Coordinate Systems in 2D

General coordinate systems can be viewed as parametric graphs in the same dimension.  In 2D we have a vector function of two parameters or "coordinates" such as

Smoothness is still the conditon

where the vector    is small when the changes in input are small,    and  .  

The term    is a linear parametric equation with paramters   and  .  We plot linear coordinates by drawing the vectors    and    and then filling in parallel lines in the directions of these vectors,

The derivative approximation says that changes in the general coordinates equal a linear term plus a term that is small compared with the change, , when and ,

If we magnify to make the norm of the input change, , appear unit size the equation becomes,

Under sufficient magnification,the difference    appears the same as the vector    and  the magnified difference    appears the same as the vector  .

        Unit scale        Nonnlinear mag    Linear & nonlinear mag

Figure 2

The area of the parallelogram with sides    and    is the determinant  , so the oriented area of the linear sector is

The nonlinear area increment is related to this by

where    when    and  .  (When the microscope is powerful enough, the linear and nonlinear increments appear to coincide and we don't see  .)

This can all be done geometrically in the case of polar coordinates:

Created by Mathematica  (July 2, 2004)