. For a
conservative vector field 
with potential function f(x,y,z) (in the
physicist's sense) so that 
. The potential energy of a
particle at position 
is 
. The Conservation of Energy
Principle says that the expression
Spring 1997
Due: Friday April 18, 1997
In this problem we derive the principle of conservation of Energy. The kinetic
energy of a particle moving with speed v is . For a
conservative vector field with potential function f(x,y,z) (in the
physicist's sense) so that . The potential energy of a
particle at position is . The Conservation of Energy
Principle says that the expression
is constant for a particle moving in the field with position vector
and velocity vector 
. Let P and Q be two points in space and
let C be a path from P to Q parameterized by 
, 
, where 
and 
.
be the velocity of the particle and 
be
the acceleration. Show that
to show that the work done by as a particle moves along C
is equal to the kinetic energy at Q minus the kinetic energy at P. is conservative as defined
above. Use the fundamental theorem of calculus for line integrals to show that
the work done by the force along the curve C is equal to the
potential energy at P minus the potential energy at Q.