"It’s one thing to check that the derivatives of a function are zero and another to feel the plaster taper to a sharp point."
Those are two very different things. A sharp point is not differentiable. A derivative of zero indicates a possible minimum or maximum of the function.
No, they are the same thing. What this sentence is referring to is the vanishing of the Jacobian determinant [1] (which is defined using the derivatives of the defining equations).
A simple example is the equations y^3 - x^2 = 0. This is a "cusp" (use wolfram alpha to see what it looks like) and has a singularity at the origin.
The jacobian is the matrix:
[ -2x, 3y^2 ]
This has rank 1 unless x and y are zero in which case it has rank zero. The fact that the rank is less than 1 indicates a singularity.
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I only read your comment and article but didn't see the video, so this comment may be off.
A simple example is the equations y^3 - x^2 = 0. This is a "cusp" (use wolfram alpha to see what it looks like) and has a singularity at the origin.
The jacobian is the matrix:
[ -2x, 3y^2 ]
This has rank 1 unless x and y are zero in which case it has rank zero. The fact that the rank is less than 1 indicates a singularity.
[1]: http://en.wikipedia.org/wiki/Singularity_(mathematics)#Algeb...