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Math 225: Calculus III	Quiz 8    4/2&4/4
:[font = input; initialization; preserveAspect]
*)
<<Calculus`Math225`;
i = {1,0,0}; j = {0,1,0}; k = {0,0,1};
(*
:[font = subsubsection; inactive; preserveAspect; startGroup]
1. Compute the Jacobian determinant of x and y with respect to u and v for the change of variables u = x y and v = x/y where x > 0 and y > 0.
;[s]
13:0,0;39,1;40,0;45,1;46,0;99,1;106,0;111,1;118,0;125,1;130,0;135,1;140,0;142,-1;
2:7,13,9,Times,1,12,0,0,0;6,13,10,Courier,1,12,0,0,0;
:[font = subsubsection; inactive; preserveAspect; endGroup]
Solution
:[font = text; inactive; preserveAspect]
First solve for x and y in terms of u and v.
;[s]
9:0,0;16,1;17,0;22,1;23,0;36,1;37,0;42,1;43,0;45,-1;
2:5,13,9,Times,0,12,0,0,0;4,13,10,Courier,1,12,0,0,0;
:[font = input; preserveAspect; startGroup]
Solve[{u == x y, v == x/y}, {x,y}]
:[font = output; output; inactive; preserveAspect; endGroup]
{{x -> -(u^(1/2)*v^(1/2)), y -> -(u^(1/2)/v^(1/2))}, 
 
  {x -> u^(1/2)*v^(1/2), y -> u^(1/2)/v^(1/2)}}
;[o]
                                  Sqrt[u]
{{x -> -(Sqrt[u] Sqrt[v]), y -> -(-------)}, 
                                  Sqrt[v]
 
                              Sqrt[u]
  {x -> Sqrt[u] Sqrt[v], y -> -------}}
                              Sqrt[v]
:[font = text; inactive; preserveAspect]
Step by step: u = x y and v = x/y implies that u v = x^2 and u/v = y^2, so
;[s]
9:0,0;14,1;21,0;25,1;33,0;47,1;56,0;61,1;70,0;75,-1;
2:5,13,9,Times,0,12,0,0,0;4,13,10,Courier,1,12,0,0,0;
:[font = input; preserveAspect]
x = Sqrt[u v]; y = Sqrt[u/v];
:[font = input; preserveAspect; startGroup]
derivs = {{D[x,u], D[x,v]},{D[y,u],D[y,v]}};
derivs//MatrixForm
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{v/(2*(u*v)^(1/2)), u/(2*(u*v)^(1/2))}, 
 
   {1/(2*(u/v)^(1/2)*v), -u/(2*(u/v)^(1/2)*v^2)}}]
;[o]


     v              u
-----------    -----------
2 Sqrt[u v]    2 Sqrt[u v]

     1              -u
-----------    ------------
       u              u   2
2 Sqrt[-] v    2 Sqrt[-] v
       v              v
:[font = input; preserveAspect; startGroup]
Det[%]
:[font = output; output; inactive; preserveAspect; endGroup]
-((u/v)^(1/2)*(u*v)^(1/2))/(2*u*v)
;[o]
       u
-(Sqrt[-] Sqrt[u v])
       v
--------------------
       2 u v
:[font = text; inactive; preserveAspect]
This simplifies to -1/(2v). In one step:
;[s]
3:0,0;18,1;27,0;41,-1;
2:2,13,9,Times,0,12,0,0,0;1,13,10,Courier,1,12,0,0,0;
:[font = input; preserveAspect; startGroup]
?J
:[font = print; inactive; preserveAspect; endGroup]
J[f,g,{u,v}] gives the Jacobian determinant of the coordinate 
   transformation x = f(u,v), y = g(u,v). J[f,g,h,{u,v,w}] gives  the
   Jacobian determinant of the coordinate transformation  x = f(u,v,w),
   y = g(u,v,w), z = h(u,v,w).
:[font = input; preserveAspect; startGroup]
J[Sqrt[u v], Sqrt[u/v], {u,v}]
:[font = output; output; inactive; preserveAspect; endGroup]
-((u/v)^(1/2)*(u*v)^(1/2))/(2*u*v)
;[o]
       u
-(Sqrt[-] Sqrt[u v])
       v
--------------------
       2 u v
:[font = subsubsection; inactive; preserveAspect; startGroup]
2. Find the volume of the portion of the solid sphere
x^2 + y^2 + z^2 <= 4 that lies above the upper nappe of the cone
z^2 == x^2 + y^2.
;[s]
5:0,0;54,1;74,0;119,1;135,0;137,-1;
2:3,13,9,Times,1,12,0,0,0;2,13,10,Courier,1,12,0,0,0;
:[font = subsubsection; inactive; preserveAspect; endGroup]
Solution
:[font = text; inactive; preserveAspect]
The cone z^2 = x^2 + y^2 has the equation phi = Pi/4 in spherical coordinates, and the sphere of radius 2 has the equation rho = 2. The solid is thus defined by the inequalities
    0 <= rho <= 2
    0 <= phi <= Pi/4
    0 <= theta <= 2Pi
The volume is given in spherical coordinates by (the volume element is rho^2 Sin[phi]).
;[s]
13:0,0;9,1;24,0;42,1;52,0;104,1;105,0;123,1;130,0;178,1;239,0;310,1;324,0;327,-1;
2:7,13,9,Times,0,12,0,0,0;6,13,10,Courier,1,12,0,0,0;
:[font = input; preserveAspect; startGroup]
Integrate[rho^2 Sin[phi],{theta,0,2Pi},{phi,0,Pi/4},{rho,0,2}]
:[font = output; output; inactive; preserveAspect; endGroup]
(8*(2 - 2^(1/2))*Pi)/3
;[o]
8 (2 - Sqrt[2]) Pi
------------------
        3
:[font = text; inactive; preserveAspect]
Step by step:
:[font = input; preserveAspect; startGroup]
Integrate[rho^2 Sin[phi],{rho,0,2}]
:[font = output; output; inactive; preserveAspect; endGroup]
(8*Sin[phi])/3
;[o]
8 Sin[phi]
----------
    3
:[font = input; preserveAspect; startGroup]
Integrate[%,{phi,0,Pi/4}]
:[font = output; output; inactive; preserveAspect; endGroup]
8/3 - (4*2^(1/2))/3
;[o]
8   4 Sqrt[2]
- - ---------
3       3
:[font = input; preserveAspect; startGroup]
Integrate[%,{theta,0,2Pi}]
:[font = output; output; inactive; preserveAspect; endGroup]
(8*(2 - 2^(1/2))*Pi)/3
;[o]
8 (2 - Sqrt[2]) Pi
------------------
        3
:[font = input; preserveAspect; startGroup]
N[%]
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4.90747298405658728
;[o]
4.90747
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:[font = input; preserveAspect]
bot = ParametricPlot3D[{r Sin[Pi/4]Cos[t],r Sin[Pi/4]Sin[t],
r Cos[Pi/4]},{r,0,2},{t,0,2Pi}]
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Show[top,bot,ViewPoint->{2.767, 1.813, 0.713}]
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