Implicit Tangents

There is a small twist in the microscopic view of an implicit graph.

The contour graph of the local linear approximation

  with gradient

at a fixed where the gradient is not zero appears the same as a highly magnified view of the nonlinear graph focused at the specific point  .

If , the implicit graph of is the whole -plane.  The microscope is broken!

A non-smooth implicit set

The function    is smooth for all    and  , with .  The implicitly defined solution set  g[x,y] = 0  consists of the  x-axis (y=0)  and the  y-axis  (x=0).  The gradient  ∇g[x,y]  is nonzero everywhere except at  (x,y) = (0,0).  The solution set is not a smooth curve at this point, but two crossing curves.

Created by Mathematica  (July 2, 2004)