#HISTORY OF INFINITESIMALS FREE#

Without getting into too much symbolism here, I’d like to mention that author Walter Rudin in his classic book REAL AND COMPLEX ANALYSIS (ISBN 0070542341), third edition, on page 150 states “The term differential is also often used for T’(x)” where T is a function of x. Now, the idea of differential is of course applicable to multi-variable functions. Clearly this does coincide with dy=f’(x)dx (where dx=h) as dx approaches (but never becomes) zero. One way to define dy is to consider it as a function of two variables, let’s say, x and h, that has the following property:ĭy(x,h) = f’(x) h + q(x,h) for some function q such that limit of q(x,h)/h is zero as h approaches zero. But it does give an intuitive sense as to what is happening. But, demanding rigor, this creates more headache in math as the word “approaches” creates a new mystery. But what is it? Some have defined it as a quantity that approaches zero but never becomes zero. But then the question is: what does dx represent? Yes, dx is the differential of x itself. Given a function f(x) of a single variable x, the differential dy is defined as dy=f’(x)dx. Once I sat in a lecture attempting to do just this. Notwithstanding the preceding, a colleague of mine has claimed that it is possible to give the notion of the infinitesimal a non-contradictory treatment in a certain logico-deductive system. In the system of real numbers, no such thing could exist. But Weierstrass made obsolete the use of infinitesimals in mathematics.Ī definition, albeit unsatisfactory, of infinitesimal is that it represents a positive quantity that is less than any other positive quantity. In the early stages of the development of the Calculus the infinitesimal was considered necessary and important. There is quite a bit (or a lot) of history behind it. What is an infinitesimal? What is a differential? – GlynĪs I noted earlier, the philosophical/logical questions surrounding the infinitesimal are, basically, beyond my expertise.

If you are still interested in receiving a copy of my resume, please let me know. It’s been fun exchanging a bit of math with you. Sorry, if I have used too many words to express my thoughts. If we treat dy and dx as infinitesimal quantities such that dx≠0, and yet exceedingly close to zero, then we may treat dy/dx as a fraction. Now, granted ∆y/∆x is indeed a fraction as long as ∆x≠0, which latter condition is the case at all times, anyway. Or, sometimes we write dy/dx=1/(dx/dy) or in the chain rule case: dy/dx = (dy/du) (du/dx).Īnyway, getting back to the point, the symbol dy/dx represents – MORE THAN ANYTHING ELSE – the limit of ∆y/∆x, where ∆y = f(x+∆x) – f(x), as ∆x approaches zero, if the limit exists. As mentioned, this happens, for example, when we want to find the differential of a function f(x) as in dy=f’(x)dx. In fact, when it comes to practicality, we treat dy/dx as if it were a fraction. This area of discussion – as exciting, rigorous and important as it is – is beyond my expertise. But there is and has been a whole lot of philosophical/logical debate as to what an infinitesimal actually is. You can treat them as (infinitesimal) differentials. That’s my point! I did say I could say more. Why do you reject dy/dx as a fraction - a ratio of differentials? You said you could say more on the topic … Cheers, Glyn Hi Albert: I accept your history but not your math.

#HISTORY OF INFINITESIMALS FREE#

Please feel free to ask further questions. But I will leave it at that at this point. This symbol was invented by Leibniz, the other contemporaneous inventor of the Calculus. So, the symbol dy/dx represents a single idea-the idea of the derivative. The symbol dy/dx represents the derivative of (the function) y with respect to x. The symbol dy/dx … does it represent a fraction or a whole symbol? Please feel free to let me know what you think about my proposal. I’d be glad to email you a copy of my resume.

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History of infinitesimals