Electrostatics and Magnetostatics
Vector calculus
Fondamental : Field in physics
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.
Fondamental : Gradient operator
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 :
Fondamental : Divergence of a vector field
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 :
GreenOstrogradsky theorem :
Where the finite closed surface surrounds the finite volume .
Fondamental : Curl (or rotation) of a vector field
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 .
Fondamental : Laplacian of a scalar field
The laplacian of a scalar field is defined by :
In a cartesian coordinate system :
Fondamental : Physical interest of vector operators
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.