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Unformatted text preview: Math 260, Spring 2012 Jerry L. Kazdan Problem Set 8 Due : Never Unless otherwise stated use the standard Euclidean norm . 1. Find a 3 3 symmetric matrix A with the property that h X, AX i =- x 2 1 + 4 x 1 x 2- x 1 x 3 + 2 x 2 x 3 + 5 x 2 3 for all X = ( x 1 , x 2 , x 3 ) R 3 . 2. Let A and B be n n real matrices. Show that ( AB ) * = B * A * . 3. Consider the surface x 2 + y 2- 4 z 2 = 1 . a) Find a vector orthogonal to the tangent plane at the point (1 , 2 ,- 1). b) Find the equation of the tangent plane to this surface at this point. 4. a) Let u ( x ) be a smooth solution of u 00 + 17 u ( x )- c ( x ) u = 0 for x R and assume that c ( x ) &gt; 0 everywhere. Show that u ( x ) cannot have a positive local maximum at any x R . b) Let u ( x,y ) be a smooth solution of u xx +3 u yy- c ( x,y ) u = 0 for ( x,y ) R 2 , where c ( x,y ) &gt; 0 everywhere. Show that u ( x,y ) cannot have a positive local maximum at any ( x,y ) R 2 ....
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- Spring '12
- Math, Continuous function, Manifold, Euclidean space