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ELECTROMAGNETIC FIELDS AND THEORY: FIELDS AND VECTORS: Spherical Coordinates

1.2.3 Spherical.

In figure [5], a point P will be identified throughout the coordinates,

rp    ,    θp        ,    φp                (19)

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

(20)

none of the unitary directions remains constant in all the point and must be,

(21)

the differential displacements along the above directions will be,

dr                r dθ             r senθ dφ           (22)

a generic displecement of point P,

(23)

In figure [6], 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  =  r2  senθ  dr  dθ  dφ                (24)

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

dSr =  r2  senθ  dθ  dφ           (25)

the area of the faces perpendicular  to direction θ,

dSθ =  r  senθ  dφ  dr          (26)

and the area  of faces perpendicular to to φ direction,

dSφ = r  dθ  dr         (27)

Like all the other systems of coordinates, any element of area described can be expressed through a vector with a magnitude equal to the area and with a direction perpendicular to the element.

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