As an engineer, approximate analytical solutions are often so much more valuable than numerical solutions. I wish approximation methods were taught more. They a

As an engineer, approximate analytical solutions are often so much more valuable than numerical solutions. I wish approximation methods were taught more. They allow playing "what if?" exercises much more quickly, or finding important scaling laws. For example, the Tsiolkovsky rockey equation says that delta v goes with the log of propellant mass. That's so useful to an engineer, to quickly estimate feasibility before diving deep into simulations! Or, the stiffness of a beam goes as thickness cubed, divided by length to the fourth power. Or the resonant frequency of an oscillator goes as the square root of 1/LC. Relationships like that are immediately useful for predicting problems and understanding what a solution looks like. Of course sometimes numerical solutions are necessary. But usually, you run numerical simulations after most of the major design parameters are already dialed-in, and you just want to refine the solution or predict its behaviour. Searching only numerically for an optimum design is kind of a blind search, and no matter how fast your simulation, it gets increasingly difficult in higher dimensions. L And it often needs to start all over again when the boss says "what if we double the payload? How much will that cost us?"