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<p>I don’t know about a particular branch as it relates to departments, but I can certainly nominate a particular topic: thermodynamics, which to this day, is still the biggest ‘mind screw’ within any engineering curriculum - perhaps more so than even quantum mechanics (which most engineers don’t take anyway). To this day, other than perhaps in academia, I have yet to meet a single engineer - including many with PhD’s - who believes they truly understand thermodynamics, and especially the mathematics behind thermodynamics, to the point that an entire cottage industry of bloggers/authors exist who write tracts trying to correct misconceptions of thermodynamics (which then begs the question of why current engineering students aren’t assigned those blogs and articles rather than the utterly baffling thermodynamics textbooks). Yet even so, confusion about even the most basic thermodynamics terms is utterly pervasive. For example, how many engineers actually understand that entropy and disorder are not the same thing, and actually know what the difference is between the two? How many engineers actually understand that entropy, strictly defined, has nothing to do with energy, temperature, or heat flow? How many engineers actually understand that entropy is specific to the observer - in the sense that a thermodynamic system that has very low (or even zero) entropy with respect to me may have very large entropy with respect to you? </p>
<p>But instead of having students understand these deeply fundamental notions (which, despite being fundamental, are nevertheless surely misunderstood by the overwhelming majority of engineers), engineering thermodynamics courses inevitably degenerate into a long slog of recondite mathematical equations that not only leaves the students baffled, but has nothing to do with the actual job. </p>
<p>As a case in point, consider the Maxwell Relations. Simply take the first one listed: that the partial derivative of temperature with respect to volume at constant entropy is equal to the negative of the partial of pressure with entropy at constant volume, and both are equal to the double partial of internal energy with entropy & volume. </p>
<p>[Maxwell</a> relations - Wikipedia, the free encyclopedia](<a href=“http://en.wikipedia.org/wiki/Maxwell_relations]Maxwell”>Maxwell relations - Wikipedia)</p>
<p>What the heck does that mean? To this day, I have not encountered a single engineer who knows what that actually means in any real-world sense. Note, it has nothing to do with the math, as many engineering students - if perhaps not practicing engineers - can solve the math. But what does the math actually mean? Does anybody in the real world actually go around calculating, or even caring what the double partial derivative of internal energy w.r.t. entropy and volume is? </p>
<p>And those are some of the more basic mathematics of thermodynamics. Even more perplexing are the Bridgman’s Equations. I defy anybody to take the last Bridgman’s Equation and tell me how to use it in any real-world engineering setting.</p>
<p>[Bridgman’s</a> thermodynamic equations - Wikipedia, the free encyclopedia](<a href=“http://en.wikipedia.org/wiki/Bridgman’s_thermodynamic_equations]Bridgman’s”>Bridgman's thermodynamic equations - Wikipedia) </p>
<p>Hence, engineering thermodynamics exams inevitably devolve into a dazed forced march of endless mathematical calculations. You don’t know what you’re actually calculating. You don’t know why you’re calculating it. And you don’t have time to find out. All you know is that you either need to do the mathematics, or you fail the class.</p>