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Date: 03/01/99 at 11:43:34
From: Eric Barning
Subject: Singularity

What is a singularity?

Date: 03/01/99 at 14:14:47
From: Doctor Rob
Subject: Re: Singularity

A little more context would be helpful here, because the meaning may 
be different in different settings.

In the case of a curve in the plane given by an equation F(x,y) = 0,
a singularity is a point where the tangent line is undefined. It can
be a point such that the partial derivatives F_x and F_y both vanish.
The tangent plane at a point (x0,y0) on the curve where both partials
exist and not both are zero is given by the equation

   F_x(x0,y0)*(x-x0) + F_y(x0,y0)*(y-y0) = 0

If both partials are zero at (x0,y0), then this equation reduces to
0 = 0, and is not the equation of a plane at all.

As an example, consider the curve F(x,y) = y^2 - x^3 - x^2 = 0. Then
the partial derivatives are F_x = -3*x^2 - 2*x, F_y = 2*y, which both
vanish when x = 0 and y = 0, which is a point on the curve. (They also
both vanish at x = -3/2, y = 0, but that is not on the curve.) Then
(0,0) is a singularity. If you draw the curve, you will see that at
this point the curve crosses itself, and that there are two distinct
tangents there, so *the* tangent there is undefined.

Another way that a singularity can occur is for one or more of the
partial derivatives not to exist at a point on the curve. As an 
example, one could have

   F(x,y) = y - x*sin(1/x) if x is nonzero
   F(x,y) = y if x = 0

F(0,0) = 0, so (0,0) is on the curve F(x, y) = 0, and F(x,y) is
continuous at (0,0). F_x does not exist at x = 0, so the curve has a
singularity at (0,0).

In the case of a surface in 3-space given by an equation F(x,y,z) = 0,
a singularity is a point where the tangent plane is undefined. It can
be a point such that all three partial derivatives F_x, F_y, and F_z
vanish, or where one or more of them is undefined. Curves in 3-space
can also have singularities. They are defined as the intersection of
two surfaces, F(x,y,z) = 0 and G(x,y,z) = 0. Any point satisfying both 
equations for which the tangent line is undefined is a singularity.

This can be extended to k-dimensional surfaces in n-space, where
0 < k < n. This can also be extended to other kinds of spaces besides
Euclidean n-space, such as 4-dimensional Lorentzian space-time.

I hope this helps. If this is not the kind of singularity you meant,
please write back with more context.

- Doctor Rob, The Math Forum   
Associated Topics:
High School Analysis

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