Maple #4 is due Thursday 5 October 2000

For FULL credit STAPLE your sheets together.
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Each plot must be rotated to a "nice" position and "look good (smooth)".
Each plot must have include axes and the title must include your name
and what was plotted. Note that the expressions below are not
necessarily in a form that Maple likes. Be sure to answer the questions
in #3.

1. Use "arrow(s)" to illustrate how the vector addition is commutative,
that is u + v is the same as v + u. (Pick u = <2, 4> and v= <5, 1>
and draw 5 arrows: u, v starting at the end of u, u+v, v, and u starting
at the end of v.  [hint: with(plottools) and display.]
2. Plot the plane P with equation, 3x-2y+5z = 12 and an normal to P
on the same plot. Have the normal plotted as an arrow with its base
at the point on the plane nearest the origin.
3. The following function is from a Matlab demo.
f(x,y)= 3(1-x)^2 e^(-x^2-(y+1)^2)
    - 10(x/5-x^3-y^5)e^(-x^2-y^2) -(1/3)e^(-(x+1)^2-y^2).
[Remember, either define e:=exp(1) or replace e^x with exp(x). A common
error is to replace e^x with exp^x, and unfortunately Maple doesn't
notice the error, and does something dumb.] Have Maple find
the both partials of f, f_x and f_y [Read f_x as f sub x.] and all
four second partials of f, f_xx, f_xy, f_yx and f_yy.] [Hint let
F:=be the expression for the function, and just do diff(F,...)]
Is f_xy = f_yx?
4. Using the matlab demo function, draw the graph together with 
the two curves we get when we hold x = the constant 1/5 and when we
hold y = the constant 3/2. [This time define F to be the "Maple"
function F:=(x,y)->ugly expression. (NOTE not F(x,y):= ugly expression.)
and do spacecurves [1/5,y,F(1/5),y],y=-3..3 and [x,3/2,F(x,3/2)],x=-3..3
with large thickness.]
5. Use Maple to find the projection of the vector u in the v direction.
Do the calculation once with u:=vector([1,2,3]) and v:=vector([3,1,3])
and once with u:=vector([u1,u2,u3]);v:=vector([v1,v2,v3]). [Hint
remember norm is norm(v,2).]