directional calculus

[AntonioLao]

Save from the need of defining a coordinate system, direction in spacetime can be determined using the ideas of infinitesimal calculus. Two of these ideas are derivative and antiderivative. Together with the theory of infinite series expansion of function, these ideas established some plausible descriptions of the physical world of the macroscopic and microscopic reality with both properties of its internal and external region of the spacetime continuum. Since the definition of derivative and antiderivative is synonymous with differential and integral, the latter can be used interchangeably without any loss of precise and accurate mathematical connotations.

If the independent variable �� marked the spacetime position of the spot where and when the theoretical treasure trove is found then the time derivative of �� represents the rate of change or speed of its location in a given direction. This direction holds linearly for all future locations of the variable. To change this direction is simply to take the second time derivative properly called linear acceleration. A continuously differentiable manifold would then imply continuous change along a direction while a continuously integrable manifold implies changes in the opposite direction. For example, the continuously differentiable transcendental function ℯⁿ is ℯⁿ. However, its continuously integrable manifold is also ℯⁿ if and only if the integrable constant is set to zero at every step, implying that the infinite series expansion of ℯⁿ with ��-th term given by ��ⁿ/��! has continuous n-derivative given by ��ⁿ⁻¹/(��-1)! and continuous ��-antiderivative given by ��ⁿ⁺¹/(��+1)!. For ��=0 of antiderivative or ��=1of derivative, the infinite series expansion remains the same.

[Graybeard]

Thanks Antonio ..... for some reason the equations were not visible ... but after reading it I'm sorta glad .... they would have been too distracting, and I needed all the concentration I could muster just to follow the words.

cool bananas ... greg

[austintorn@aol.com]

Yes. me. too; so I'll just say "and then what happened?"

[AntonioLao]

Gentlemen, nowadays, people simply use the scientific graphing calculators to find the nth derivative or antiderivative and taking the directional invariance property of both for granted such that to take the derivative and then followed by the antiderivative of the same function means simply the same as doing nothing at all, that is knotting and then unknotting is same as no knotting.