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ELECTROMAGNETIC FIELDS AND THEORY: FIELDS AND VECTORS: Cylindrical Coordinates

1.2.2 Cylindrical

Another system, as indicated in figure [3], here a point P can be identified by using coordinates

rp , φp , zp  (10)

the unitary vectors that indicate the direction in which the coordinates grow are,

(11)

Now, the unitary directions aren’t constant in all the points, only coordinate z will always grow in the same direction. The directions will be so that,

(12)

the differential displacements along the above directions will be,
dr , r dφ , dz  (13)

a generic displacement for point P,

(14)

In figure [4], a differential volume element is shown as well as the differential elements of area that enclose it. The differential element of volume is equal to,

dV = r dr dφ dz  (15)

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

dSz = r dφ dr  (16)

The area of the faces perpendicular to direction r,

dSr = r dφ dz  (17)

and the area of the faces perpendicular to direction φ,

dSφ = dr dz  (18)

Also, in this system of coordinates, any element of area can be expressed through a vector with a magnitude equal to the area and with a direction perpendicular to the element.