If y is a function of x, then the differential dy of y is related to dx by the formulaĭ y = d y d x d x. Using calculus, it is possible to relate the infinitely small changes of various variables to each other mathematically using derivatives. The idea of an infinitely small or infinitely slow change is, intuitively, extremely useful, and there are a number of ways to make the notion mathematically precise. The differential dx represents an infinitely small change in the variable x. For example, if x is a variable, then a change in the value of x is often denoted Δ x (pronounced delta x). The term differential is used nonrigorously in calculus to refer to an infinitesimal ("infinitely small") change in some varying quantity. 3.3 Differentials as germs of functions.3.2.3 Differentials as linear maps on a vector space.3.2.2 Differentials as linear maps on R n.3.2.1 Differentials as linear maps on R.He is the most extreme of the bunch though. People like Poincare, Brouwer, Bishop, Weil, Von Neumann (surprisingly, even though he worked in set theory earlier in his life) sorta agree with him. People are going to bash Wildberger, but I'd like to say that his line of thinking does have a pedigree. He knows this, but suggests that things like Fourier analysis are yet to be justified using means he considers adequate, and hopes that mathematicians of the future might justify Fourier analysis etc more computationally. Wildbergers approach doesn't justify the math that even electrical engineers use. With Fourier analysis, and results like the infinite sum of continuous function being discontinuous, we moved to a more general notion of a function. Roughly, this means functions which can be written down as a formula like cos(x), sin(x), e x etc. However, his idea of a function is an analytic function from complex analysis. He wrote a book called "Treatise on Analytic functions" or something similar (the book title is French). What many people don't realize, is that between Newton/Leibniz and Weirstrass/Cauchy there was Lagrange. I've never heard of this "method of Lagrange," so maybe someone on r/math can shed light on this? Are there less magical explanations of how this method was derived? Can you get by in differential geometry using only polynomials? Are his concerns about infinitesimals legitimate?ĮDIT: I just watched the next lecture, and Lagrange's technique generalizes to partial derivatives of polynomials of any number of variables. I don't see how this is anything more than a fancy way of calculating the Taylor series for a polynomial.Īdmittedly, the large advantage of this method is that it holds for all fields, as it only deals with algebraic numbers. He just shows that you get the same result as you would get using limits. The problem is that he never gives any kind of proof that his way of finding derivatives actually resembles tangent approximations. The resulting expression is what he calls the "k-th tangent to p(x) at r." Uses synthetic differential geometry and. Finally, he substitutes every x with x-r. Bell: A Primer of Infinitesimal Analysis, Cambridge University Press, 1998. The p_n(r) polynomials turn out to be terms in the Taylor series of the original polynomial. Then he factors out all the x powers so the expression looks like this: Then, using the binomial theorem, he expands the expression. The way it works is he substitutes x with x+r in his polynomial of choice, p(x). In any case, he continues in his lecture to define derivatives algebraically using something like the Taylor series, and he points to Lagrange as the method's inventor. In the comments of the above video, he seems to oppose the existence of infinite sets, though the set of rationals is infinite (maybe he means only countably infinite sets exist?). In the previous lectures, he made it clear that he is not a fan of transcendentals (sin, cos, etc.) and prefers parametrizations of curves that are polynomials with rational coefficients. Wildberger begins by challenging the traditional approach to calculus education which uses limits and infinitesimals. A glance through the comments will give you a rough impression of what's going on. I'm not saying it was bad, but it was definitely controversial. I've recently begun watching a lecture series on differential geometry by Norman Wildberger, and I thoroughly enjoyed the first two lectures.
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