A deepened understanding of mathematics provides anyone with an added dimensional perspective on the universe as whole. Such is the sentiment that was shared by the famed Charles Darwin who stated "*I have deeply regretted that I did not proceed far enough at least to understand something of the great leading principles of mathematics; for men thus endowed seem to have an extra sense*".

As this regards sport, and preparation as whole well beyond sport, I have found extraordinary utility in not only the analogs to be found from derivatives and integrals, but also a near literal translation of their mathematical roles to an applied setting in, for example, sport coaching.

In the mathematical sense, understand that derivatives are slopes of functions (rates of change) and with each subsequent derivative that is calculated another rate of change is found (and then it becomes a matter of how many dimensions one is working in as this determines the nature of the derivative). Alternatively, integrals are anti-derivatives in that they are sums/areas under curves in addition to three dimensional, and beyond, sums that account for volumes of solids and more.

What's to be understood here is that both derivatives and integrals are relative to functions from the standpoint that the first derivative taken, of a given function, is one step removed and an integral of that derivative brings you back to the original function. In this way, I draw your attention to any conceivable sport, or military tactical, motion and recognize this motion as a function. Preparatory motions may then arise as both derivatives taken from the sport motion in question as well as integrals that work their way towards the same sport motion. The significance is that the derivatives and integrals, in this example, are contextualized by the sport motion (function).

Sport jargon, while clearly helpful is no where near as universal as mathematical language, has brought us Dynamic Correspondence, Transfer of Training, and Special Strength Training; however, what each describes are processes predicated upon what they all share in common; which is their mathematical common ground. The closer the derivative or integral (preparatory motion) is to the original function (sport motion), in terms of a new set of criteria to be explained forthcoming, the greater the impact on improving the intended result on the respective sport motion.

The work of the esteemed Anatoly Bondarchuk and late Yuri Verkhoshansky gave us Training Transfer and Dynamic Correspondence, respectively; and to this I will ad criteria that are of apex significance in contexts in which they are intrinsic to the execution of the sport motion. The work of Bondarchuk and Verkhoshansky gave as all physical motion substrates to consider in terms of kinematics and kinetics. That stated, however, does not account for the psychomotor and sensorimotor aspects of motion execution that are often definitive in their realization. As a result, I propose that the criteria expand from biodynamic, bioenergetic, and biomotor considerations to:

- psychomotor- the implications of conscious mental activity on motion
- sensorimotor- the implications of sensory processing on motion
- biodynamic- mechanical
- bioenergetic- phsyiological
- biomotor- output/kinesthetic (which relates to sensorimotor)

Of paramount importance is that the psychomotor and sensorimotor aspects of preparation be elaborated upon, and instructed, based upon the subtleties of each that underpin the motion in question.

The result of applying this expanded criteria, mathematically influenced from my perspective, to sport/military preparation is one that will ensure an even higher assurance that what's done in preparation will most significantly impact the preparatory objective.

You have my theoretical offering, experimentalists...the floor is yours.

Email James@globalsportconcepts.net for consulting information

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