+ ORTHOGONAL and RIGHT-HANDED coordinates:
rectangular ( ) ( , , )
cylindrical ( ) ( , , )
Example 1.8: Position vector
Example 1.9: Application of position vector
On the xy plane, (r in text) -- radius of a circle AND -- angle between x and axes.
spherical ( ) ( )
(R in text) -- radius of a sphere, -- angle between and z axes.
Note: , , and are not pointing in fixed directions. They vary with angles.

+ Position vector : a vector from the ORIGIN to a POINT, e.g. cylindrical coordinates: .
Scalar field, e.g. or
Vector field, e.g.

+ Application of dot product: find component of a vector in certain direction, e.g. find , component of in direction of , i.e.
Given two vectors and , with .

+ Coordinates point conversion: identity coordinate systems and apply formula accordingly.

+ Coordinate transform: convert unit vectors and then the coordinate axis variables.
E.g. convert in cylindrical system into rectangular system such that .
convert unit vectors .
convert axis variable
Example 1.10: Coordinate transform

+ Why coordinate transform?
Some operation is not obvious in certain coordinates, e.g. .
Operation with vectors in mixed coordinate systems, e.g. where is in spherical coordinates.

+ Differential line elements, surface elements, and volume elements
All differential line elements have unit length. For angles , need to multiple with certain radii to get arc length of corresponding circles,
e.g. in cylindrical sys., in spherical sys.
_________________________________

HW #3	due 2/1/07