Dyadics are mathematical objects, representing linear functions of vectors . Dyadic notation was first established by Gibbs in 1884.
Definition
Dyadic A is formed by two vectors a and b (complex in general). Here, upper-case bold variables denote dyads whereas lower-case bold variables denote vectors.
A
=
a
b
{\displaystyle \mathbf {A} =\mathbf {a} \mathbf {b} }
In matrix form :
A
=
a
⇀
T
.
b
⇀
=
(
a
1
a
2
)
.
(
b
1
b
2
)
=
(
a
1
b
1
a
1
b
2
a
2
b
1
a
2
b
2
)
.
{\displaystyle \mathbf {A} ={\overset {\rightharpoonup }{a}}^{T}.{\overset {\rightharpoonup }{b}}=\left({\begin{array}{c}a_{1}\\a_{2}\end{array}}\right).\left({\begin{array}{cc}b_{1}&b_{2}\end{array}}\right)=\left({\begin{array}{cc}a_{1}b_{1}&a_{1}b_{2}\\a_{2}b_{1}&a_{2}b_{2}\end{array}}\right).}
In general algebraic form:
A
=
=
∑
i
,
j
a
i
b
j
x
i
⇀
x
j
⇀
{\displaystyle {\overset {=}{A}}=\sum _{i,j}a_{i}b_{j}{\overset {\rightharpoonup }{x_{i}}}{\overset {\rightharpoonup }{x_{j}}}}
where
x
i
⇀
{\displaystyle {\overset {\rightharpoonup }{x_{i}}}}
are basics ( also known as coordinate axes) and i ,j goes from 1 to space dimension, in general = 3.
A more general definition for multiple vectors
a
⇀
i
,
b
⇀
i
{\displaystyle {\overset {\rightharpoonup }{a}}_{i},{\overset {\rightharpoonup }{b}}_{i}}
A
=
=
∑
i
a
⇀
i
b
⇀
i
=
a
⇀
1
b
⇀
1
+
a
⇀
2
b
⇀
2
+
a
⇀
3
b
⇀
3
+
⋯
{\displaystyle {\overset {=}{A}}=\sum _{i}{\overset {\rightharpoonup }{a}}_{i}{\overset {\rightharpoonup }{b}}_{i}={\overset {\rightharpoonup }{a}}_{1}{\overset {\rightharpoonup }{b}}_{1}+{\overset {\rightharpoonup }{a}}_{2}{\overset {\rightharpoonup }{b}}_{2}+{\overset {\rightharpoonup }{a}}_{3}{\overset {\rightharpoonup }{b}}_{3}+\cdots }
A dyadic which cannot be reduced to a sum of less than 3 dyads is said to be complete. In this case, the forming vectors are non-coplanar, see Chen (1983) .
The following table classify dyadic:
Dyadics algebra
Dyadic with vector
There are 4 operations for a vector with a dyadic
c
⋅
a
b
=
(
c
⋅
a
)
b
(
a
b
)
⋅
c
=
a
(
b
⋅
c
)
c
×
(
a
b
)
=
(
c
×
a
)
b
(
a
b
)
×
c
=
a
(
b
×
c
)
{\displaystyle {\begin{aligned}\mathbf {c} \cdot \mathbf {a} \mathbf {b} &=\left(\mathbf {c} \cdot \mathbf {a} \right)\mathbf {b} \\\left(\mathbf {a} \mathbf {b} \right)\cdot \mathbf {c} &=\mathbf {a} \left(\mathbf {b} \cdot \mathbf {c} \right)\\\mathbf {c} \times \left(\mathbf {ab} \right)&=\left(\mathbf {c} \times \mathbf {a} \right)\mathbf {b} \\\left(\mathbf {ab} \right)\times \mathbf {c} &=\mathbf {a} \left(\mathbf {b} \times \mathbf {c} \right)\end{aligned}}}
Dyadic with dyadic
There are 5 operations for a dyadic to another dyadic:
Simple-dot product
(
a
b
)
⋅
(
c
d
)
=
a
(
b
⋅
c
)
d
=
(
b
⋅
c
)
a
d
{\displaystyle \left(\mathbf {ab} \right)\cdot \left(\mathbf {cd} \right)=\mathbf {a} \left(\mathbf {b} \cdot \mathbf {c} \right)\mathbf {d} =\left(\mathbf {b} \cdot \mathbf {c} \right)\mathbf {ad} }
For 2 general dyadics A and B :
A
=
=
∑
i
a
⇀
i
b
⇀
i
{\displaystyle {\overset {=}{A}}=\sum _{i}{\overset {\rightharpoonup }{a}}_{i}{\overset {\rightharpoonup }{b}}_{i}}
B
=
=
∑
j
c
⇀
j
d
⇀
j
{\displaystyle {\overset {=}{B}}=\sum _{j}{\overset {\rightharpoonup }{c}}_{j}{\overset {\rightharpoonup }{d}}_{j}}
A
=
.
