|3) Field Operators in Curvilinear Coordinates|
As it is generally known, gradient of a scalar function is a vector the direction of which points the maximum growth of the function and its magnitude is equal to the derivative of that function along that direction. To obtain the formula for the gradient of a scalar function in curvilinear coordinate system (u1, u2, u3) let us consider the function j(u1,u2,u3). For the differential of the function one can write
Eq. (3.1) express the increment of the function along the elementary path . However, since result of applying of gradient operator is a vector, it can be written in the sum of its components
Therewithal, for the differential dj holds
what can be rewritten into the form of a dot product
Comparing of (3.3) and (3.4) it is possible to write
Equation (3.7) represents the expression for the gradient in general orthogonal curvilinear coordinate system. Mathematically, grad represents the operator which, when applied to the function of space coordinates j(u1,u2,u3), assigns to this function a new vector function grad j(u1,u2,u3). In cartesian coordinate system is gradient operator usually written by means of so called Ń operator (read “del” operator). In this special coordinate system the general curvilinear gradient operator
The divergence of a vector function is defined by the formula
where the integration is performed on the closed surface enclosing the volume DSV, see Fig.3.1. In fact equation (3.8) the outflow of the vector field through the surface SV divided by the volume of the enclosed region. The volume of the considered region is expected to be shrunk to a very small elementary volume. In this case one can speak about the volume density of the outflow of the vector as a function of point position in the space.
Fig.3.1, To the definition of the “divergence” operator
The physical meaning of the divergence will be treated in more details later.
To obtain the formula for divergence in curvilinear coordinate system we use the relations for elementary surface and volume elements introduced in beforegoing sections.
Let us calculate the surface integral in relation (3.8) for the case of an elementary volume in an orthogonal curvilinear coordinate system, see Fig.3.2. The integration on the surface of that element can be replaced by a sum of the particular components of the scalar product taken on the walls of the elementary volume according to the Fig.3.2.
Fig.3.2, To the calculation of the outflow of through the surface of the elementary volume
First, we can estimate the contribution to the outflow through the elementary surfaces perpendicular to the unit vector . For the outflow through the surfaces at u1 and (u1+Du1) we can write
The difference of the two terms in brackets can be expressed using the Taylor series for the function A(u1,u2,u3)
|higher order terms.|
Higher order terms can be neglected due to limiting the volume to zero. So the contribution of the two surfaces can be written in the form
The procedure for the calculation of the contributions of the remaining couples of surfaces perpendicular to and is similar to one performed above. For these contributions we can write
Finally, replacing the surface integral (3.8) by the sum of components (3.15), (3.16), (3.17) and expressing the volume element in (3.8) by using metric coefficients we can write for the divergence in orthogonal curvilinear coordinate system the following relation
In cartesian coordinate system this operator can be written in the form
or using the "del" operator gives
The dot means that this vector operator should be applied as a scalar product on the vector function.
Similarly as in the case of “gradient” the “divergence” ca be considered as a differential operator acting on the
vector function of space coordinates. The physical interpretation of divergence is often useful in obtaining
qualitative information about the divergence of a vector field without resorting to a mathematical investigation.
For instance let us consider the divergence of the velocity vector in a bath tube after the drain has been opened.
The net outflow of water through any closed surface lying entirely within the water must be zero, for water is
essentially incompressible and the water entering and leaving different regions of the closed surface must be
equal. Hence the divergence of velocity is zero.
If, however, we consider the velocity of the air in a tube, which has just been punctured by nail, we realize that the air is expanding as the pressure drops and that consequently there is a net outflow from any closed surface lying within the tube. The divergence of this velocity vector is therefore grater then zero. A positive divergence for any vector quantity indicates a source of that quantity at that point. Similarly, a negative divergence indicates a sink. Since the divergence at the water velocity above is zero, no source or sink exists. The expanding air, however, produces a positive divergence of the velocity and each interior point may be considered a source.
The results of well known Faraday’s experiments with the concentric sphere electrodes make possible to state that the electric flux passing through any imaginary surface lying between the two conducting charged spheres is equal to the charge enclosed by this imaginary surface. This enclosed charge is distributed on the surface of the inner sphere, or it might be concentrated as a point charge at the center of the imaginary sphere. However, since one coulomb of electric flux is produced by one coulomb of charge, the inner conductor might just as well have been a cube or a brass door key and the total induced charge on the outer sphere would still be the same. Certainly the flux density would change from its previous symmetrical distribution to some unknown configuration, but +Q coulombs on any inner conductor would produce an induced charge of –Q coulombs on the surrounding sphere. Going on step further, we could e.g. now replace the two outer hemispheres by an empty completely closed soup can Q coulombs on the brass door key would produce Q lines of electric flux and would induce –Q coulombs on the tin can.
