Vector field

Representation of a vector field based on selected points. The vectors are shown as arrows, which indicate the direction and amount (arrow length)

3-dimensional vector field (-y, z, x)

In multi-dimensional analysis and differential geometry , a vector field is a function that assigns a vector to every point in space .

Continuous vector fields are of great importance in physical field theory , for example to indicate the speed and direction of a particle in a moving liquid , or to describe the strength and direction of a force such as magnetic or gravity . The field sizes of these vector fields can be illustrated by field lines .

Vector fields in Euclidean space

definition

A vector field on a set is a mapping that assigns a vector to each point . An “arrow” is clearly attached to each point in the crowd . Most of the time, it is tacitly assumed that the vector field is smooth , i.e. a mapping. If a mapping is- times differentiable , one speaks of a -vector field . ${\displaystyle v}$ ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ ${\displaystyle x\in \Omega }$ ${\displaystyle v(x)\in \mathbb {R} ^{n}}$ ${\displaystyle \Omega }$ ${\displaystyle C^{\infty }}$ ${\displaystyle v}$ ${\displaystyle k}$ ${\displaystyle v\colon \Omega \to \mathbb {R} ^{n}}$ ${\displaystyle C^{k}}$

Examples

• Gradient field: If there is a differentiable function on an open set , the gradient field of is defined by the assignment ${\displaystyle f\colon \Omega \rightarrow \mathbb {R} }$ ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ ${\displaystyle \operatorname {grad} f\colon \Omega \rightarrow \mathbb {R} ^{n}}$ ${\displaystyle f}$

  ${\displaystyle x\mapsto \operatorname {grad} f(x)=\left({\frac {\partial f}{\partial x_{1}}}(x),\dotsc ,{\frac {\partial f}{\partial x_{n}}}(x)\right)}$.

Often you write it with the Nabla symbol : . If a vector field is the gradient field of a function , that is , it is called a potential . It is also said that it has a potential. ${\displaystyle \operatorname {grad} f=\nabla f}$ ${\displaystyle v}$ ${\displaystyle f}$ ${\displaystyle v=\nabla f}$ ${\displaystyle f}$ ${\displaystyle v}$

Examples of gradient fields are the field of a current flowing uniformly in all directions from a point source and the electric field around a point charge.

• Central fields: Let be an interval containing zero and a spherical shell. Central fields on the spherical shell are defined by ${\displaystyle I}$ ${\displaystyle K(I)=\{x\in \mathbb {R} ^{n}:\|x\|\in I\}\subset \mathbb {R} ^{n}}$

${\displaystyle v(x)=a(\|x\|)\cdot x}$with . ${\displaystyle a\colon I\rightarrow \mathbb {R} }$

• In the gravitational field is such a central field. ${\displaystyle \mathbb {R} ^{3}\backslash \{0\}}$ ${\displaystyle v(x)=-{\frac {x}{\|x\|^{3}}}}$
• Further examples are the mathematically more difficult so-called " vortex fields ". They can be described as the rotation of a vector potential according to the formula (see below). ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle \mathbf {A} }$ ${\displaystyle \mathbf {v} (\mathbf {r} )=\mathbf {rot\,\,} \mathbf {A} }$

A concise example of a vortex field is the flow field swirling in circular lines around the outflow of a "bathtub", or the magnetic field around a wire through which current flows.

Source-free and vortex-free vector fields; Decomposition kit

At least twice continuously differentiable vector field in is free of sources (or wirbelfrei ) when its source density ( divergence ) or vortex density ( rotation ) is everywhere zero. Under the further prerequisite that the components of in infinity vanish sufficiently quickly, the so-called decomposition theorem applies : Each vector field is uniquely determined by its sources or eddies, namely the following decomposition into an eddy-free or source-free part applies: ${\displaystyle \mathbf {v} (\mathbf {r} )}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle \mathbf {v} }$ ${\displaystyle \mathbf {v} (\mathbf {r} )}$

${\displaystyle \mathbf {v} (\mathbf {r} )\equiv \mathbf {-grad_{\mathbf {r} }\,\,} \int _{\mathbb {R} ^{3}\,}\,d^{3}\mathbf {r} '\,{\frac {\mathrm {{div'}\,\,} \mathbf {v} (\mathbf {r} ')}{4\pi |\mathbf {r} -\mathbf {r} '|}}+\mathbf {rot_{\mathbf {r} }\,\,} \int _{\mathbb {R} ^{3}\,}\,d^{3}\mathbf {r} '\,\,{\frac {{\mathbf {rot'\,\,} }\mathbf {v} (\mathbf {r} ')}{4\pi |\mathbf {r} -\mathbf {r} '|}}}$.

This corresponds to the decomposition of a static electromagnetic field into the electrical or magnetic component (see electrodynamics ). It is precisely the gradient fields (i.e. the "electric field components") that are free of eddies or precisely the eddy fields (i.e. the "magnetic field components") are source-free. Thereby and are the known operations formed with the Nabla operator ( ) of vector analysis . ${\displaystyle \mathbf {grad\,\,} \phi (\mathbf {r} ):=\nabla \phi \,,}$    ${\displaystyle \mathrm {div\,\,} \mathbf {v} :=\nabla \cdot \mathbf {v} }$ ${\displaystyle \mathbf {rot\,\,} \mathbf {v} :=\nabla \times \mathbf {v} }$ ${\displaystyle \nabla }$

Vector fields on manifolds

definition

Let be a differentiable manifold . A vector field is a (smooth) cut in the tangential bundle . ${\displaystyle M}$ ${\displaystyle TM}$

In more detail, this means that a vector field is a mapping such that with holds. So each is assigned a vector . The figure is using the natural projection . ${\displaystyle v}$ ${\displaystyle v\colon M\to TM}$ ${\displaystyle \pi \circ v=\operatorname {id} _{M}}$ ${\displaystyle x\in M}$ ${\displaystyle v(x)\in T_{x}M}$ ${\displaystyle \pi }$ ${\displaystyle \pi \colon TM\to M}$ ${\displaystyle (p,v)\mapsto p}$

Remarks

This definition generalizes the vector fields in Euclidean space. It applies namely and . ${\displaystyle \mathbb {R} ^{n}\cong T_{p}\mathbb {R} ^{n}}$ ${\displaystyle T\mathbb {R} ^{n}\cong \mathbb {R} ^{n}\times \mathbb {R} ^{n}}$

In contrast to vector fields is a scalar field every point of a manifold, a scalar associated.

Vector fields are precisely the contravariant first order tensor fields .

Applications

Vector and force fields have except in physics and chemistry also of great importance in many fields of engineering : electrical engineering , geodesy , mechanics , atomic physics , Applied Geophysics .

See also

• Flow of a vector field

literature

• Konrad Königsberger : Analysis 2. 5th corrected edition. Springer, Berlin a. a. 2004, ISBN 3-540-20389-3 .
• R. Abraham, JE Marsden , T. Ratiu: Manifolds, Tensor Analysis, and Applications . 2nd Edition. Springer, Berlin 1988, ISBN 3-540-96790-7 (English).
• John M. Lee: Introduction to smooth manifolds (= Graduate Texts in Mathematics 218). Springer, New York a. a. 2003, ISBN 0-387-95495-3 .

Individual evidence

1. ↑ See u. a. U. Krey, A. Owen, Basic Theoretical Physics - A Concise Overview , Berlin, Springer 2007, ISBN 978-3-540-36804-5 , part II