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The field with one element is then defined to be F 1 = {0, 1}, the multiplicative monoid of the field with two elements, which is initial in the category of multiplicative monoids. A monoid ideal in a monoid A is a subset I that is multiplicatively closed, contains 0, and such that IA = { ra : r ∈ I , a ∈ A } = I .
The fundamental theorem of vector calculus states that any vector field can be expressed as the sum of an irrotational and a solenoidal field. The condition of zero divergence is satisfied whenever a vector field v has only a vector potential component, because the definition of the vector potential A as: = automatically results in the identity ...
In addition, there is a class EnumSet, which represents a Set of values of an enumerated type internally as a bit vector, as a safer alternative to bit fields. The .NET Framework supplies a BitArray collection class. It stores bits using an array of type int (each element in the array usually represents 32 bits). [5]
Under zero-based numbering, the initial element is sometimes termed the zeroth element, [1] rather than the first element; zeroth is a coined ordinal number corresponding to the number zero. In some cases, an object or value that does not (originally) belong to a given sequence, but which could be naturally placed before its initial element ...
A vector field is a vector-valued function that, generally, has a domain of the same dimension (as a manifold) as its codomain, Conservative vector field, a vector field that is the gradient of a scalar potential field; Hamiltonian vector field, a vector field defined for any energy function or Hamiltonian
The zero rig, which is the zero ring, consisting only of a single element 0 = 1 is a terminal object. In Field, the category of fields, there are no initial or terminal objects. However, in the subcategory of fields of fixed characteristic, the prime field is an initial object.
Another method of deriving vector and tensor derivative identities is to replace all occurrences of a vector in an algebraic identity by the del operator, provided that no variable occurs both inside and outside the scope of an operator or both inside the scope of one operator in a term and outside the scope of another operator in the same term ...
A vector field V defined on an open set S is called a gradient field or a conservative field if there exists a real-valued function (a scalar field) f on S such that = = (,,, …,). The associated flow is called the gradient flow , and is used in the method of gradient descent .