space-time derivative

[AntonioLao]

space-time derivative
A widely used mathematical concept that is also taken for granted for more than 300 years is the conventional definition of a derivative. Together with the integral, derivative was the crowning achievement of Sir Isaac Newton (1642-1727) while the integral was that of Gottfried Wilhelm Leibniz (1646-1716). For Newton, the method of derivatives (fluxions) was applied for finding tangents of curves, maxima and minima of functions, curvature of curves, and points of inflection of curves. However, Jean Le Rond d’Alembert was the first to see the correct notion of the derivative in 1737. D’Alembert explicitly asserted that the derivative must be based on the limit of the ratio of the differences of dependent and independent variables. In 1817, Bernhard Bolzano ( 1781-1848 ), who carefully studied the properties of functions and gradually initiated the distinction between functional continuity and discontinuity, was the first to define the derivative of f(x) as the quantity f’(x) which the ratio [f(x+∆x)-f(x)]/∆x approaches indefinitely closely as ∆x approaches zero thru positive and negative values. He emphasized that f’(x) was not a quotient of zeros or a ratio of evanescent quantities but a number which the given ratio approached.

Although Augustin-Louis Cauchy (1789-1857) was able to clarify the relation between ∆y/∆x and f’(x) using the mean value theorem where ∆y=f’(x+θ∆x)∆x and 0<θ<1, his proof of the mean value theorem used the continuity of f’(x) in the interval ∆x and he believed as nearly all mathematicians of his era for another 50 years that a continuous functions must be differentiable (derivatives exist) except of course at isolate points such as x=0 for y=1/x, Bolzano as early as 1834 gave an example of a continuous function which has no finite derivative at any point. The historical significance of Bolzano’s discovery is that differentiable space-time manifolds need not necessarily imply the existence of a space-time continuum, vice versa. On the contrary, it suggests the viability of space-time quantization. Main reference: Morris Kline, Mathematical Thought: from Ancient to Modern Times, 3 volumes, Oxford University Press, 1972.