FOWLES CASSIDAY 4.3

1. 1
     
2 3
2
Find the value of the constant
c such that each of the
following forces is conservative:
/ / /
ixy jcx kz
i z y cj xz y k x y
  
  
F
F

2. 2
Solution
Conservative Field
In summary, the curl of a conservative force field is
zero everywhere because a field being conservative
is equivalent to having zero curl, which can be
derived in any basic text on vector analysis.
Additionally, a conservative force field can be
written as the gradient of a scalar field.

3. 3
2
2 3
3 4
( )
( 1) 0
( 1) 0 ( 1) (2 )
(2 1)
,
Conservative (2 -1) 0
1
2 1
2
a
i j k
X i X
x y z
xy cx z
j X k cx x
kx c
Now
kx c
c c
  
   
  
   
 
 
   
F


4. 4
2 2
2
2 2
2 2
2 2
( )
( )
1 1
( ) ( )
,
Conservative ( ) 0 1
, ( ) 0 1
b
i j k
x cx
X i
x y z y y
z xz x
c
y y y
cz z
j k
y y y y
Now
x cx
c
y y
cz z
Also c
y y
   
   
  

  

     

    
 F
Analytical Mechanics, Grant R. Fowles, George L. Cassiday