This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*
**Unformatted text preview: **Math 260, Spring 2012 Jerry L. Kazdan Class Outline: Feb. 28, 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. The Second Derivative Test for Maxima, Minima, and Saddles 2. Examples http://www.math.upenn.edu/ ∼ kazdan/260S12/notes/max-min-notesJan09/max-min.pdf 3. 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 x 2 . This is not pathological since after expanding, f ( x, y ) = y 2 − 3 x 2 y + x 3 so at the origin the second derivative matrix is not invertible. The second derivative test fails unless the second derivative matrix at that critical point is invertible. 4. Nonlinear maps F : R n → R k . Intuitive Examples Velocity V ( x, y, z, t ) of a fluid in a region Ω ⊂ R 3 Gravity force field. A factory. The derivative, F ′ ( X ) at X as the best linear approximation F ( X ) − [ F ( X ) + F ′ ( X )( X − X )] bardbl X − X bardbl → . Computational Examples 1 a) The function w = x 2 1 + x 2 2 whose graph is a paraboloid, is a map from R 2 into R 1 . It can also be regarded as a map from R 2 into R 3 by a useful artifice. Let y 1 = x 1 , y 2 = x 2 , and y 3 = w = x 2 1 + x 2 2 . Then y 1 = x 1 y 2 = x 2 y 3 = x 2 1 + x 2 2 is a map F from R 2 into R 3 ....

View
Full Document