Function to generate the i,j th scalar component for a dyad in a particular basis. The i,j scalar component of dyad T is equal to bi*T*bj (basis vector i dot T dot basis vector j.
(<=> (Dyad-Component ?T ?I ?J ?Basis)
(And (Dyad ?T)
(Positive-Integer ?I)
(Positive-Integer ?J)
(Orthonormal-Basis ?Basis)
(Scalar-Quantity ?S)
(= (Quantity.Dimension ?S) (Quantity.Dimension ?T))
(= (Spatial.Dimension ?T) (Basis.Dimension ?B))
(=< ?I (Spatial.Dimension ?T))
(=< ?J (Spatial.Dimension ?T))
(= ?S
(Dot (Basis.Vec ?Basis ?I) (Dot ?T (Basis.Vec ?B ?J))))))
(Nth-Domain Dyad-Component 5 Scalar-Quantity) (Nth-Domain Dyad-Component 4 Orthonormal-Basis) (Nth-Domain Dyad-Component 3 Positive-Integer) (Nth-Domain Dyad-Component 2 Positive-Integer) (Nth-Domain Dyad-Component 1 Dyad)
(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)))))
(<=> (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))))))
(=> (= (Tensor-To-Matrix ?T ?Basis) ?M)
(=> (= (Tensor-Order ?T) 2)
(Square-Matrix ?M)
(= (Size ?M) (Spatial.Dimension ?T))
(Forall (?I ?J)
(= (Value ?M ?I ?J) (Dyad-Component ?T ?I ?J ?Basis)))))
(=> (= (Dyad-Component ?T ?I ?J ?Basis) ?S)
(= ?S (Dot (Basis.Vec ?Basis ?I) (Dot ?T (Basis.Vec ?B ?J)))))
(=> (= (Dyad-Component ?T ?I ?J ?Basis) ?S)
(=< ?J (Spatial.Dimension ?T)))
(=> (= (Dyad-Component ?T ?I ?J ?Basis) ?S)
(=< ?I (Spatial.Dimension ?T)))
(=> (= (Dyad-Component ?T ?I ?J ?Basis) ?S)
(= (Spatial.Dimension ?T) (Basis.Dimension ?B)))
(=> (= (Dyad-Component ?T ?I ?J ?Basis) ?S)
(= (Quantity.Dimension ?S) (Quantity.Dimension ?T)))
(=> (= (Dyad-Component ?T ?I ?J ?Basis) ?S) (Scalar-Quantity ?S))
(=> (= (Dyad-Component ?T ?I ?J ?Basis) ?S)
(Orthonormal-Basis ?Basis))
(=> (= (Dyad-Component ?T ?I ?J ?Basis) ?S) (Positive-Integer ?J))
(=> (= (Dyad-Component ?T ?I ?J ?Basis) ?S) (Positive-Integer ?I))
(=> (= (Dyad-Component ?T ?I ?J ?Basis) ?S) (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))))))))