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2 Vector and Tensor Analysis
Here, we note two important properties of dyadics (and polyadics). First, the
tensor product is generally not commutative, i.e., ab = ba. In manipulating
dyads, therefore, maintaining the order of the vectors is essential. Second, because
its component vectors are invariant, a dyadic is invariant relative to a change in
coordinates.
Example 2.2 Show that, unlike the case for vectors, the dot product of two dyads
is not commutative, i.e., ab · cd = cd · ab.
Solution
Mathematical operations involving dyads are straightforward so long as care is taken
to maintain the order of the vectors. The dot product operates on the two vectors
immediately to the left and right of the dot. (Any intervening scalars are ignored by
the dot and can be moved anywhere within a term.) Thus, the dot products of the
dyads ab and cd are given by
ab · cd = a(b · c)d = (b · c) ad
cd · ab = c(d · a)b = (d · a) cb,
which show that ab ·cd = cd ·ab. Note also that the dot product of two dyads yields
another dyad.
The double-dot product of two dyads is defined by
ab : cd = (a · c)(b · d) = (c · a)(d · b)
= cd : ab.
(2.11)
This equation shows that double-dotting two dyads produces a scalar, and the
operation is commutative (since single-dotting two vectors is commutative).
The tensor product can be treated essentially the same as multiplying two
algebraic functions. For example, if the vectors a and b are written in component
form, then
ab = (a 1 e 1 + a 2 e 2 + a 3 e 3 )(b 1 e 1 + b 2 e 2 + b 3 e 3 )
= a 1 b 1 e 1 e 1 + a 1 b 2 e 1 e 2 + a 1 b 3 e 1 e 3
+ a 2 b 1 e 2 e 1 + a 2 b 2 e 2 e 2 + a 2 b 3 e 2 e 3
+ a 3 b 1 e 3 e 1 + a 3 b 2 e 3 e 2 + a 3 b 3 e 3 e 3 .
(2.12)
2 Vector and Tensor Analysis
Here, we note two important properties of dyadics (and polyadics). First, the
tensor product is generally not commutative, i.e., ab = ba. In manipulating
dyads, therefore, maintaining the order of the vectors is essential. Second, because
its component vectors are invariant, a dyadic is invariant relative to a change in
coordinates.
Example 2.2 Show that, unlike the case for vectors, the dot product of two dyads
is not commutative, i.e., ab · cd = cd · ab.
Solution
Mathematical operations involving dyads are straightforward so long as care is taken
to maintain the order of the vectors. The dot product operates on the two vectors
immediately to the left and right of the dot. (Any intervening scalars are ignored by
the dot and can be moved anywhere within a term.) Thus, the dot products of the
dyads ab and cd are given by
ab · cd = a(b · c)d = (b · c) ad
cd · ab = c(d · a)b = (d · a) cb,
which show that ab ·cd = cd ·ab. Note also that the dot product of two dyads yields
another dyad.
The double-dot product of two dyads is defined by
ab : cd = (a · c)(b · d) = (c · a)(d · b)
= cd : ab.
(2.11)
This equation shows that double-dotting two dyads produces a scalar, and the
operation is commutative (since single-dotting two vectors is commutative).
The tensor product can be treated essentially the same as multiplying two
algebraic functions. For example, if the vectors a and b are written in component
form, then
ab = (a 1 e 1 + a 2 e 2 + a 3 e 3 )(b 1 e 1 + b 2 e 2 + b 3 e 3 )
= a 1 b 1 e 1 e 1 + a 1 b 2 e 1 e 2 + a 1 b 3 e 1 e 3
+ a 2 b 1 e 2 e 1 + a 2 b 2 e 2 e 2 + a 2 b 3 e 2 e 3
+ a 3 b 1 e 3 e 1 + a 3 b 2 e 3 e 2 + a 3 b 3 e 3 e 3 .
(2.12)