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CALCULUS 265 Honors
Fall Semester 2000
8:00 to 8:50 MTTF
Carver 0002

Instructor: Hentzel
Office: 432 Carver
Office Hours: 9:00-10:00 MTTF

E-mail  hentzel@iastate.edu
phone  294-8141
Website

http://www.math.iastate.edu/hentzel/honors.265

Monday, October 30:   Test Tomorrow 


Main idea:  TEST on Tuesday, October 31.


F[x,y] = (x^2 y, - y^2)

The path is the rectangular path (0,0) to (4,0) to (4,1) to (0,1) to (0,0) 

1.  Use Green's theorem to evaluate INT F.T ds along the path.

x=4 y=1
INT INT  -x^2 dy dx = -64/3
x=0 y=0





2.  Use Green's theorem to evaluate INT F.N ds along the path.


x=4 y=1
INT INT  2 x y - 2 y dy dx  = 4
x=0 y=0





F[x,y,z] = ( Sin[x],  -y Cos[x], x+y+z )


Surface is the top, bottom, and curved walls of the cylinder

x^2+y^2 = 1 from z=0 to z=4.  N is the outward directed normal.



3.  Find   INT INT F.N dS.

INT INT INT Cos[x]-Cos[x]+1 dV = 4 Pi 





Assume F(x,y,z) =  (y, -x, 8 ).

S is that portion of the paraboloid z = 9 - x^2 -y^2 above the xy-plane

and N is directed upwards.   

4.  Evaluate  INT INT  F.N dS


S = (x, y, 9 - x^2 - y^2)

    |  i        j            k        |
    |  1        0           -2x       |  = (2x, 2y, 1)
    |  0        1           -2y       |

                       (2x,2y,1)
INT INT  (y,-x,8). ------------------   Sqrt[4x^2+4y^2+1] dA
                    Sqrt[4x^2+4y^2+1]


INT INT 8 dA = 72 Pi



Let C be the intersection of the plane y=z with the cylinder x^2 + y^2 = 1.

(C is an ellipse oriented clockwise as viewed from above.)

Let F(x,y,z) = (-Cos[z], x Sin[y], Sin[y]-2x ).


5.  Evaluate INT F.T ds





         i            j                k
        d/dx         d/dy             d/dz
       -Cos[z]     x Sin[y]         Sin[y]-2x


=       (Cos[y], 2+Sin[z],Sin[y])


Surface = ( r Cos[t], r Sin[t], r Sin[t] )

             i          j            k
            Cos[t]    Sin[t]       Sin[t]
           -rSin[t]   r Cos[t]    r Cos[t]


      = (0, -rCos[t]^2+r Sin[t]^2, rCos[t]^2+rSin[t]^2

      = (0,-r,r)

r=1 t=2Pi                            (0,-r,r)
INT INT     (Cos[y],2+Sin[z],Sin[y]) ----------  r Sqrt[2] dt dr
r=0 t=0                              r Sqrt[2]


INT INT      -2r -r Sin[z] +r Sin[y] dt dr

INT INT      -2r -r Sin[r Sin[t] ] + r Sin[r Sin[t] ] dt dr

INT INT       -2r dt dr

= -2 Pi






Previous Review Questions.

What type of things do these constructions calculate.

(a) INT F.T ds    
    curve


(b) INT F.N ds
    closed
    curve
    in plane


(c) INT INT F.N dS
    surface


(d) INT INT INT f(x,y,z) dV
    volume 


Green's Theorem has two forms;

(a)  What is the Divergence theorem form?



(b)  What is the Stoke's theorem form?



The Divergence theorem.

(a)  What does the Divergence theorem say in English?


(b)  Write out the Divergence theorem in mathematics.


Stoke's Theorem.

(a)  What does Stoke's theorem say in English?


(b)  Write out Stoke's theorem in mathematics.



Independence of path

(a) Which type of thing is independent of path, F.T ds or F.N ds?


(b) What property does F(x,y) or F(x,y,z) have which makes it

independent of path?  (Use the word "gradient" in yout answer.)



(c) How do you test to see if F is the gradient of something.


(d) How do you construct f such that grad g = F when such a function

f does exist?



Parameterization of lines.

If the line is given by  R(t) = (f(t),g(t))  or R(t) = (f(t),g(t),h(t)):

(a) what is ds?


(b) what is T?


(c) what is N?  (This question has meaning in several different

interpretations.  I can justify at least three answers). 



Parameterization of surfaces.

It the surface is given by S(u,v) = (f(u,v),g(u,v),h(u,v)

(a)  Why are there two parameters?  Why not just 1 or perhaps 3?


(b)  What is dS.


(c)  What is N.


(d)  What is T (This is a trick question.  Why does it not make sense?).



Confusing terminology for line integrals.  

(a)  What is INT F.T ds?


(b)  How does one get  INT f dx + g dy  +h dz  from INT F.T ds?


(c)  How does one get INT F.dr from INT F.T ds?





Surface of a sphere, top half.

(a) Express the surface by parameterization in Cartesian coordinates.


(b) Express the surface by parameterization in cylindrical coordinates. 


(c) Express the surface by parameterization in spherical coordinates.


(d) Express as a level surface.


When a surface is expressed as a level surface, how do you get the normal?






Surfaces.  How do you write the surface of a cone?


(a) in Cartesian coordinates?


(b) in cylindrical coordinates?


(c) in spherical coordinates?

 



Key words:   INT F.T ds   
             INT INT F.N ds   
             INT INT curl F.N dS
             INT INT INT  grad F d V 

             Stokes Theorem, Divergence theorem



Example 1 page 1059.

Find INT F.T ds along the boundary of the triangle in the

coordinate planes of 2x+2y+z = 6.

The force is  F = (-y^2, z, x)


You can use the three line segments or you can compute the curl and

integrate the curl F.N over the "planer" surface.




Example 2:   F(x,y,z) = (2z,x,y^2)

Find    INT F.T ds around the circle in the x,y plane of radius 4.


Use Stokes theorem for 

(a)  the paraboloid z = 4-x^2-y^2 

(b)  the circle

(c)  the sphere.