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In 3-dimensional geometry and vector calculus, an area vector is a vector combining an area quantity with a direction, thus representing an oriented area in three dimensions. Every bounded surface in three dimensions can be associated with a unique area vector called its vector area .
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 .
Note: This page uses common physics notation for spherical coordinates, in which is the angle between the z axis and the radius vector connecting the origin to the point in question, while is the angle between the projection of the radius vector onto the x-y plane and the x axis. Several other definitions are in use, and so care must be taken ...
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 ...
For vector flux, the surface integral of j over a surface S, gives the proper flowing per unit of time through the surface: = ^ =, where A (and its infinitesimal) is the vector area – combination = ^ of the magnitude of the area A through which the property passes and a unit vector ^ normal to the area. Unlike in the second set of equations ...
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
A vector field defines a direction and magnitude at each point in space. A field line is an integral curve for that vector field and may be constructed by starting at a point and tracing a line through space that follows the direction of the vector field, by making the field line tangent to the field vector at each point.
The idea of a vector flow, that is, the flow determined by a vector field, occurs in the areas of differential topology, Riemannian geometry and Lie groups. Specific examples of vector flows include the geodesic flow, the Hamiltonian flow, the Ricci flow, the mean curvature flow, and Anosov flows.