1. Energy cannot be created or destroyed. In other words, in a closed system, the total amount of energy that can be taken out of the system will be equal to the total amount of energy that was put into the system.

2. In any given exchange of energy, there will always be energy lost. This is referred to as entropy. This basically means that in any system, energy will always be lost in some means, be it friction, or some random quantum effect.

This also implies that there can be no such thing as a perpetual motion machine as energy will always be lost in some form.

Note that this lost isn't mean to say "destroyed". It rather means unusable. That lost energy is transferred to the microscopical degrees of freedom of the system, as are molecular vibrations, place in which we call it "heat".

3. No system can reach absolute zero temperature. This is due to the fact that at absolute zero, a system has no energy, and thus does not move. Although this does not cause any problems in the sense of classical mechanics; it does cause problems on the quantum level. If a particle had no movement at all, its speed would be exactly known (zero, exactly), which is forbidden by Heisenberg's uncertainty principle.

Engineering is an area in which mathematics is applied to actual real world problems. The basic fundamentals of Engineering all require Calculus to understand. The more advanced topics require knowledge of both ordinary and partial differential equations. Actual applications require the usage of multiple forms of mathematics in tandem with each other. Frequently, multiple partial differential equations are solved at the same time. To solve these equations effectively, a combination of linear algebra and differential equations is used.

One example is in solving the gas equation in chemistry. Standard equations of the "ideal" gas law is

PV = nRT.

Engineers understand that the "ideal" gas does not exist in reality, and a compression ratio must usually be used. As such the equation is often used as " PV = znRT " where z is used as a compression ratio.

The compression ratio though is a mathematically derived constant which is based on experimental data. The data is taken and has curvefitting applied to it. Basic algebraic curve fitting does not do an adequate job, so a differential approach is applied to it. In order to do this Maxwell's equations must be applied to get the ideal Gas law into a partial differential format.

where T is temperature, H is enthalphy, S is entropy, and p is pressure.

So, in essence, the temperature is equal to the change in enthalpy with respect to the change in entropy, assuming that pressure is constant. The $\partial$ operator states that the enthalpy (H) is dependent on multiple variables (S,p) and that S is just one of them. By making pressure a constant, the derivative of H just depends on S, and not on p.

By making substitutions like this into the ideal gas equation, the compression factor "z" can be determined for each gas.