 2 + B 3 A 3 b e 3 b e 3
are different dyadics.
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The scalar dot product of a dyad with a vector C is defined for both pre and post multiplication as
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φ · C = AB · C =A(B · C)
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C · φ = C · AB =(C · A)B
These products are, in general, not equal.