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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 .
Finite fields (also called Galois fields) are fields with finitely many elements, whose number is also referred to as the order of the field. The above introductory example F 4 is a field with four elements. Its subfield F 2 is the smallest field, because by definition a field has at least two distinct elements, 0 and 1.
Muladhara Chakra (मूलाधार चक्र) Muladhara (Sanskrit: मूलाधार or मूलाधारा; IAST: Mūlādhāra, lit. "root of Existence." Mula means root and dhara means flux.) or the root chakra is one of the seven primary chakras according to Hindu tantrism. It is symbolized by a lotus with four petals and ...
In the study of mathematics, and especially of differential geometry, fundamental vector fields are instruments that describe the infinitesimal behaviour of a smooth Lie group action on a smooth manifold. Such vector fields find important applications in the study of Lie theory, symplectic geometry, and the study of Hamiltonian group actions.
Interchanging the vector field v and ∇ operator, we arrive at the cross product of a vector field with curl of a vector field: = () , where ∇ F is the Feynman subscript notation, which considers only the variation due to the vector field F (i.e., in this case, v is treated as being constant in space).
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If F is a subfield of E then E is an extension field of F. We then also say that E/F is a field extension. Degree of an extension Given an extension E/F, the field E can be considered as a vector space over the field F, and the dimension of this vector space is the degree of the extension, denoted by [E : F]. Finite extension
A field can be classified as a scalar field, a vector field, a spinor field or a tensor field according to whether the represented physical quantity is a scalar, a vector, a spinor, or a tensor, respectively. A field has a consistent tensorial character wherever it is defined: i.e. a field cannot be a scalar field somewhere and a vector field ...