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As I've progress through my undergraduate engineering program, I've noticed a shift away from physical intuition - understanding why certain phenomena or equations make sense because of basic patterns we are already familiar with - to mathematical intuition - showing why equations and their phenomena should occur by mathematical derivation.

An example of this I came across today in aerodynamics:

For Isentropic Flow: $\frac{p_0}{p} = (1 + \frac{\gamma-1}{2}M^2)^{\gamma/(\gamma-1)}$

I can usually follow the derivation of the equation, identify the variables in it, and apply it to appropriate cases. But, I cannot describe why this equation is justified by simply looking at it - I must follow the mathematical derivation. There is no way I'd be able to describe this equation to a layperson in a way that could satisfy them. I can't come up with an explanation that satisfies myself; I want to know why $1+$ is there, why $\gamma- 1$ should be divided by $2$, and why the Mach number should be squared, without solely relying on the mathematical derivation. I have this problem with many formulas I have learned.

I know a fundamental part of engineering is understanding the equations and phenomena we use to to describe physical systems. Now, it seems like I should just trust the equations and make sure I'm not using them improperly.

Should I, as an engineer-in-training hoping to complete research, focus on trying to understand equations to my satisfaction, or should I instead just become well acquainted the equations, their use cases, and their general behavior?

Note: I acknowledge questions on this site should have answers and not just be discussed. I hope this question has an answer that has some consensus among experienced engineers.

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  • $\begingroup$ To some extent, the straight answer to this depends on the particular engineering degree programme you're studying. I've taught on a programme where both being able to prove that particular equation from first principles and being able to explain in natural language what it represents were among the published intended learning outcomes; other engineering programmes prioritize different things. $\endgroup$ Dec 11 '20 at 17:31
  • $\begingroup$ The way I got to understand the physical meaning of that equation was first to reproduce the mathematical derivation, then go through that derivation step-by-step, labelling each term with the natural-language name of the term in the steady flow energy equation from which it originates... $\endgroup$ Dec 11 '20 at 17:44
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    $\begingroup$ However, there are also clues in the equation itself. With practice, the Mach number will become familiar as a "dimensionless velocity", in which case squaring it and dividing by $2$ is a natural way to get a "dimensionless specific kinetic energy". Similarly, $\gamma-1$ will become familiar as a "dimensionless specific gas constant", and $1/\gamma$ as a "dimensionless specific heat capacity at constant volume". Finally, it will become a familiar idea that, when one non-dimensionalizes by dividing through by X, the term "$1$" is a "dimensionless X" (X here being specific static enthalpy). $\endgroup$ Dec 11 '20 at 17:52
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Should I, as an engineer-in-training hoping to complete research, focus on trying to understand equations to my satisfaction, or should I instead just become well acquainted the equations, their use cases, and their general behavior?

Engineers apply equations to predict behavior. The critical requirement is to know the constraints of the equation. The constraints are either from first principles or from practical applications. The application demand is to appreciate the trends that arise from the equation. This arises in cases where we need "seat of the pants" estimates or "back of the envelope" calculations.

Consider the ideal gas law. From practice, it has general constraints to be at low pressure and high temperature relative to the critical point of the gas. From first principles, it has specific constraints for molecules to have little to no molecular interactions and to be small spheres. The trends it predicts are encapsulated in such statements as the Boyles or Charles laws.

Consider Bernoulli's equation. From its fundamental derivation, it has constraints to be applied only for incompressible fluids under inviscid flow. The trends it predicts are encapsulated in statements that say as the height of the discharge increases, the pressure at the outlet decreases in proportion.

What then should you do when you confront an equation for the first time? As an engineer who wants to make accurate predictions, you must not fail to learn all of the constraints of that equation. Correspondingly, as an engineer who wants to move efficiently through a practice of problem solving, you must not fail to deconstruct the various trends of that equation.

