The class of second order tensors (aka tensors). Given an orthornormal basis a dyad can be decomposed into scalar components. These components can be represented as a square matrix. The matrix is not unique but it a 'view' of the tensor for a particular basis.
(<=> (Dyad ?D)
(And (Constant-Quantity ?D)
(Tensor-Quantity ?D)
(= (Tensor-Order ?D) 2)
(Forall (?B ?I ?J)
(=> (And (Orthonormal-Basis ?B)
(= (Basis.Dimension ?B)
(Spatial.Dimension ?D))
(Positive-Integer ?I)
(=< ?I (Spatial.Dimension ?D))
(Positive-Integer ?J)
(=< ?J (Spatial.Dimension ?D)))
(And (Defined (Dyad-Component ?D ?I ?J ?B))
(= (Quantity.Dimension (Dyad-Component ?D
?I
?J
?B))
(Quantity.Dimension ?D)))))
(Forall (?U)
(=> (And (Unit-Of-Measure ?U)
(= (Quantity.Dimension ?U)
(Quantity.Dimension ?D)))
(Numeric-Tensor (Magnitude ?D ?U))))))
(Forall (?U)
(=> (And (Unit-Of-Measure ?U)
(= (Quantity.Dimension ?U) (Quantity.Dimension ?D)))
(Numeric-Tensor (Magnitude ?D ?U))))
(Forall (?B ?I ?J)
(=> (And (Orthonormal-Basis ?B)
(= (Basis.Dimension ?B) (Spatial.Dimension ?D))
(Positive-Integer ?I)
(=< ?I (Spatial.Dimension ?D))
(Positive-Integer ?J)
(=< ?J (Spatial.Dimension ?D)))
(And (Defined (Dyad-Component ?D ?I ?J ?B))
(= (Quantity.Dimension (Dyad-Component ?D ?I ?J ?B))
(Quantity.Dimension ?D)))))
(Tensor-Quantity ?D)
(Constant-Quantity ?D)
(<= (Tensor-Order $X 2) (Dyad $X))
(<=> (Dyad ?D)
(And (Constant-Quantity ?D)
(Tensor-Quantity ?D)
(= (Tensor-Order ?D) 2)
(Forall (?B ?I ?J)
(=> (And (Orthonormal-Basis ?B)
(= (Basis.Dimension ?B)
(Spatial.Dimension ?D))
(Positive-Integer ?I)
(=< ?I (Spatial.Dimension ?D))
(Positive-Integer ?J)
(=< ?J (Spatial.Dimension ?D)))
(And (Defined (Dyad-Component ?D ?I ?J ?B))
(= (Quantity.Dimension (Dyad-Component ?D
?I
?J
?B))
(Quantity.Dimension ?D)))))
(Forall (?U)
(=> (And (Unit-Of-Measure ?U)
(= (Quantity.Dimension ?U)
(Quantity.Dimension ?D)))
(Numeric-Tensor (Magnitude ?D ?U))))))
(Nth-Domain Dyad-Component 1 Dyad)
(=> (= (Dyad-Component ?T ?I ?J ?Basis) ?S) (Dyad ?T))
(Nth-Domain The-Dyad 3 Dyad)
(=> (= (The-Dyad ?M ?B) ?T) (Dyad ?T))
(=> (And (Dyad ?X) (Dyad ?Y))
(=> (+ ?X ?Y ?Z)
(And (Dyad ?Z)
(Forall (?B ?I ?J)
(=> (And (Orthonormal-Basis ?B)
(= (Spatial.Dimension ?X)
(Basis.Dimension ?B))
(Positive-Integer ?I)
(Positive-Integer ?J)
(=< ?I (Spatial.Dimension ?X))
(=< ?J (Spatial.Dimension ?X)))
(= (Dyad-Component ?Z ?I ?J ?B)
(+ (Dyad-Component ?X ?I ?J ?B)
(Dyad-Component ?Y ?I ?J ?B))))))))
(=> (And (Vector-Quantity ?V1) (Dyad ?T1))
(<=> (Dot ?V1 ?T1 ?V)
(And (Vector-Quantity ?V)
(Forall (?B)
(= ?T
(The-Vector-Quantity (* (Tensor-To-Matrix ?V1
?B)
(Tensor-To-Matrix ?T1
?B))
?B)))
(Forall
(?B ?I ?J)
(=> (= (Basis.Dimension ?B)
(Spatial.Dimension ?V1))
(= ?T
(Summation
(Lambda
(?I)
(* (Basis.Vec ?B ?I)
(Summation
(Lambda (?J)
(* (Vector-Component
?V1
?J
?B)
(Dyad-Component ?T1
?J
?I
?B)))
1
(Spatial.Dimension ?V1))))
1
(Spatial.Dimension ?V1))))))))
(=> (And (Dyad ?T1) (Vector-Quantity ?V1))
(<=> (Dot ?T1 ?V1 ?V)
(And (Vector-Quantity ?V)
(Forall
(?B)
(= ?T
(The-Vector-Quantity
(Transpose (* (Tensor-To-Matrix ?T1 ?B)
(Transpose (Tensor-To-Matrix
?V1
?B))))
?B)))
(Forall
(?B ?I ?J)
(=> (= (Basis.Dimension ?B)
(Spatial.Dimension ?V1))
(= ?T
(Summation
(Lambda
(?I)
(* (Basis.Vec ?B ?I)
(Summation
(Lambda (?J)
(* (Dyad-Component ?T1
?I
?J
?B)
(Vector-Component
?V1
?J
?B)))
1
(Spatial.Dimension ?V1))))
1
(Spatial.Dimension ?V1))))))))
(=> (And (Dyad ?T1) (Dyad ?T2))
(<=> (Dot ?T1 ?T2 ?T)
(And (Dyad ?T)
(Forall (?B)
(= ?T
(The-Dyad (* (Tensor-To-Matrix ?T1 ?B)
(Tensor-To-Matrix ?T2 ?B))
?B)))
(Forall
(?B ?I ?J)
(=> (= (Basis.Dimension ?B)
(Spatial.Dimension ?T1))
(= ?T
(Summation
(Lambda
(?I)
(Summation
(Lambda (?J)
(* (* (Basis.Vec ?B ?I)
(Basis.Vec ?B ?J))
(* (Dyad-Component ?T1
?I
?J
?B)
(Dyad-Component ?T2
?J
?I
?B))))
1
(Spatial.Dimension ?T1)))
1
(Spatial.Dimension ?T1))))))))
(Nth-Domain The-Zero-Dyad-Of-Type 3 Dyad)
(=> (= (The-Zero-Dyad-Of-Type ?Spatdim ?Physdim) ?V0)
(Forall (?V)
(=> (Dyad ?V)
(= (Dot ?V ?V0)
(The-Zero-Scalar-For-Dimension ?Physdim)))))
(<- (Dyad-Of-Dimensions ?Physim ?Spatdim)
(If (And (Physical-Dimension ?Physim)
(Positive-Integer ?Spatdim))
(Kappa (?Vq)
(And (Dyad ?Vq)
(= (Spatial.Dimension ?Vq) ?Spatdim)
(= (Quantity.Dimension ?Vq) ?Physim)))))