unreal up-down slope

[AntonioLao]

The slope of absolute vertical is undefined, meaning infinity. It is unreal. On the other hand, the slope of horizontal is always zero. The product of slopes of 2 perpendicular lines is -1. But product of infinity and zero is undefined. Mathematical logic does have a limit to which few care to investigate and mostly taken for granted.

The x, y, and z axes of all Cartesian systems are orthogonal. Proving this statement requires a priori assumption that at least one base system has orthogonal axes. Then relative to this base the rest are proved by infinitesimal rotations and linear orthogonal transformations such that there exists a square matrix L satisfying |Lx|=|x| where x is a given multi-dimensional vector. The ups of Arctic circles are the downs of Antarctic circles, the left-right or fore-back of equatorial circles. Therefore relative verticals can be proved by inner products and outer products if and only if the directions are predefined on the absolute base system.

The entire edifice of differential calculus stands on the concept of slope made possible by the theory of limits and Cauchy’s completeness theorem. The result is the derivative defined as the limit of the ratio of a continuous variation of a function over the variation of its domain as the value approaches zero. However, another noted application of derivative is dimension contraction or expansion. For examples, the volume of a sphere is a function of its radius (R) given by V=4pR³/3. The 1st derivative of volume with respect to R is 4pR². This is simply the surface area of the sphere. The 2nd derivative of volume gives 8pR or 4 times the circumference. The 3rd derivative of volume is a constant of 8p.The area of a circle is pR². Its 1st derivative is 2pR which is just the circumference. Its 2nd derivative is 2p. These suggest that the 3rd derivative constant of volume is equal to four times the 2nd derivative of area: d³V/dR³=4d²A/dR². Furthermore, by the Fundamental Theorem of Calculus, integration increases the number of dimensions. Single integration transforms a point into a curve, double integrations transform a point into a surface, and triple integrations transform a point into a volume.

[mkirkpatrick]

At the very bottom of the slope,at great depth lies the number of all keys?The mind can
unlock this riddle!



regards michael.