Function to generate the ith scalar component for a vector-quantity in a particular basis. The ith scalar component is the dot (inner) product of the vector-quantity with the ith basis vector of the given basis.
(<=> (Vector-Component ?V ?I ?B)
(And (Vector-Quantity ?V)
(Positive-Integer ?I)
(Orthonormal-Basis ?B)
(Scalar-Quantity ?S)
(= (Quantity.Dimension ?S) (Quantity.Dimension ?V))
(= (Spatial.Dimension ?V) (Basis.Dimension ?B))
(=< ?I (Spatial.Dimension ?V))
(= ?S (Dot ?V (Basis.Vec ?B ?I)))))
(Nth-Domain Vector-Component 4 Scalar-Quantity) (Nth-Domain Vector-Component 3 Orthonormal-Basis) (Nth-Domain Vector-Component 2 Positive-Integer) (Nth-Domain Vector-Component 1 Vector-Quantity)
(Forall (?B ?I)
(=> (And (Orthonormal-Basis ?B)
(= (Basis.Dimension ?B) (Spatial.Dimension ?V))
(Positive-Integer ?I)
(=< ?I (Spatial.Dimension ?V)))
(And (Defined (Vector-Component ?V ?I ?B))
(= (Quantity.Dimension (Vector-Component ?V ?I ?B))
(Quantity.Dimension ?V)))))
(<=> (Vector-Quantity ?V)
(And (Constant-Quantity ?V)
(Tensor-Quantity ?V)
(= (Tensor-Order ?V) 1)
(Forall (?B ?I)
(=> (And (Orthonormal-Basis ?B)
(= (Basis.Dimension ?B)
(Spatial.Dimension ?V))
(Positive-Integer ?I)
(=< ?I (Spatial.Dimension ?V)))
(And (Defined (Vector-Component ?V ?I ?B))
(= (Quantity.Dimension (Vector-Component ?V
?I
?B))
(Quantity.Dimension ?V)))))
(Forall (?U)
(=> (And (Unit-Of-Measure ?U)
(= (Quantity.Dimension ?U)
(Quantity.Dimension ?V)))
(Numeric-Tensor (Magnitude ?V ?U))))))
(=> (= (Tensor-To-Matrix ?T ?Basis) ?M)
(=> (= (Tensor-Order ?T) 1)
(= (Row-Dimension ?M) 1)
(= (Column-Dimension ?M) (Spatial.Dimension ?T))
(Forall (?I)
(= (Value ?M 1 ?I) (Vector-Component ?T ?I ?Basis)))))
(=> (= (Vector-Component ?V ?I ?B) ?S)
(= ?S (Dot ?V (Basis.Vec ?B ?I))))
(=> (= (Vector-Component ?V ?I ?B) ?S)
(=< ?I (Spatial.Dimension ?V)))
(=> (= (Vector-Component ?V ?I ?B) ?S)
(= (Spatial.Dimension ?V) (Basis.Dimension ?B)))
(=> (= (Vector-Component ?V ?I ?B) ?S)
(= (Quantity.Dimension ?S) (Quantity.Dimension ?V)))
(=> (= (Vector-Component ?V ?I ?B) ?S) (Scalar-Quantity ?S))
(=> (= (Vector-Component ?V ?I ?B) ?S) (Orthonormal-Basis ?B))
(=> (= (Vector-Component ?V ?I ?B) ?S) (Positive-Integer ?I))
(=> (= (Vector-Component ?V ?I ?B) ?S) (Vector-Quantity ?V))
(=> (And (Vector-Quantity ?X) (Vector-Quantity ?Y))
(=> (+ ?X ?Y ?Z)
(And (Vector-Quantity ?Z)
(Forall (?B ?I)
(=> (And (Orthonormal-Basis ?B)
(= (Spatial.Dimension ?X)
(Basis.Dimension ?B))
(Positive-Integer ?I)
(=< ?I (Spatial.Dimension ?X)))
(= (Vector-Component ?Z ?I ?B)
(+ (Vector-Component ?X ?I ?B)
(Vector-Component ?Y ?I ?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 (Vector-Quantity ?V1) (Vector-Quantity ?V2))
(<=> (Dot ?V1 ?V2 ?S)
(And (Scalar-Quantity ?S)
(Forall (?B)
(= ?S
(Value (* (Tensor-To-Matrix ?V1 ?B)
(Transpose (Tensor-To-Matrix ?V2
?B)))
11)))
(Forall (?B ?J)
(=> (= (Basis.Dimension ?B)
(Spatial.Dimension ?V1))
(= ?S
(Summation (Lambda (?J)
(* (Vector-Component
?V1
?J
?B)
(Vector-Component
?V2
?J
?B)))
1
(Spatial.Dimension ?V1))))))))
(=> (And (Scalar-Quantity ?S) (Vector-Quantity ?V1) (* ?S ?V1 ?V))
(And (Vector-Quantity ?V)
(= (Spatial.Dimension ?V1) (Spatial.Dimension ?V))
(Forall (?B ?I)
(=> (= (Basis.Dimension ?B) (Spatial.Dimension ?V1))
(= (Vector-Component ?V ?I ?B)
(* ?S (Vector-Component ?V1 ?I ?B)))))))