M2521608 - 16.8.10 Defining a Saddle Point (SP) The Harper...

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16.8.9 PART E: FOOTNOTES Extending the 2 nd Derivative Test If you have a nice function of n variables, you will construct an n x n real symmetric matrix consisting of n th-order partial derivatives; such a matrix only has real eigenvalues. When classifying a critical point (CP), we consider the signs of the determinants of all the upper left square submatrices (1 x 1, 2 x 2, etc.). • If they are all positive, the matrix is called positive definite, and all of its eigenvalues are positive. 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. The 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.
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Unformatted text preview: 16.8.10 Defining a Saddle Point (SP) The Harper Collins Dictionary of Mathematics : A point on a surface that is a maximum in one planar cross-section and a minimum in another. Visualizing a hyperbolic paraboloid helps. The definition may vary. Are degenerate "ties" allowed along a cross-section, like for horizontal lines? Also, for example, are the points along the y-axis saddle points if we have the graph of the "snake cylinder" f x , y ( ) = x 3 ? Orientation of axes: < x y That's debatable. Using the Harper Collins definition, I don't believe they would be; the thing just doesn't look like a "saddle" along the y-axis. But it is true that there are higher and lower points "immediately around" those points. Incidentally, D = 0 everywhere for this function, so the 2 nd Derivative Test says nothing. See: http://en.wikipedia.org/wiki/Saddle_point...
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M2521608 - 16.8.10 Defining a Saddle Point (SP) The Harper...

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