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One of the goals of science is to explain the world. * While computational tools will symbolically solve a lot of integrals, they won't solve them all. Resorti
by abdullahkhalids 5y ago
One of the goals of science is to explain the world.
* While computational tools will symbolically solve a lot of integrals, they won't solve them all. Resorting to numerics often means you have lose some understanding along the way, because you no longer have a closed form expression to analyze.
* One general strategy in Physics is to take a complicated expression and make different sets of simplifying assumptions to reduce it to simpler forms. This adds explanation to your model because you understand how the system is said to behave under different limitating scenarios. But if you are not adept at manipulating complicated expressions, you won't be able to use the strategy fully. Computer solvers are really bad at writing mathematical expressions in the nicest way possible so that the simplifying assumptions pop out naturally.
Full disclosure: I am a physicist, who uses Mathematica quite a lot to solve various expressions (but I know the limitations of the tool).
- sillysaurusx 5y agoYour second point is very interesting! Can you point me to any problems of that type? I’d be interested in learning how to make a mathematical model of something, and then simplify parts of it with various assumptions. The one I can think of offhand is the pendulum problem, where sin(theta) is approximately equal to theta for small values. But you made it sound like there are problems with multiple parts, and many different simplifications.
- petschge 5y agoThere definitely are. The simple pendulum (or rather harmonic oscillators) are the problem we want to transform harder stuff to, because we know how to solve that one. In other words, you might have to make a bunch of assumptions till you have reduced your problem to simple harmonic oscillations around a steady state that you have found separately.
- bernulli 5y agoCheck out Prandtl's Boundary Layer Theory https://en.wikipedia.org/wiki/Boundary_layer https://en.wikipedia.org/wiki/Boundary_layer It's a set of smart physical observations/assumptions that allow you to find closed form solutions for the Navier Stokes equations in fluid mechanics https://en.wikipedia.org/wiki/Navier–Stokes_equations https://en.wikipedia.org/wiki/Navier–Stokes_equations for the important case of flow close to some body, such as an airfoil.