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Note, that the dotted line represents the equilibrium position of the
spring, where the spring is nonstretched . Once again, the possible
presence of other forces acting on the particle is irrelevant to this
discussion, since we are only focusing on the work done by the spring.

The work done by the spring on the particle as it moves from A to
B is given by the following scalar equation:

 kx 2   kx 2   kx 2 kx 2 
A    2     1     2  2  . (5.27)
 2   2   2 2 
     
Therefore, the work done by the spring on the particle depends
only on the position of A and B (relative to the position of point O),

since this is what determines the amount of stretch or compression in the
spring (s and s ). Therefore, an elastic spring is conservative force so
1
2
that potential energy stored in spring is equal to
kx 2
E  . (5.28)
P
2
and work done by elastic spring force is possible to write down as
A    E , similar to the work done by gravity force.
P
Considering properties of electric field we can prove , that
electrostatic forces are conservative forces as well. In general, the work

done by a conservative force F is equal to
A   dE . (5.29)
P
Let us consider the stationary case system consisting of one material
point on which the conservative force F acts:
 
A  F ( r d  )  F x dx  F y dy  F z dz. (5.30)

At the same time
F  x   E P , F  y   E P , F  z   E (5.31)
P
y
z
x
Hence
 E P  E P  E P
F   , F   , F   (5.32)
y
z
x
x  y  z 
Using a definition of the operator of gradient
     
grad    i  j  k (5.33)
x  y  z 
it is possible to obtain:

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