Vector calculus

In physics, the term field denotes a physical quantity that is continuous function of position, within a certain region of space.

We distinguish scalar fields (such as temperature or electrostatic potential) from vector fields (such as the velocity of a moving fluid or the electric field ).

A vector field assigns a direction, as well as a magnitude, to each point in space.

Since a vector can be represented by three components, a vector field is specified analytically by a set of three functions of position.

Since scalar and vector fields are functions of positions, they must have spatial derivatives and integrals.

We consider first a scalar field .

The rate of change in for small step away from the point depends upon the direction of the step : the derivative operation has vectorlike properties.

Specifically, for an infinitesimal step of magnitude in the direction specified by the unit vector  :

The change in is :

And :

Where we introduce the gradient (or Nabla ) operator :

Thus the derivated of in the direction is :

The vector field is the gradient of  :

The gradient is thus a vector derivative, having the magnitude and direction of the greatest space rate of change of the scalar field .

The derivative in an arbitrary direction is simply the component of the gradient vector in that direction.

The gradient is directed perpendicular to the surface .

If the field is expressed in cylindrical or spherical coordinate systems, the forms of derivatives are altered but the meaning of the gradient is unchanged.

• Cylindrical coordinate system :

  

• Spherical coordinate system :

  

The divergence of a vector field :

is defined by (in a cartesian coordinate system) :

It can be proved that the divergence is the flux generation per unit volume.

• Cylindrical coordinate system :

  

• Spherical coordinate system :

  

Green-Ostrogradsky theorem :

Where the finite closed surface surrounds the finite volume .

The curl of a vector field is defined by (in a cartesian coordinate system) :

It means :

• Cylindrical coordinate system :

  

• Spherical coordinate system :

  

Stokes' theorem :

Where the closed loop bounds the finite open surface .

The laplacian of a scalar field is defined by :

In a cartesian coordinate system :

We can illustrate the terms of divergence and curl for some types fields : (see figure below)

The divergence and the curl are zero for the field whose field lines are parallel.

The divergence is negative for fields whose field lines converge to a point. It is positive for a diverging field.

The curl operator of the last field (fields lines rotate around point O in the positf side) is positif.

The considered field can be that of velocity of a solid rotating about the(Oz) axis .

The speed of a point of the solid is, if denotes the rotation vector of the solid,, .

Using Cartesian coordinates :

This result allows to associate the curl operator to the idea of rotation.