Math 127(3) - Warren D. Smith - Homework #4 - due Tues 18 Feb 2003
------------------------------------------------------------------
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   
         2
ANS = (3x  , 2y , 0)
and the directional derivative in the direction 
(which is a unit vector) (4,3,0)/5.
ANS =      2
      (12 x  + 6y + 0)/5
                                              2   3
2:   Same things, but for   F(x,y,z) = exp(y z / x ).

               2 -4   2  -3          -3         2   3
gradF = ( -3y z x  , z  x  ,  2 y z x  ) exp(y z / x ) = vector.

               2 -4    2  -3            2   3
DD =    (-12y z x  + 3z  x   +0) exp(y z / x ) / 5 = scalar.

                                        6    7
3:   Same things, but for   F(x,y,z) = x  + y  + sin(z) x.
            5             6
gradF = (6 x + sin(z), 7 y  , cos(z) x)
               5                6                         
DD =    = (24 x + 4sin(z) + 21 y  + 0) / 5
                                                        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.
ANS.
               2   2   2
r= SquareRoot(x + y + z ) is a scalar length of (x,y,z) [which is a
vector].  We are here taking the gradient-vector of the nth power of r (a scalar).

note as a special case n=1:
          (2x, 2y, 2z)
grad(r) = ------------
               r
the general case is:
      n            2   2   2 n/2             2 n/2-1               n-2
grad(r ) = grad( (x + y + z )    ) = (n/2) (r )    (2x,2y,2z) = n r   (x,y,z).

                                             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.)

ANS.   The "laplacian" is the sum of the unmixed 2nd partials.
These are (by applying the product rule to take the deriv of each
component in the above gradient [which is list of 1st partials])
                n              n-4                 n-2
     Laplacian(r ) =  (n-2) n r    (xx+yy+zz) + n r    (1+1+1).

                               n-2        n-2
                   =  (n-2) n r    + 3 n r .
We have harmonicness if this is 0 everywhere, so that (in 3 dimensions) either:
   * n=0
   * or n is nonzero and   n - 2 + 3 = 0   so   n = -1.
In D dimensions, the answer would be n=0 or n-2+D=0 so n=2-D.

                         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?

gradF = (1-2x+y,  x-4y-8)

this is (0,0) when   x = -4/7,  y = -15/7.  
This leads to   F(-15/7, -4/7) = 79/7 = 11.28571429
This is a max. Sanity checks: 
 F(-4/7+0.001 ,-15/7+ 0.0034) = 11.28569356
 F(-4/7-0.001 ,-15/7+ 0.0034) = 11.28568676
 F(-4/7-0.001 ,-15/7- 0.0034) = 11.28569356
each are indeed a little smaller than the max.

                                                           3
7. What is the (signed) curvature of the plane curve  y = x  
as a function of x?

ANS. Use curvature formula      
          F"(x)                 6 x                6 x
curv = ----------------- = ------------------ = ---------------
                  2  3/2             2 2  3/2            4  3/2
       ( 1 + F'(x)  )      ( 1 + (3 x )  )      ( 1 + 9 x  )

(the absolute value of this would be the unsigned curvature).