These generalizations of Faraday’s experiment lead to the following statement, which is known as Gauss’s law:
Mathematical formulation of Gauss’s law is
where Y is the electric flux, is the electric displacement vector (electric induction or electric flux density), Q is the total electric charge enclosed by surface SV.
Nothing is stated about the distribution of charge in a considered volume enclosed by surface SV. It can be literally arbitrary. This charge can be described by the volume density r of that charge
Using this expression Gauss’s law may be written in terms of the charge distribution as
This mathematical statement means simply that total electric flux through any closed surface is equal to the charge enclosed. If now the volume of integration on the right side of (3.21) is getting smaller and smaller, we can obtain the definition of divergence
or more explicitly
Relation (3.23) represents the Maxwell equation as it is applied to electrostatic fields. It states that the "electric flux per unit volume leaving a vanishingly small volume unit is exactly equal to the volume charge density there". This equation is suitably called the "point form of the Gauss's law". Gauss's law relates the flux leaving any closed surface to the charge enclosed and Maxwell's equation makes an identical statement on a per-unit volume basis for a vanishingly small volume at a point. Remembering that Maxwell's equation is also described as the differential-equation form of Gauss's law, and conversely, Gauss's law is recognized as the integral form of Maxwell's equation.
The third and last of the special differential operator analysis which is frequently used for the characterizing of a special physical vector fields is curl-operator. It is a vector which roughly speaking characterizes the vector field as for its charge in space in the plane perpendicular to the "curl" of that vector field. First we shall define the projection of the curl on to the vector of elementary surface , which is perpendicular on this surface, see Fig.3.3.
Fig.3.3, To the definition of curl operator
Surface vector represents the surface enclosed by the curve . The mutual orientation of and is defined by the rule of the right-handed screw. The projection of is defined by the following relation
It is the ratio of the line integral of the vector along the closed path enclosing the area and the absolute value of the surface of that area. It is proposed that the area is shrunk to zero. All this operations are performed at the point . One can imagine the possibility of choosing the orientation of at the point in all possible directions and calculating corresponding projections of . Among all different projections there is only one to be maximum. That maximum projection is taken to be . If we call the line integral the circulation of the vector then it is also possible to characterize the as the vector with the magnitude equal to the maximum surface density of the circulation of and with the direction which is perpendicular to the surface enclosed by the path of the maximum circulation. It is possible to express the curl as a sum of its projections onto the basic unit vectors of the local curvilinear coordinate system at point as
Let us first calculate (curl )1:
Fig.3.4, To the calculation of curl in curvilinear coordinate system
Integral along the closed curve lS will be approximated by the summation using the Taylor series of at the point . Limitation allows neglecting of higher of higher order terms. The calculation of along the lS laying in the plane perpendicular to can be written in the form (see Fig.3.4)
Hence taking into account (3.24) and (3.26) one can write for the component (curl )1
Similarly the estimation o f(curl )2 and (curl )3 leads to
Finally, according to (3.24) and (3.27-3.29) in an orthogonal curvilinear system is given by
The general relation (3.30) can be written also by using the determinant formulation
From the mathematical point of view the curl can be considered as a differential operator that assigns to a vector function of space coordinates a new one defined by (3.30) or (3.31). It should be noted that the above definition of curl does not refer to any particular coordinate system. In the cartesian coordinate system and using the operator (3.31) can be written in the form
In the following we shall try to enlighten the physical meaning of the curl operator. Let us repeat well known Ampere’s circuital law, which states that the line integral of magnetic field intensity about any closed path is exactly equal to the direct current enclosed by that path, that is
Here it is necessary to stress the relation between the direction of and the “positive” current I. Positive current is one flowing in the direction of advance of right-handed screw turned in the direction in which the closed path is traversed. It is to remind that we have introduced the same relation between the integration path and elementary surface orientation when defining the projection of curl above. Total positive current can be described by the current density distributed on the area enclosed by the integration path in (3.33) so that we can write (3.33) in the form
If the surface Sl approaches “zero”, shrinking to the point, is nearly constant in the vicinity of that “point”. If is perpendicular to the area dSl or if then
Relation (3.36) is the Maxwell equation for stationary magnetic field. Physically it states that the source of magnetic field is non-zero curl of that field.
Although we have described curl as a line integral pre unit area, this does not provide every one with a satisfactory physical picture of the nature of the curl operation for the closed line integral itself require physical interpretation.
Shilling suggest the use of a very small paddle wheel as a “curl meter”. Our vector quantity, then must be thought of as capable of applying a force to each blade of the paddle wheel, the force being proportional to the component of the field normal to the surface of that blade. In order to test a field for curl we dip our paddle wheel into the field with the axis of the paddle wheel lined up with the direction of the component of the curl desired and note the action of the field on the paddle. No rotation means no curl. A reversal in the direction of the spin means a reversal in the sign of the curl. I order to find the direction of the vector curl and not merely to establish the presence of any particular component, we would place our paddle wheel in the field and hunt around for the orientation which produces the greatest torque. The direction of the curl is then along the axis of the paddle wheel as given by the right-handed rule. As an example consider the flow of water in river, see Fig.3.5.
