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Vector field reconstruction [1] is a method of creating a vector field from experimental or computer-generated data, usually with the goal of finding a differential equation model of the system. Definition
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 .
For a tensor field of order k > 1, the tensor field of order k is defined by the recursive relation = where is an arbitrary constant vector. A tensor field of order greater than one may be decomposed into a sum of outer products, and then the following identity may be used: = ().
Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, . [1] The term vector calculus is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration.
The primary methods for visualizing two-dimensional (2D) scalar fields are color mapping and drawing contour lines. 2D vector fields are visualized using glyphs and streamlines or line integral convolution methods. 2D tensor fields are often resolved to a vector field by using one of the two eigenvectors to represent the tensor each point in ...
Depiction of the Arc/INFO coverage data model, a geo-relational topological vector data model based on the early POLYVRT data model. Vector data structures can also be classified by how they manage topological relationships between objects in a dataset: [22] A topological data model incorporates topological relationships as a core part of the ...
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.
A scalar is an element of a field which is used to define a vector space.In linear algebra, real numbers or generally elements of a field are called scalars and relate to vectors in an associated vector space through the operation of scalar multiplication (defined in the vector space), in which a vector can be multiplied by a scalar in the defined way to produce another vector.