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Was (wer) ist vector field - definition

Vector field

ASSIGNMENT OF A VECTOR TO EACH POINT IN A SUBSET OF EUCLIDEAN SPACE

Vector fields; Vector field on a manifold; Vector-field; Tangent bundle section; Tangent Bundle Section; Gradient vector field; Gradient flow; Vector plot; Vector Field; Vectorfield; Index of a vector field; F-related; Vector point function; Operations on vector fields; Complete vector field

In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with a given magnitude and direction, each attached to a point in the plane.

https://en.wikipedia.org/wiki/Vector_field

Conservative vector field

$The above vector field <math>\mathbf{v} = \left( - \frac{y}{x^2 + y^2},\frac{x}{x^2 + y^2},0 \right)</math> defined on <math>U = \R^3 \setminus \{ (0,0,z) \mid z \in \R \}</math>, i.e., <math>\R^3</math> with removing all coordinates on the <math>z</math>-axis (so not a simply connected space), has zero curl in <math>U</math> and is thus irrotational. However, it is not conservative and does not have path independence.$

CONCEPT IN VECTOR CALCULUS

Irrotational; Irrotational field; Gradient field; Potential vector field; Conservative field; Curl free field; Curl-free vector field; Irrotational vector field; Irrotational flow; Irrotational Flow

In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of any path between two points does not change the value of the line integral.

https://en.wikipedia.org/wiki/Conservative_vector_field

Killing vector field

VECTOR FIELD ON A RIEMANNIAN MANIFOLD THAT PRESERVES THE METRIC

Killing vector; Killing vector fields; Killing vectors; Killing symmetry; Killing Vectors; Killing equation

In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold.

https://en.wikipedia.org/wiki/Killing_vector_field