MATH 1AA3 Chapter Notes - Chapter 16: Harpercollins, Paraboloid, Symmetric Matrix
30 Jul 2018
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If you have a nice function of n variables, you will construct an n x n real symmetric matrix consisting of nth-order partial derivatives; such a matrix only has real eigenvalues. The cp corresponds to a local min: if they alternate in sign from negative to positive, etc. , the matrix is called negative definite, and all of its eigenvalues are negative. Cp corresponds to a local max: if they are all nonzero, and neither of the two above configurations occur, then the cp corresponds to a saddle point (sp). Observe that the notes on 16. 8. 4 are consistent with all of this. A point on a surface that is a maximum in one planar cross-section and a minimum in another. Also, for example, are the points along the y-axis saddle points if we have the graph of the "snake cylinder"
