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Feb23-12 - Math 260 Spring 2012 Jerry L Kazdan Class...

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Math 260, Spring 2012 Jerry L. Kazdan Class Outline: Feb. 2, 2012 Scalar Functions of Several Variables Reading: Marsden and Tromba, Vector Calculus , Chapter 2, Chapter 3.1–3.4 and Sections 8.1-8.2 in http://www.math.upenn.edu/ kazdan/260S12/notes/math21/math21-2012-2up.pdf 1. Taylor’s Theorem Generalization of the Best Linear Approximation. Motivation: the result for functions of one variable: f ( x ) = f ( x 0 ) + f ( x 0 )( x x 0 ) + 1 2 f ′′ ( x 0 )( x x 0 ) 2 + · · · 2. The Second Derivative Test for Maxima, Minima, and Saddles 3. Examples http://www.math.upenn.edu/ kazdan/260S12/notes/max-min-notesJan09/max-min.pdf 4. Quadratic Polynomials The second derivative test boils down to under- standing quadratic polynomials. Here are some notes that may be useful: http://www.math.upenn.edu/ kazdan/260S12/notes/quadratic/quadratic.pdf See also Section 3.3 of Marsden-Tromba. Interesting Example The seemingly innocuous function f ( x,y ) := ( y 2 x 2 )( y x 2 ) is an example of a function that has a local minimum at the origin if you approach the origin along any straight line y = cx . However the origin is not a local min since the function is negative in the region between the curves y = x 2 and y = 2
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