Vector calculus

Vector analysis is a branch of mathematics that deals mainly with vector fields in two or more dimensions and thereby significantly generalizes the already treated in school mathematics areas of differential and integral calculus. The area consists of a set of formulas and problem solving techniques that include the essential equipment by engineers and physicists, but are usually learned at the respective universities until the second or third semester.

Are considered vector fields that map each point in space a vector and scalar fields that map each point in space a scalar. The temperature of a swimming pool is a scalar: scalar value to every point of his temperature is assigned. In contrast, the water movement corresponds to a vector field, since each point a velocity vector is assigned, the amount and direction has.

The results of vector analysis can be generalized by means of differential geometry and abstraction, a mathematical branch, which comprises the vector analysis. The physical main applications are in electrodynamics.

• 4.1 consequence of the disappearance of the rotation
• 4.2 Conclusion from the vanishing of the divergence

The three covariant differential operators

Three computational operations are in the vector analysis of particular importance because they produce fields which rotate in three-dimensional rotation of the original field. Operative words: At Gradient, divergence and rotation, it does not matter whether they are applied before or after a rotation. This property follows from the coordinate- independent definitions ( see the respective main article) and is not self-evident. For example, is from a partial derivative with respect to x at 90 - degree rotation, a partial derivative with respect to y. The following is the operator of the partial derivative and the nabla operator.

• Gradient of a scalar field: Specifies the direction and strength of the steepest ascent of a scalar field. The gradient of a scalar field is a vector field.

• Divergence of a vector field: Indicates the tendency of a vector field from points to flow away ( this applies to positive sign and a negative sign when there are accordingly a tendency to points added flow ). The divergence of a vector field is a scalar field. Namely, it follows from the Gaussian integral theorem (see below) that the divergence describes the local source density of a vector field.

• Rotation of a vector field: indicates the tendency of a vector field in order to rotate points; the rotation of a vector field is a vector of pseudo- field vectors. Namely, it follows from the Stokes' integral theorem (see below) that describes the rotation of the local vortex density of a vector field.

Integral theorems

Integral theorem of Gauss

Below is the " integration volume " -dimensional.

The volume integral of the gradient of a scalar quantity can be converted over the edge of this volume in a surface integral ( or integral hypersurface ):

On the right side is remembered by the symbol in the center of the integral because one has to do it because of the edge forming a closed surface (or a closed hypersurface ).

Conversion surface integral is also possible for the divergence of a vector quantity: the integral of the divergence of the entire volume is equal to the integral of the flow from the surface,

This is the actual Gaussian integral theorem. He is - as I said - not only for.

Stokes' theorem

Below is and it is the spelling used with multiple integrals.

The closed line integral of a vector quantity (right side), by means of rotation bounded into a surface integral over a closed path of integration of the not necessary flat surface are converted (left side). The usual orientation properties provided - Here, - as well as the Gaussian set. The following applies:

The vector is equal to the sum of the observed surface or belonging to the infinitesimal surface elements multiplied by the corresponding normal vector. On the right side is remembered by the circle symbol in the integral sign because one assumes a closed curve.

Fundamental decomposition

The fundamental theorem of vector analysis, also called Helmholtz shear decomposition theorem, describes the general case. He states that can be described as a superposition of a source component and a vortex component of each vector field. The former is the negative gradient of a superposition of scalar Coulomb -type potentials, as determined by the source of density as a formal " charge density ", as in static electric fields; the latter is the rotation of a vector potential, Biot- Savart law as in the case of magnetostatics, determined by the vortex density as a formal " current density "

One can clearly track the validity of such a decomposition on the course of a brook: At locations with large gradients and rectilinear course the flow is dominated by the Gradientenanteil while on flat places, especially if the stream is a " corner " or a small island around flows, the eddy proportion prevails.

And that is true if the components of the vector are everywhere twice continuously differentiable (otherwise you have to at the interfaces of volume integrals replaced by surface integrals) and sufficiently rapidly disappear at infinity, the following formula, which corresponds exactly to the above-mentioned combination of electrodynamics and all mentioned operators includes:

A general vector field with respect to its physical meaning, therefore, only be clearly specified if both statements about the source and vortex densities and possibly the necessary boundary values ​​are available.

Identities

These identities often prove to be useful in transformations:

• Or

• For all scalar fields.
• For all vector fields.
• For all scalar fields.

In the next two sections are common names instead of in a different context ( electrodynamics ) and used:

Consequence of the disappearance of the rotation

If it follows a scalar potential. This is given by the first part of the above fundamental separation and identical to the corresponding triple integral, and is therefore determined by the source density.

Or are the usual in electrostatics names for the electric field and its potential. There the specified condition is met.

Consequence of the disappearance of the divergence

If it follows a so-called vector potential. This is given by the second part of the above fundamental decomposition and identical to the corresponding triple integral, that is determined by the vortex density.

Or are customary in magnetostatics names for the magnetic induction or the vector potential. There again, the condition is fulfilled.