(*
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 

Thursday, October 5:    Section  14.2
Friday, October 6 also

p1006:5,9,13
p1007:41,43
p1008:51,55

Main idea:  Testing what the book is saying. 

Key words:  Stokes theorem, Divergence theorem. Green's Theorem
            F.T ds  F.N ds   F.dr  Pdx+Qdy+Rdz	 

Goal:  Let us just try one to see what happens. 

Remark:  There are three types of problems.  

(a)  Simply weight each section of wire.   INT w ds.  

where w tells how much extra the particular segment ds should be

counted.

(b)  INT F.T ds gives the work done.   F is the force.

(c)  INT F.N ds gives the material oozing through, crossing somehow

the boundry segment ds.  F is the flow.


Of course, since the dot product reduces vectors to numbers, 

F.T and F.N are actually also qualify to be considered in form (a).


But I find it easier to keep these three problems as different 

concepts.  Now there are several notations in your book and in

all books which make this section incomprehensible.  The other

two notations are F.dr and Pdx+Qdy+Rdz.  There are probably more,

there seems to be an endless supply of notations.  


 






 

Integrate over the region  y = x^2  -1<=x<=1. and the top y=1, -1<=x<=1.


F[x,y] =   (x+y, x^2+y^2)
__
\/.F = 1+2y

__     |  i     j       k  |
\/xF = | d/dx  d/dy   d/dz | = (0,0,2x-1)
       | x+y  x^2+y^2   0  |

    __
INT \/.F dA = Integrate[1+2 y,{x,-1,1},{y,x^2,1}] = 44/15 

    __
INT \/xF.N dA = Integrate[2x-1,{x,-1,1},{y,x^2,1}] = -4/3




The two paths are (t,t^2)  -1<=t<=1       and         (-t,1)  -1<=t<=1
                  
                    (1,2t)                              
             T= -------------                        T =  (-1,0) 
                 Sqrt[1+4t^2]

             ds = Sqrt[1+4t^2] dt                   ds = dt

                    (2t,-1) 
             N = ---------------                    N = (0,1)
                Sqrt[4t^2+1]
                

INT F.T ds = Integrate[ {t+t^2,t^2+t^4}.{1,2t},{t,-1,1}]    

            +Integrate[{-t+1,(-t)^2+1}.{-1,0},{t,-1,+1}]  = 2/3 -2 = -4/3

           

INT F.N ds = Integrate[ {t+t^2,t^2+t^4}.{+2 t,-1},{t,-1,1}]   

            +Integrate[ {-t+1,(-t)^2+1}.{0,1} ,{t,-1,+1}] =  +4/15+8/3 = 44/15