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e. In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.
Field (physics) Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. In science, a field is a physical quantity, represented by a scalar, vector, or tensor, that has a value for each point in space and time. [1][2][3] A weather map, with the surface temperature described by assigning a number to each ...
Field (computer science) In data hierarchy, a field (data field) is a variable in a record. [1] A record, also known as a data structure, allows logically related data to be identified by a single name. Identifying related data as a single group is central to the construction of understandable computer programs. [2]
Definition of a field. A field is a commutative ring (F, +, *) in which 0 ≠ 1 and every nonzero element has a multiplicative inverse. In a field we thus can perform the operations addition, subtraction, multiplication, and division. The non-zero elements of a field F form an abelian group under multiplication; this group is typically denoted ...
Field extension. In mathematics, particularly in algebra, a field extension (denoted ) is a pair of fields , such that the operations of K are those of L restricted to K. In this case, L is an extension field of K and K is a subfield of L. [1][2][3] For example, under the usual notions of addition and multiplication, the complex numbers are an ...
Gradient. The gradient, represented by the blue arrows, denotes the direction of greatest change of a scalar function. The values of the function are represented in greyscale and increase in value from white (low) to dark (high). In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field ...
A finite field is a finite set that is a field; this means that multiplication, addition, subtraction and division (excluding division by zero) are defined and satisfy the rules of arithmetic known as the field axioms. The number of elements of a finite field is called its order or, sometimes, its size.
A portion of the vector field (sin y, sin x) In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space . [1] A vector field on a plane can be visualized as a collection of arrows with given magnitudes and directions, each attached to a point on the plane.