B
=
=
∑
i
,
j
a
⇀
i
b
⇀
i
.
c
⇀
j
d
⇀
j
=
∑
i
,
j
(
b
⇀
i
.
c
⇀
j
)
a
⇀
i
d
⇀
j
=
(
b
⇀
1
.
c
⇀
1
)
a
⇀
1
d
⇀
1
+
(
b
⇀
1
.
c
⇀
2
)
a
⇀
1
d
⇀
2
+
...
+
(
b
⇀
2
.
c
⇀
1
)
a
⇀
2
d
⇀
1
+
(
b
⇀
2
.
c
⇀
2
)
a
⇀
2
d
⇀
2
+
...
{\displaystyle {\overset {=}{A}}.{\overset {=}{B}}=\sum _{i,j}{\overset {\rightharpoonup }{a}}_{i}{\overset {\rightharpoonup }{b}}_{i}.{\overset {\rightharpoonup }{c}}_{j}{\overset {\rightharpoonup }{d}}_{j}=\sum _{i,j}\left({\overset {\rightharpoonup }{b}}_{i}.{\overset {\rightharpoonup }{c}}_{j}\right){\overset {\rightharpoonup }{a}}_{i}{\overset {\rightharpoonup }{d}}_{j}=\left({\overset {\rightharpoonup }{b}}_{1}.{\overset {\rightharpoonup }{c}}_{1}\right){\overset {\rightharpoonup }{a}}_{1}{\overset {\rightharpoonup }{d}}_{1}+\left({\overset {\rightharpoonup }{b}}_{1}.{\overset {\rightharpoonup }{c}}_{2}\right){\overset {\rightharpoonup }{a}}_{1}{\overset {\rightharpoonup }{d}}_{2}+{\text{...}}+\left({\overset {\rightharpoonup }{b}}_{2}.{\overset {\rightharpoonup }{c}}_{1}\right){\overset {\rightharpoonup }{a}}_{2}{\overset {\rightharpoonup }{d}}_{1}+\left({\overset {\rightharpoonup }{b}}_{2}.{\overset {\rightharpoonup }{c}}_{2}\right){\overset {\rightharpoonup }{a}}_{2}{\overset {\rightharpoonup }{d}}_{2}+{\text{...}}}
Double-dot product
There are two ways to define the double dot product. Many sources use a definition of the double dot product rooted in the matrix double-dot product,
a
b
:
c
d
=
(
a
⋅
d
)
(
b
⋅
c
)
{\displaystyle \mathbf {ab} \colon \mathbf {cd} =\left(\mathbf {a} \cdot \mathbf {d} \right)\left(\mathbf {b} \cdot \mathbf {c} \right)}
whereas other sources uses a definition unique (usually referred to as the "colon product") to Dyads:
(
a
b
)
:
(
c
d
)
=
c
⋅
(
a
b
)
⋅
d
=
(
a
⋅
c
)
(
b
⋅
d
)
{\displaystyle \left(\mathbf {ab} \right):\left(\mathbf {cd} \right)=\mathbf {c} \cdot \left(\mathbf {ab} \right)\cdot \mathbf {d} =\left(\mathbf {a} \cdot \mathbf {c} \right)\left(\mathbf {b} \cdot \mathbf {d} \right)}
One must be careful when deciding which convention to use. As there are no analogous matrix operations for the remaining dyadic products, no ambiguities in their definitions appear.