Fig.3.5, a) The velocity field in a river and its curl
b) Solenoidal magnetic streamlines of the line current
Fig.3.5a shows the longitudinal section of a wide river taken at the middle of the river. The water velocity is zero at
the bottom and increases linearly as the surface is approached. A paddle wheel placed in the position shown, with
its axis perpendicular to the screen, will turn in a clockwise direction, slowing the presence of a component of cure
in the direction of an inward normal to the surface of the screen. If the velocity of the water does not change as we
go-up or downstream and also shows no variation as we go across the river (or even if it decreases in the same
fashion towards either bank), then this component is the only component present at the center of the stream, and the
curl of the water velocity has a direction into the page.
In Fig.3.5b the streamlines of the magnetic field intensity about an infinitely long filament conductor are shown. The curl meter placed in this field of curved lines shows that a larger number of blades have a clockwise force exerted on them but that force is in general smaller than the counter clockwise force exerted on the smaller number of blades closer to the wire. It seems possible that if the curvature of the streamlines is correct and also if the variation of the field strength is just right, the net torque on the paddle wheel may be zero. Actually the paddle wheel does not rotate in this case for since , we may calculate curl in the cylindrical coordinate system (see Table below) what gives
d) Laplace operator
Laplace operator is generally defined as a divergence of the gradient of a scalar field. It is frequently used in
the wave equations in different areas of physics. As we remember from the section concerning the divergence of
a vector the non-zero divergence meant that in the particular point there was a source of that vector field. In the case
of Laplace operator the vector field is represented by the gradient of scalar function. Laplace operator is a scalar
Let us consider the static electric field represented by the scalar potential j(u1,u2,u3). The gradient of that potential is proportional to the electric field intensity that is proportional to the electric flux density vector . Reminding the Gauss´s law describing the relation between and volume charge density r we can state that for the static electric field the application of Laplace operator to the scalar electric potential gives the volume charge density of the full electric charge. This statement follows directly from the well know Poisson´s equation for the potential of a static electric field. In other words for the case of electric field the Laplace operator applied on the electric potential function gives the scalar function that describes the space distribution of the electric intensity source and vice versa. The physical meaning of Laplace operator in other cases of physical fields depends on the properties of those fields. The relation for the Laplace operator in general curvilinear coordinate system can be obtained by a straightforward way trough application already derived formulas for gradient and divergence.
Hence combining these equations one can obtain
Finally, using the “del” operator Ń or D for Laplace operator we obtain
In a special case for cartesian coordinate system it is
e) The review of basic field operators for three frequently used orthogonal coordinates systems – cartesian, circular cylindrical and spherical one
|Coordinate system||Operator||Explicite form|
(x, y, z)
u1 = u2 = u3 = 1
(r, y, z)
u1 = u3 = 1
u2 = r
(r, y, J)
u1 = 1
u2 = r
u3 = r.sin J
f) Gauss´s and Stoke´s theorems
To this point we have introduced generals definitions of the basic field operators in general curvilinear and orthogonal coordinates. We have tried to bring some imaginations concerning the physical meaning of these operators. Defining the divergence and curl we have already used two important integral relations, which related surface and volume integrals and line and surface integrals. We have taken them, as experimentally verified lows. There were Gauss´s law for electric flux and Ampere’s circular law for the magnetic field of filament electric current. These integral relations can be generalized. Avoiding the particular physical meaning they can be formulated as very useful mathematical theorems.
f.1.) Gauss´s theorem
where volume and surface of integration on both sides of (3.43) is apparent from Fig.3.6
Fig.3.6, The volume and surface elements in integration according to the Gauss´s theorem
According to the (3.43) Gauss´s theorem makes possible to replace equally the volume integral of the divergence of vector field by the surface integral of that vector on the surface that encloses the volume of integration at left side of (3.43). We have omitted here intentionally some conditions that the function and the shape of surface must fulfil.
f.2.) Stoke's theorem
where the ls of integration and the surface of integration are evident from Fig.3.7
Fig.3.7, The volume and surface elements in integration according to the Gauss´s theorem
Stockes´s theorem allows to replace equally the circulation of a vector field along the closed path ls that encloses the smooth surface Sl (see. Fig.3.7), by the outflow of the curl of that field through the surface Sl. The orientation of the elementary surface element and the enclosing line are coupled by a rule of right- handed screw.
Note If one takes the existence of the above theorems as primary knowledge, the definitions of divergence and curl can be derived as the straightforward consequences of those theorems.