Vector field

In the multi-dimensional analysis and of the differential geometry, a vector field is a function which assigns to each point of a vector space.

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

• 2.1 Definition
• 2.2 Notes

Vector fields in Euclidean space

Definition

Under a vector field in an amount refers to an image, which assigns to each point of a vector. If k - times differentiable, then it is called a vector field. Clearly that is a will, on every point of the set "arrow attached ".

Examples

• Gradient: Is a differentiable function on an open set, so the gradient is defined by the assignment of.

• Central fields: Be an interval containing zero, and a spherical shell. Central fields on the spherical shell defined by

• In the gravitational field is such a central field.
• Other examples are in the mathematical diffizileren so-called " vortex fields ". They can be described as a rotation of a vector potential, according to the formula (see below).

Source -free and irrotational vector fields; decomposition theorem

A least twice continuously - differentiable vector field in the source is called free (or irrotational ) if its source density ( divergence) or vortex density (rotation) is everywhere zero. Under the further assumption that the components at infinity sufficiently rapidly disappear, the so-called decomposition theorem applies: Each vector field is uniquely determined by its source or vortex, namely the following decomposition applies in a non-turbulent or source-free content:

This corresponds to the decomposition of a static electromagnetic field in the electric or magnetic content (see electrodynamics). This means that exactly the gradient ( ie, the " electric field components " ) irrotational or just the vortex fields ( ie, the " magnetic field components " ) source free. In this case, and the known, formed with the operator of vector analysis operations.

Vector fields on manifolds

Definition

Let be a differentiable manifold. A vector field is a (smooth ) section in the tangent bundle.

Expanded means that a vector field is a mapping, such that with. Each a vector so it is assigned. The picture is the natural projection.

Comments

This definition generalizes the vector fields in Euclidean space. It is namely and.

As opposed to a scalar vector fields is assigned by means of a scalar field every point of a manifold.

Vector fields are just the contravariant tensor of first order.

Applications

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