The double-dot product is commutative .
A
:
B
=
B
:
A
{\displaystyle \mathbf {A} \colon \!\mathbf {B} =\mathbf {B} \colon \!\mathbf {A} }
There is special double dot product with transpose
A
:
B
T
=
A
T
:
B
{\displaystyle \mathbf {A} \colon \!\mathbf {B} ^{\mathrm {T} }=\mathbf {A} ^{\mathrm {T} }\colon \!\mathbf {B} }
Another identity is:
A
=
.
.
B
=
=
(
A
=
.
B
=
T
)
.
.
I
=
=
(
B
=
.
A
=
T
)
.
.
I
=
{\displaystyle {\overset {=}{A}}{\begin{array}{c}.\\.\end{array}}{\overset {=}{B}}=\left({\overset {=}{A}}.{\overset {=}{B}}^{T}\right){\begin{array}{c}.\\.\end{array}}{\overset {=}{I}}=\left({\overset {=}{B}}.{\overset {=}{A}}^{T}\right){\begin{array}{c}.\\.\end{array}}{\overset {=}{I}}}
Dot–cross product
(
a
b
)
⋅
×
(
x
y
)
=
(
a
⋅
x
)
(
b
×
y
)
{\displaystyle \left(\mathbf {ab} \right)\!\!\!{\begin{array}{c}_{\cdot }\\^{\times }\end{array}}\!\!\!\left(\mathbf {x} \mathbf {y} \right)=\left(\mathbf {a} \cdot \mathbf {x} \right)\left(\mathbf {b} \times \mathbf {y} \right)}
Cross–dot product
(
a
b
)
×
⋅
(
x
y
)
=
(
a
×
x
)
(
b
⋅
y
)
{\displaystyle \left(\mathbf {ab} \right)\!\!\!{\begin{array}{c}_{\times }\\^{\cdot }\end{array}}\!\!\!\left(\mathbf {xy} \right)=\left(\mathbf {a} \times \mathbf {x} \right)\left(\mathbf {b} \cdot \mathbf {y} \right)}
Double-cross product
(
a
b
)
×
×
(
x
y
)
=
(
a
×
x
)
(
b
×
y
)
{\displaystyle \left(\mathbf {ab} \right)\!\!\!{\begin{array}{c}_{\times }\\^{\times }\end{array}}\!\!\!\left(\mathbf {xy} \right)=\left(\mathbf {a} \times \mathbf {x} \right)\left(\mathbf {b} \times \mathbf {y} \right)}
We can see that, for simple dyadics, that only formed by 2 vectors a and b ,
S
=
a
b
{\displaystyle \mathbf {S} =\mathbf {a} \mathbf {b} }
Its double cross product is zero.
(
a
b
)
×
×
(
a
b
)
=
(
a
×
a
)
(
b
×
b
)
=
0
{\displaystyle \left(\mathbf {ab} \right)\!\!\!{\begin{array}{c}_{\times }\\^{\times }\end{array}}\!\!\!\left(\mathbf {ab} \right)=\left(\mathbf {a} \times \mathbf {a} \right)\left(\mathbf {b} \times \mathbf {b} \right)=0}
However, for 2 general dyadics, there double-cross product is defined as:
A
=
×
×
B
=
=
∑
i
,
j
(
a
⇀
i
×
c
⇀
j
)
(
b
⇀
i
×
d
⇀
j
)
{\displaystyle {\overset {=}{A}}\!\!\!{\begin{array}{c}_{\times }\\^{\times }\end{array}}\!\!\!{\overset {=}{B}}=\sum _{i,j}\left({\overset {\rightharpoonup }{a}}_{i}\times {\overset {\rightharpoonup }{c}}_{j}\right)\left({\overset {\rightharpoonup }{b}}_{i}\times {\overset {\rightharpoonup }{d}}_{j}\right)}
For double-cross product on itself, the result will not be zero. For a special dyadic A :
A
=
=
∑
i
=
1
3
a
⇀
i
b
⇀
i
