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 http://mathforum.org/dr.math/
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