2.3 Dyadics and Tensors
25
The manipulations in Eq. (2.7) illustrate another important shortcut. We can
immediately obtain the result ij l in the last line simply by replacing k by l in ij k δ lk
of the first line and removing δ lk . This operation involving the Kronecker delta is a
form of tensor contraction, which often comes in handy for simplifying equations.
Consider, for example, the expression a ij δ ij , in which summation is implied over
both i and j , giving nine terms. To simplify the math, we can contract over either
i or j . Contracting over i (replacing i by j and removing δ ij ) gives a jj , whereas
contracting over j (replacing j by i) gives a ii . Both procedures yield the same
result, i.e., a ij δ ij = a ii = a jj = a 11 + a 22 + a 33 , showing that only three of the
nine terms survive.
The above relations can be used to write vector dot and cross products in terms
of components. The dot product of two vectors a = a i e i and b = b i e i is given by
a · b = (a i e i ) · (b j e j ) = a i b j (e i · e j ) = a i b j δ ij
or
a · b = a i b i .
(2.9)
It is important to note that the dummy indices in the above expression for b were
changed from i to j without altering its meaning, i.e., b = b i e i = b j e j . This step
is necessary to avoid using more than two i’s in the same term, since more than two
of any subscript in a single term are meaningless. The cross product is
a × b = (a i e i ) × (b j e j ) = a i b j (e i × e j ),
and Eq. (2.4) 2 gives
a × b = a i b j ij k e k .
(2.10)
Note how both sides of Eqs. (2.9) and (2.10) contain no unpaired indices, i.e., the
subscripts balance.
2.3 Dyadics and Tensors
2.3.1 Dyadics
The tensor product of two vectors a and b is called a dyad and is denoted simply
by ab. 8 A linear combination of dyads is called a dyadic, e.g., ab + 2cd. In general,
tensor products of multiple vectors are referred to as polyads, e.g., abc or abcd. 9
8 Some authors use the notation a ⊗ b for the tensor product.
9 Drew (1961) is an excellent resource for dyadic and polyadic analysis.
25
The manipulations in Eq. (2.7) illustrate another important shortcut. We can
immediately obtain the result ij l in the last line simply by replacing k by l in ij k δ lk
of the first line and removing δ lk . This operation involving the Kronecker delta is a
form of tensor contraction, which often comes in handy for simplifying equations.
Consider, for example, the expression a ij δ ij , in which summation is implied over
both i and j , giving nine terms. To simplify the math, we can contract over either
i or j . Contracting over i (replacing i by j and removing δ ij ) gives a jj , whereas
contracting over j (replacing j by i) gives a ii . Both procedures yield the same
result, i.e., a ij δ ij = a ii = a jj = a 11 + a 22 + a 33 , showing that only three of the
nine terms survive.
The above relations can be used to write vector dot and cross products in terms
of components. The dot product of two vectors a = a i e i and b = b i e i is given by
a · b = (a i e i ) · (b j e j ) = a i b j (e i · e j ) = a i b j δ ij
or
a · b = a i b i .
(2.9)
It is important to note that the dummy indices in the above expression for b were
changed from i to j without altering its meaning, i.e., b = b i e i = b j e j . This step
is necessary to avoid using more than two i’s in the same term, since more than two
of any subscript in a single term are meaningless. The cross product is
a × b = (a i e i ) × (b j e j ) = a i b j (e i × e j ),
and Eq. (2.4) 2 gives
a × b = a i b j ij k e k .
(2.10)
Note how both sides of Eqs. (2.9) and (2.10) contain no unpaired indices, i.e., the
subscripts balance.
2.3 Dyadics and Tensors
2.3.1 Dyadics
The tensor product of two vectors a and b is called a dyad and is denoted simply
by ab. 8 A linear combination of dyads is called a dyadic, e.g., ab + 2cd. In general,
tensor products of multiple vectors are referred to as polyads, e.g., abc or abcd. 9
8 Some authors use the notation a ⊗ b for the tensor product.
9 Drew (1961) is an excellent resource for dyadic and polyadic analysis.