{\displaystyle {\overset {=}{A}}=\sum _{i=1}^{3}{\overset {\rightharpoonup }{a}}_{i}{\overset {\rightharpoonup }{b}}_{i}}
A
=
×
×
A
=
=
∑
i
,
j
(
a
⇀
i
×
a
⇀
j
)
(
b
⇀
i
×
b
⇀
j
)
=
(
a
⇀
1
×
a
⇀
2
)
(
b
⇀
1
×
b
⇀
2
)
+
(
a
⇀
1
×
a
⇀
3
)
(
b
⇀
1
×
b
⇀
3
)
+
(
a
⇀
2
×
a
⇀
1
)
(
b
⇀
2
×
b
⇀
1
)
+
(
a
⇀
2
×
a
⇀
3
)
(
b
⇀
2
×
b
⇀
3
)
+
(
a
⇀
3
×
a
⇀
1
)
(
b
⇀
3
×
b
⇀
1
)
+
(
a
⇀
3
×
a
⇀
2
)
(
b
⇀
3
×
b
⇀
2
)
{\displaystyle {\overset {=}{A}}\!\!\!{\begin{array}{c}_{\times }\\^{\times }\end{array}}\!\!\!{\overset {=}{A}}=\sum _{i,j}\left({\overset {\rightharpoonup }{a}}_{i}\times {\overset {\rightharpoonup }{a}}_{j}\right)\left({\overset {\rightharpoonup }{b}}_{i}\times {\overset {\rightharpoonup }{b}}_{j}\right)=\left({\overset {\rightharpoonup }{a}}_{1}\times {\overset {\rightharpoonup }{a}}_{2}\right)\left({\overset {\rightharpoonup }{b}}_{1}\times {\overset {\rightharpoonup }{b}}_{2}\right)+\left({\overset {\rightharpoonup }{a}}_{1}\times {\overset {\rightharpoonup }{a}}_{3}\right)\left({\overset {\rightharpoonup }{b}}_{1}\times {\overset {\rightharpoonup }{b}}_{3}\right)+\left({\overset {\rightharpoonup }{a}}_{2}\times {\overset {\rightharpoonup }{a}}_{1}\right)\left({\overset {\rightharpoonup }{b}}_{2}\times {\overset {\rightharpoonup }{b}}_{1}\right)+\left({\overset {\rightharpoonup }{a}}_{2}\times {\overset {\rightharpoonup }{a}}_{3}\right)\left({\overset {\rightharpoonup }{b}}_{2}\times {\overset {\rightharpoonup }{b}}_{3}\right)+\left({\overset {\rightharpoonup }{a}}_{3}\times {\overset {\rightharpoonup }{a}}_{1}\right)\left({\overset {\rightharpoonup }{b}}_{3}\times {\overset {\rightharpoonup }{b}}_{1}\right)+\left({\overset {\rightharpoonup }{a}}_{3}\times {\overset {\rightharpoonup }{a}}_{2}\right)\left({\overset {\rightharpoonup }{b}}_{3}\times {\overset {\rightharpoonup }{b}}_{2}\right)}
A
=
×
×
A
=
=
2
{
(
a
⇀
1
×
a
⇀
2
)
(
b
⇀
1
×
b
⇀
2
)
+
(
a
⇀
2
×
a
⇀
3
)
(
b
⇀
2
×
b
⇀
3
)
+
(
a
⇀
3
×
a
⇀
1
)
(
b
⇀
3
×
b
⇀
1
)
}
{\displaystyle {\overset {=}{A}}\!\!\!{\begin{array}{c}_{\times }\\^{\times }\end{array}}\!\!\!{\overset {=}{A}}=2\left\{\left({\overset {\rightharpoonup }{a}}_{1}\times {\overset {\rightharpoonup }{a}}_{2}\right)\left({\overset {\rightharpoonup }{b}}_{1}\times {\overset {\rightharpoonup }{b}}_{2}\right)+\left({\overset {\rightharpoonup }{a}}_{2}\times {\overset {\rightharpoonup }{a}}_{3}\right)\left({\overset {\rightharpoonup }{b}}_{2}\times {\overset {\rightharpoonup }{b}}_{3}\right)+\left({\overset {\rightharpoonup }{a}}_{3}\times {\overset {\rightharpoonup }{a}}_{1}\right)\left({\overset {\rightharpoonup }{b}}_{3}\times {\overset {\rightharpoonup }{b}}_{1}\right)\right\}}
