{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

finalrecommend

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

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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 (
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern