
We have the intuitive relationships
This set of equations is known as the Galilean Transformation. They
enable us to
relate a measurement in one inertial reference frame to another. For
example,
suppose we measure the velocity of a vehicle moving in the in
direction in system S,
and we want to know what would be the velocity of the vehicle in S'.
(1) 
We have stated the we would like the laws of physics to be the same in all inertial reference frames, as this is indeed our experience of nature. Physically, we should be able to perform the same experiments in different reference frames, and find always the same physical laws. Mathematically, these laws are expressed by equations. So, we should be able to ``transform'' our equations from one inertial reference frame to the other inertial reference frame, and always find the same answer.
Suppose we wanted to check that Newton's Second Law is the same in two
different reference frames. (We know from experiment that this is the
case.)
We put one observer in the unprimed frame, and the other
in the primed frame, moving with velocity relative to the
unprimed frame.
Consider the vehicle of the previous
case undergoing a constant acceleration in the direction,
(2)  
Exercise 1.3
In the tutorial, you show that the Law of Momentum Conservation holds
regardless of the inertial frame a
given collision is viewed in. This is done by specialising to a
collision where all velocities are in the
direction. How would you do this for a more general collision ?
So far so good !
We have Classical Mechanics, a beautiful theory, as it has an elegant
independence of how you observe it. There is a sense of poetry in how
ugly terms, arising from
observation in a different frame, eventually drop away, until we are
left with
physical laws which are invariant under the Galilean Transformation.
But .... as time passes, it becomes clear we are in a fools paradise !
The first problem ...
Experiments on electric and magnetic fields, as well as induction of
one type of
field from changes in the other, lead to the collection of a set of
equations, describing all
these phenomena, known as Maxwell's Equations. You are already
familiar with them. In
vacuum they are
(3) 
Now, these equations are considered to be rock solid, arising from and verified by many experiments. Amazingly, they imply the existence of a previously not guessed at phenomenon. This is the electromagnetic wave. Every electrical engineer, following Marconi, must appreciate this !
To see this in detail,
take the time derivative of the second last equation and the curl of
the last.
(4) 
Now note that space and time derivatives commute
(5) 
(6) 
Now, we use the identity
(7) 
(9) 
(10) 
It is clear therefore that Maxwell's Equations are highly predictive.