Unit dyadic
For any vectors a , there exist a unit dyadic I , such that
I
⋅
a
=
a
⋅
I
=
a
{\displaystyle \mathbf {I} \cdot \mathbf {a} =\mathbf {a} \cdot \mathbf {I} =\mathbf {a} }
For any base of 3 vectors a , b and c , with reciprocal base
a
^
{\displaystyle {\hat {\mathbf {a} }}}
,
b
^
{\displaystyle {\hat {\mathbf {b} }}}
and
c
^
{\displaystyle {\hat {\mathbf {c} }}}
, the unit dyadic defined by
I
=
a
a
^
+
b
b
^
+
c
c
^
{\displaystyle \mathbf {I} =\mathbf {a} {\hat {\mathbf {a} }}+\mathbf {b} {\hat {\mathbf {b} }}+\mathbf {c} {\hat {\mathbf {c} }}}
In Cartesian coordinates,
I
=
x
x
^
+
y
y
^
+
z
z
^
{\displaystyle \mathbf {I} =\mathbf {x} {\hat {\mathbf {x} }}+\mathbf {y} {\hat {\mathbf {y} }}+\mathbf {z} {\hat {\mathbf {z} }}}
For orthonormal base
x
i
⇀
=
x
i
′
⇀
{\displaystyle {\overset {\rightharpoonup }{x_{i}}}={\overset {\rightharpoonup }{x_{i}'}}}
,
I
=
∑
i
x
i
⇀
x
i
⇀
{\displaystyle \mathbf {I} =\sum _{i}{\overset {\rightharpoonup }{x_{i}}}{\overset {\rightharpoonup }{x_{i}}}}
The corresponding matrix is
I
=
(
1
0
0
0
1
0
0
0
1
)
{\displaystyle \mathbf {I} =\left({\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\\\end{array}}\right)}
Rotation dyadic
For any vector a ,
a
×
I
{\displaystyle \mathbf {a} \times \mathbf {I} }
is a 90 degree right hand rotation dyadic around a .
Some operations with unit dyadics
(
a
×
I
)
⋅
(
b
×
I
)
=
a
b
−
(
a
⋅
b
)
I
{\displaystyle \left(\mathbf {a} \times \mathbf {I} \right)\cdot \left(\mathbf {b} \times \mathbf {I} \right)=\mathbf {ab} -\left(\mathbf {a} \cdot \mathbf {b} \right)\mathbf {I} }
I
×
⋅
(
a
b
)
=
b
×
a
{\displaystyle \mathbf {I} \!\!{\begin{array}{c}_{\times }\\^{\cdot }\end{array}}\!\!\!\left(\mathbf {ab} \right)=\mathbf {b} \times \mathbf {a} }
I
×
×
A
=
(
A
×
×
I
)
I
−
A
T
{\displaystyle \mathbf {I} \!\!{\begin{array}{c}_{\times }\\^{\times }\end{array}}\!\!\mathbf {A} =\left(\mathbf {A} \!\!{\begin{array}{c}_{\times }\\^{\times }\end{array}}\!\!\mathbf {I} \right)\mathbf {I} -\mathbf {A} ^{\mathrm {T} }}
I
:
(
a
b
)
=
(
I
⋅
a
)
⋅
b
=
a
⋅
b
=
Trace
(
a
b
)
{\displaystyle \mathbf {I} \;\colon \left(\mathbf {ab} \right)=\left(\mathbf {I} \cdot \mathbf {a} \right)\cdot \mathbf {b} =\mathbf {a} \cdot \mathbf {b} ={\text{Trace}}\left(\mathbf {ab} \right)}
See also
References
ISMO V. Lindell (1992). Methods for Electromagnetic Field Analysis . ISBN 019856239 . .
Hollis C. Chen (1983). Theory of Electromagnetic Wave - A Coordinate-free approach . ISBN 0070106886 . .