What Math Equations Look Like in 3-D(wired.com)
wired.com
What Math Equations Look Like in 3-D
http://www.wired.com/2014/06/math-equations-models/
5 comments
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.
[1]: http://en.wikipedia.org/wiki/Singularity_(mathematics)#Algeb...
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...
This seems to explain the sentence. Thank you, now I have something to read about for the next few hours.
In 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.
I only read your comment and article but didn't see the video, so this comment may be off.
Fixed headline: "This is what 3d graphs of math equations look like".
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.