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# finalrecommend - f x,y,g x,y G x,y = v f x,y,g x,y are also...

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RECOMMENDED PROBLEMS - FINAL EXAM 1. Point Set Topology Prove rigourously that the set of matrices where A + A 2 is invertible is open in Mat n × n . 2. Limits Problem 1.19, page 158. 3. Continuity Using and δ s, show that if f and g are continuous real valued functions, then f + g and fg are also continuous. 4. Directional derivatives, gradient Consider the function f ( x, y ) = e - 3 x +2 y 2 x + 1 . (i) Calculate the gradient of f at (0 , 0) . (ii) Find the directional derivative of f at (0 , 0) in the direction u = i + j 2 . (iii) What is the unit direction for which the rate of increase of f at (0 , 0) is maximal? What is the rate of increase? 5. Pathological functions Problem 1.33 page 159. 6. Taylor polynomials Problem 3.7, page 390. 7. Chain rule Assume that u and v are harmonic conjugates and that f and g are also harmonic conjugates. Show that F ( x, y ) = u ( f (

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Unformatted text preview: f ( x,y ) ,g ( x,y )) , G ( x,y ) = v ( f ( x,y ) ,g ( x,y )) are also harmonic conjugates. 8. Functions of matrices Problem 1.31 page 159. 9. Critical points and second derivative test Problem 3.19, page 391. 10. Functions on compact sets. Find the global minimum and global maximum of the function f ( x,y,z ) = x 2 + y 2 + z 2-2 x-4 y-6 z over the compact set x 2 + y 2 + z 2 ≤ 15 . 1 11. Lagrange multipliers Problem 3.22, page 392. 12. Inverse function theorem Problem 2.30, page 282. 13. Implicit function theorem Problem 3.11(a), page 390. Also calculate the derivative D 1 g r ( r, 0) . 14. Manifolds Problem 3.1, page 389. 15. Tangent spaces Problem 3.4, page 389....
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