Math 127(3) - Warren D. Smith - Homework #4 - due Tues 18 Feb 2003
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Start reading ch 15.

Upcoming quiz: will be Tues feb 18 (same day this HW due).
Will cover: everthing up to and including ch 14. Also will
cover just a little of ch 15, namely the idea of "gradient"
of a real-valued function of a vector, and the idea
that optimizing it requires setting  gradient = 0  
as a vector equation.

                                     3   2
1:  find the gradient of F(x,y,z) = x + y   
and the directional derivative in the direction 
(which is a unit vector) (4,3,0)/5.

                                              2   3
2:   Same things, but for   F(x,y,z) = exp(y z / x ).

                                        6    7
3:   Same things, but for   F(x,y,z) = x  + y  + sin(z) x.

                                                        n
4: See eq 15.1.5 p950 in book, saying the gradient of  r ,
where r = |(x,y,z)|, is
                 n             n-2
           grad(r )  =     n  r    (x,y,z).
Prove this is true for any n.
                                             n
5: For which n is  the function  f(x,y,z) = r
"harmonic"?  (Defn of "harmonic" is in exercises 14.6 p939, see ex 21
"Laplace's equation".  A Harmonic function is a function such that the
unmixed 2nd partials add up to 0, i.e. derive with respect to x twice,
or y twice, or z twice, add up the 3 results, get 0.)

                         2            2
6. Let F(x,y) = 3 + x - x + x y  - 2 y  - 8 y .

What is the maximum possible value of F(x,y), and at what (x,y)
pair does it happen?
                                                           3
7. What is the (signed) curvature of the plane curve  y = x  
as a function of x?