Class TENSOR-QUANTITY

• Defined in theory: Tensor-quantities
• Source code: tensor-quantities.lisp

Slots on this class:

Documentation:

Every tensor has an associated spatial dimension and tensor order. Each tensor-quantity can be decomposed into scalar components for a given orthonormal basis of the proper dimension (except of . The physical-dimension of a scalar component of the tensor-quantity is equivalent to the physical-dimension of the tensor-quantity.

Instance-Of: Class

Subclass-Of: Constant-quantity

Domain-Of: Spatial.dimension, Tensor-order

Superclass-Of: Dyad, Numeric-tensor, Vector-quantity

---

Slots on instances of this class:

Spatial.Dimension:

Slot-Cardinality: 1

Slot-Value-Type: Non-negative-integer

Tensor-Order:

Slot-Cardinality: 1

Slot-Value-Type: Non-negative-integer

Equivalence Axioms:

(<=> (And (Tensor-Quantity ?T) (= (Tensor-Order ?T) 0))
     (Scalar-Quantity ?T))

Other Related Axioms:

(<=> (And (Tensor-Quantity ?T) (= (Tensor-Order ?T) 0))
     (Scalar-Quantity ?T))

(=> (Spatial.Dimension $X $Y) (Tensor-Quantity $X))

(=> (Tensor-Order $X $Y) (Tensor-Quantity $X))

(<=> (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))))))

(<=> (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 Tensor-To-Matrix 1 Tensor-Quantity)

(=> (= (Tensor-To-Matrix ?T ?Basis) ?M) (Tensor-Quantity ?T))

(=> (And (Tensor-Quantity ?X) (Tensor-Quantity ?Y) (+ ?X ?Y ?Z))
    (And (Tensor-Quantity ?Z)
         (= (Spatial.Dimension ?X) (Spatial.Dimension ?Y))
         (= (Spatial.Dimension ?X) (Spatial.Dimension ?Z))))

(Nth-Domain Dot 3 Tensor-Quantity)

(Nth-Domain Dot 2 Tensor-Quantity)

(Nth-Domain Dot 1 Tensor-Quantity)

(Distributes Dot + Tensor-Quantity)