|1) General Curvilinear Coordinates|
One of the main advantages of the vector calculus is the possibility to formulate physical laws without using some particular coordinate system. If one has to solve a special case of electromagnetic field defined by given boundary conditions it is necessary to decompose corresponding vector equations into coordinate components and to solve them in that coordinate system. The choice of a suitable coordinate system can significantly simplify the solution. In the opposite the case construction of a suitable analytical solution is rather tedious. The appropriate coordinate system is chosen in such a way that its coordinate surfaces, obtained by putting the particular coordinates equal to constants, coincide with the boundary surfaces of the region in which the solution is to be seek.
In this respect the definition and utilization of curvilinear coordinate system is one of the important topics that are necessary to cope with for a proper understanding of the formulation and solution of electromagnetic problems
Let us consider three independent, unambiguous and smooth functions f1(x,y,z), f2(x,y,z), f3(x,y,z), of the three independent space variables x,y,z in the cartesian coordinate system (x,y,z). Setting these functions equal to constant parameters u1, u2, u3 defines three surfaces, that can be labeled by these numbers, see Fig. 1.1. Common intersection of the surfaces u1=const1, u2=const2, u3=const3 defines one point in the space to which a set of three unique numbers (u1, u2, u3) can be assigned. These numbers are called curvilinear coordinates of that point, see Fig. 1.1.
Fig.1.1, To the definition of the curvilinear coordinates of a point A(u1,u2,u3) in the space
The set of equations
can be solved and the solution can be written in the form
It defines the position of the point A in the cartesian system (x,y,z) using coordinates (u1, u2, u3), where is the position vector of the point A and are the unit vectors aligned in the space with the coordinate axes of the cartesian system, see Fig. 1.1. An elementary displacement of the point A in the space can be described by the differential formula
are called the basic vectors of the general curvilinear coordinate system at point A(u1,u2,u3). It is to be noted that the absolute values of are not equal to 1, they are generaly not unit vectors. If we have the orthogonal curvilinear coordinate system. In the following we shall consider only orthogonal coordinate systems. The relation (1.4) can be rewritten into the form
where we have put
are now the units vectors of the same directions as vectors . The functions h1, h2, h3 are usually called the metric coefficients. The physical meaning of these coefficients can be understood when defining the length elements along the particular directions given by vectors in the local curvilinear coordinate system at point corresponding with the elementary displacements of A(u1,u2,u3) by . The elementary displacement from point to can be described by
One can see that the products represent the lengths of projections of the elementary displacement onto the vectors respectively. Consequently the change of the curvilinear coordinate dui can be transformed into corresponding displacement in space by multiplying it by hi, corresponding metric coefficient see Fig. 1.2.
Fig.1.2, To the definition of the elementary displacement , surface and volume respectively.
Similarly one can define elementary coordinate surface corresponding with the elementary changes of a couple of coordinates. It is explicitly defined by the
Using expressions for elementary displacements from (1.7) it is possible to write
In (1.9) the relations , and were used. According to (1.9) a general elementary surface is composed of the three elementary surfaces oriented along the unit vectors , see Fig. 1.2. Elementary volume element dV can be described by
or using metric coefficients we obtain