![]() ![]() While Wolfram|Alpha knows (and allows you to compute with) this simple formula, it also knows the correct general result. For instance, many physics textbooks offer a simple formula for the inductance of a circular coil with a small radius: ![]() ![]() Textbooks often contain simple formulas that correspond to a simplified version of a general physical system-sometimes even without explicitly stating the implicit simplifying assumptions! However, it is often possible to give a more precise and correct result in terms of special functions. Special functions turn up in diverse areas ranging from the spherical pendulum in mechanics to inequivalent representations in quantum field theory, and most of them are solutions of first- or second-order ordinary differential equations. It includes a few hundred members, and can be viewed as an extension of the so-called elementary functions such as exp(z), log(z), the trigonometric functions, their inverses, and related functions. Over the last 200+ years, mathematicians and physicists have found a large, fascinating, and insightful world of phenomena that can be described exactly using these so-called special functions (also commonly known as “the special functions of mathematical physics”), the class of functions that describe phenomena between being difficult and complicated. On the other hand, some systems, such as the force of a simple spring, can be described by formulas involving simple low-order polynomial or rational relations between the relevant problem variables (in this case, Hooke’s law, F = k x): A simple example is provided by free fall from large heights: Unfortunately, if one uses a sufficiently realistic physical model that incorporates all potentially relevant variables (including things like friction, temperature dependence, deformation, and so forth), the resulting equations typically become complicated-so much so that in most cases, no exact closed-form solution can be found, meaning the equations must be solved using numerical techniques. For example, for a pendulum: Frechet derivative of Integrate^2/2 – Cos], ] wrt x. The equations describing the extremality condition of a functional are frequently low-order ordinary and/or partial differential equations and their solutions. Typically this happens with the action for time dependent processes and quantities such as the free energy for static configurations. Virtually all known processes occur in such a way that certain functionals that describe them become extremal. While the use of computations to predict the outcomes of scientific experiments, natural processes, and mathematical operations is by no means new (it has become a ubiquitous tool over the last few hundred years), the ease of use and accessibility of a large, powerful, and ever-expanding collection of such computations provided by Wolfram|Alpha is. ![]()
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