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ELECTROMAGNETIC FIELDS AND THEORY: Rectangular Coordinates

1.2.1 Rectangular

In this system, as indicated in figure [1], any point P can be identified using it’s three coordinates,

xp , yp , zp    (1)

 
     
 

       
   


The unitary vectors that indicate the direction in which the coordinates grow are:

       (2)

It’s important that the three axis are oriented toward space in a way that

       (3)

the differential displacement along the mentioned directions will be,

dx , dy , dz   (4)

a generic displacement of point P


      
  (5)

In figure [2], a differential volume element is shown as well as the differential elements of area that encloses it. The differential element of volume is equal to,

dV = dx dy dz   (6)

 


the perpendicular faces to axis z have a differential area equal to,

dSz = dx dy   (7)

frequently, the elements of area are expressed with a vector quantity, in which case, the module of the vector is the element’s area and the direction of the vector is along the  perpendicular and oriented towards the exterior side. For example, the area element furthest from the center and perpendicular to axis z will have the following vector representation,


(8)

it’s opposite element will be

(9)

 

 
     
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