Points and Their Classification

Each topological feature of p(r), whether it be a maximum, a minimum or a saddle, has associated with it a point in space called a critical point, where the first derivatives of p(r) vanish. Thus at such a point, denoted by the position vector r., Vp(rc) = 0, where Vp denotes the operation 

Whether a function is a maximum or a minimum at an extremum is, of course, determined by the sign of its second derivative or curvature at this point. 

In general, for an arbitrary choice of coordinate axes, one will encounter nine second derivatives of the form d p/dxdy in the determination of the curvatures of p at a point in space. Their ordered 3x3 array is called the Hessian matrix of the charge density, or simply, the Hessian of p. This is a real, symmetric matrix and as such it can be diagonalized. The new coordinate axes are called the principal axes of curvature. The trace of the Hessian matrix, the sum of its diagonal elements, is invariant to a rotation of the coordinate system. Thus the value of the quantity V p, called the Laplacian of p, 

There are just four possible signature values for critical points of rank three. They are  

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List stationary points and their classification (maximum, minimum, saddle point) of... 

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