How did people discover the conservation of mechanical energy? I want to write about the history of energy, and am focusing on mechanical energy at the moment. My question is:

Can I attribute the conservation of mechanical energy to the power of maths, especially vector calculus?

From a pure theoretical point of view, given a total force $\vec F$ acting on a particle, I define the work done as $$W=\int_{\vec r_1}^{\vec r_2}\vec F \ {\cdot} \ \vec {dr}=\int_{t_1}^{t_2}m{d\vec v \over {dt}} \cdot \vec v \ dt = \int_{t_1}^{t_2}{1\over 2}m{d \over dt}(v^2) \ dt = {1\over 2}mv_2^2-{1\over 2}mv_1^2 \tag{1}$$
Thus if I could define the potential energy as $$U(t_1)=-\int_{\vec r_0}^{\vec r_1} \vec F \cdot \vec {dr} \tag{2}$$
I could write (1) as $${1\over 2}mv_1^2+U(t_1)={1\over 2}mv_2^2+U(t_2)$$
Thus if I define the kinetic energy to be $K={1\over 2 }mv^2$, I could claim that there exists a so-called mechanical energy, which is the potential energy defined above plus the kinetical energy, which does not change over time.

Is this the true story behind the process during which people discover conservation of mechanical energy? Is there a better way of arriving at the conclusion?

 A: Historically, conservation of energy may have been discovered by Julius Robert Mayer in 1842.  When he was a youngster he tried to build a water wheel that drove an Archimedean screw to lift water back up to the top of the wheel and keep it turning.  He found this to be impossible.  The lesson stayed with him 'til later in life he became a doctor and studied the conversion of food to mechanical work.  He hit upon an equivalence between mechanical energy and heat energy.
He was followed by James Joule, an Englishman who tried to create an electric motor run by a battery that would equal the efficiency of a coal fired engine.  He wasn't successful, but along the way he found that heat appearing in a current-carrying wire was directly proportional to mechanical work done by a dynamo.  This led to the equivalency of mechanical energy and heat.
This is the historical story of Mayer and Joule, and how their discoveries were picked up by William Thomson, and Rudolph Clausius, and became the basis of an idea of local conservation of energy: http://galileo.phys.virginia.edu/classes/152.mf1i.spring02/MayerJoule.htm.
A: 
Can I attribute the conservation of mechanical energy to the power of maths, especially vector calculus?

No.

Is this the true story behind the process during which people discover conservation of mechanical energy? 

No, energy conservation was historically a long debate over centuries.

Is there a better way of arriving at the conclusion?

For that you should start with a Lagrangian, possibly with some time symmetry, then learn how to get dynamics from the Lagrangian.  Then find out there is a constant associated with that time symmetry, so that even though dynamics means things change there is something that doesn't, something associated with that symmetry in time.
Then we can go a step further. And ask our self the reason the Lagrangian depended on time when it depended on time. And the answer truly is because something else changed, and if something is changing then it has its own dynamics. So if Lagrangian mechanics is the way the world is, then there should be a Lagrangian for the larger system.
And there is. And that Lagrangian determines the dynamics for that larger system, and then with the time symmetry, there is a conserved quantity.
So when there was a symmetry it indicated we had a closed system, evolving on its own. When the Lagrangian didn't have that symmetry it indicated an external interaction and energy was not conserved because of the interaction.
This idea of open and closed system of external and internal energy and interactions is totally key. And the idea that you have failed to truly describe the total system if energy is not conserved.
That is the true story of energy conservation. It is aspirational. It becomes like Newton's third law a judgement on a theory that if found wanting makes the theory seem incomplete.
Sure you could have an external force but it will seem incomplete. And you can increase or decrease your total energy but that will also seem incomplete.
That said: energy is not actually conserved in the world we live in.
When you include fields all you really get is a kind of local conservation law for energy density. And local conservation doesn't mean global conservation if your space is flat and bounded or if it not flat.  And to get energy from energy density you need volume, hence geometry.  And once you bring geometry into it things get an additional complication.
First, energy is geometrically just part of energy-momentum and breaking it into four pieces is as arbitrary as picking a direction for a coordinate axis. And further, enrrgy-momentum itself has a flux as well as a density. So you have an entire stress-energy tensor for it all.
And at this point potential energy has finally died a quiet death because now the energy flows and the energy sits throughout space instead of being stored abstractly in the idea of things being part of a system or not a system.
So no, potential energy is not the proper way to understand conservation of energy. It doesn't correctly assign energy a place or a correct flow. It can make it seem like energy is conserved when in our universe energy is not conserved.