How do you go about the first requirement? You study the roots of the equation. It may arise from empirical analysis. A case in point is the derivation of the ideal gas law from the Avagadro, Boyles, and Charles laws. Alternatively, it may arise from first principles. A case in point is the derivation of the ideal gas law from kinetic theory of gases. Each approach will provide insights to the practical limits or conceptual assumptions behind the equation.

How do you go about the second requirement? You take the equation and do a thorough unit analysis. You also take the equation and (literally if not figuratively) plot the trends it should predict. In essence, as an engineer in training, you have tools that transcend the hand-held calculator (that was the revolution from the slide-rule of my days). Go plot the left side of the equation as a function of the various parameters it contains!

In summary, when your goal is to be know for accuracy and efficiency in your engineering practice, you cannot just focus on trying to understand an equation to your satisfaction. This nebulous, subjective statement will fail you in the end. You must become well-acquainted with the equation and its trends. This goal, and the habit it demands of you, will stand you well through your coming career.

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Knowing the derivation is important because it usually tells you what initial assumptions were made in the derivation and what the limits of applicability of the resulting equation are. Understanding both of these things is an essential skill all engineers must master.

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    $\begingroup$ And I had an example of this where 2 engineers couldn't get the calcs to come out right... Until a 3rd one went back to basics to show the formula they were using made an assumption they had not considered which was relevant in that case. $\endgroup$
    – Solar Mike
    May 17 '19 at 6:35
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    $\begingroup$ @SolarMike Yeah it happens. Undergrads tend not to think this way though. It comes as a surprise to them that you need to know when a model is valid as often as you need to compute stuff. Problem is that this part is handled in a sentence (or 10 seconds) versus the equation, its derivation, and application which you get hours uppon hours of practice on. $\endgroup$
    – joojaa
    May 17 '19 at 15:22
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As with a lot of things in engineering (and life in general), you have to weight the cost vs benefit associated with the application. Sometimes it’s enough to treat the equation like a black box, needing only to know the input output behavior. Other times, knowing the how to derive equations enables you to derive new things from new assumptions. You won’t memorize how to derive every equation that you’ll end up using only because there just too much to remember. But having and actively practicing analytical derivations will help you to read research papers in case you need to.

How deep you need to go in understanding each equation is really a function of how much you’ll need it in your application.

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  • $\begingroup$ So, how deep you need to go to learn an equation is based on being able to make predictions that you have absolutely no experience to make at that point in time? This seems to promise nothing but future ignorance. It certainly does not promise a development of good learning habits. $\endgroup$ May 17 '19 at 15:10
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But, I cannot describe why this equation is justified by simply looking at it - I must follow the mathematical derivation. There is no way I'd be able to describe this equation to a layperson in a way that could satisfy them.

Every student starts with equations in their math classes, at school. With a thorough understanding of the various interrelated concepts in math, she is able to explain these equations to laymen. By the time we start college, we have the ability to explain why some variables are present and why it is present specifically in that exact form (atleast for the concepts we have been very familiar with).

But can we do that outside math? Is it possible in engineeering? I have viewed a lot of videos where a very comprehensive understanding of concepts have been presented, which can be presented to a general audience. But now i (have come to) believe that some equations are only empirical. Intuition can be developed, but i am not sure as to what extend.

Maybe simulations at the micro-scale so that our analysis becomes perfect? Quantum mechanics, molecular/DFT simulations could provide more insight.

P.S. I am not an experienced engineer, but an engineering student. I have gone through a futile search to explain some equations i have learned. It involves a lot more work. The deeper i go into theory, the point is lost, because an engineer provides real-life solutions, and i feel going deeper into theory is science rather than engineering; are we supposed to do that as engineering students (i await an answer to that). Developing such intuition has been easier for me in math, rather than engineering.

This is not an answer to the question, this is more intended as a comment, but i do not have the ability to comment at the moment. Perhaps people with similar experience could share their learning here.

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