Tutorial03-solutions

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Unformatted text preview: ues. Moreover, note that tr H = 4 + 3 + 6 = 13, which we know also equals to the sum of the three eigenvalues of H . If H has two negative eigenvalues, then the only positive eigenvalue of H must be greater than 13. Now, it’s easy to check that φ(13) < 0. This together with the observation that lim +∞ φ( ) = −∞ implies that φ( ) = 0 can not just have one positive root > 13. Hence H must have no negative eigenvalues. 1 :) Thus the Morse index is 0 and 2 1 1 is a local minimum. 3 E Questions on logical thinking Consider the function ( )= 2 2 2 2 + 0 2 + + 2 2 =0 =0 In the last tutorial we showed that the restriction of this function to a line parallel to any coordinate axis, is continuous on the entire real line, while on the other hand the unrestricted function ( ) is not continuous at (0 0) (as it has no limit at this point). (a) Find the partial derivatives (b) Is the second derivative S and . (0 0) defined? (a) It’s easy to see that 2 ( 2 − 2) ( 2 + 2 )2 2 ( 2 − 2) = ( 2 + 2 )2 = Note that ( 0) = (0 2 2 + + 2 2 =0 =0 ) = 0 by definition, so it follows that 4 (0 0) = (0...
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This document was uploaded on 02/10/2014.

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