2-4.3 Spherical Coordinates

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2-4.3 Spherical Coordinates
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• A vector in spherical coordinates
• The local base vectors from a right handed system

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• The differential length
• The differential areas are
• The differential volume

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On many occasions the differential areas are
vectors
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Table 2-1 Basic Orthogonal Coordinates

• Cartesian Cylindrical
  Spherical

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• Cartesian coordinates
• and are vectors.
• is a scalar.

Differential displacement Differential normal
area Differential volume
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• The differential surface element may be
  defined as
• we need to remember only !

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• Cylindrical coordinates

Differential displacement
Differential normal area
Differential volume
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• Coordinate transforms
• Example 2-11. Convert a vector in spherical
  coordinates (SPC)
• 
  
• into the Cartesian coordinates (CRT).
• Solution. The general form of a vector in the
  CRT is
• We need
• In fact

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• The other eight dot-products can be worked out.
• A faster and better way to represent the
  transformation is based on the del operator.

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• Example 2-12
• Sphare chell
• ra2 cm
• rb5 cm
• The charge density
• Find the total charge Q

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• Solution

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2-5 Integrals Containing Vector Functions

• .

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• The line integral around a close path C is
  denoted as
• In the Cartesian coordinates (CRT)

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• Example 2-13
• a) along the straight line OP, where P(1,1,0)

y
P(1,1,0)
P1
x
0
P2
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• b). Along path OP1P
• Solution. Using (2-52) of cylindrical
• a).