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ELECTROMAGNETIC FIELDS AND THEORY: FIELDS AND VECTORS: Rotational/Laplacian

1.6.3 Rotational.

In this case the operator has as an argument a vector function and produces as a result another vector function. Using the following expression, the rotational of the vector function is indicated.

 
     
 

        (45)

Using the concept of the line integral, we can define the component in direction   of the rotational, the following way,


       
(46)

This expression tells us: in a particular point the component in direction of the rotational of function E is equal to the limit of the circuitation of E along the closed path of C (C is in a perpendicular plane to ) divided by area Su (Su is the area enclosed by C) when Su tends to zero.
 
   

Must be stated that the orientation to solve the line integral on curve C,  and the positive direction of ; must obey the rule of the right hand (or the screw). This is, if we were to put our or right index finger along C in the positive direction, our right thumb would tell us which was the positive direction in for the component of the rotational.

Lets again take as an example, the vector field constituted by the speed of water that moves throughout a canal. Only that this time, we will suppose that speed varies lineally with and only with depth; lets suppose that it is zero in the bottom of the canal and maximum at the surface. Without getting into much details, lets suppose that the force water can apply to an obstacle is proportional to the velocity.

Lets now imagine we have a thin wheel that can be held by an axis; and that in the perimeter of this wheel, we can put several small blades with the same distance between them; each one put perpendicularly to the wheel’s plane.

If we introduce this device in the canal with it’s axis in a vertical position; the pressure on both sides of the wheel would be equal and  there for the wheel wouldn’t spin. It would be the same as to believe that the lineal integral of the applied force by the water; throughout the perimeter was zero.  In other words, the vertical component of the rotational of the velocity would be zero.

If we introduce this device in the canal with its axis in a horizontal position and towards the velocity; water would pass through tangentially among the blades and wouldn’t pressure them. In mathematical terms, the force would be perpendicular to the path and logically, the circuitation would be equal to zero. In this case, the component parallel to the velocity of the rotational of the force would also be zero.


If we now introduce the wheel with its  axis horizontally and in a perpendicular direction to velocity, water would pressure the upper blades more than the lower ones; because in the upper part, the speed of water is greater. In this case, the wheel would spin, the circuitation would be different from zero an we would have a rotational component in this direction, also different from zero.

In all three systems of coordinates that we have used, the rotational is given by the following expressions,  


Rectangular:

       
(47)


It may also be written the following way by using the rules to calculate a determinant


       
(48)


Cylindrical:

        
(49)



Spherical:

        (50)
 



1.6.4 Laplacian.


A double operator that is very useful is the Laplacian. Here, we will only refer to the scalar Laplacian, which operates on a scalar function and produces another scalar function. This operator, is indicated and define as it goes,


     
(51)

In different systems, we would have the following expressions for the scalar Laplacian:


Rectangular:

    
  (52)


Cylindrical:

   
(53)


Spherical:

    (54)



 

 
     
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