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"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 thin
by MrMeker 12y ago
"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.
- arithma 12y agoIn a parametric curve, a derivative of zero may indicate a cusp, which I feel is where the comparison is coming from. I only read your comment and article but didn't see the video, so this comment may be off.
- ansjmels 12y agoNo, 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. [1]: http://en.wikipedia.org/wiki/Singularity_(mathematics)#Algebraic_geometry_and_commutative_algebra http://en.wikipedia.org/wiki/Singularity_(mathematics)#Algeb...
- MrMeker 12y agoThis seems to explain the sentence. Thank you, now I have something to read about for the next few hours.