Use the Leibniz-Maclaurin method to determine the power series solution for the differential equation given th?
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How "Original" of a contribution was Calculus?
The two founding notions of the calculus, the direction and thus the slope of a curve, and the area under a curve, can easily be demonstrated to preschoolers. Working out the formulas for derivatives and for elementary integrals is now advanced high school math. Many topics in foundations of mathematics in general and analysis in particular remain at the graduate level.A great many mathematicians from the Greeks onward contributed ideas and methods that would eventually be combined into calculus as we understand it in modern science and mathematics. Here are some of the most important of them.Eudoxus and Archimedes, method of exhaustion applied to various areas and volumes, and to evaluating some infinite seriesAlhazen, volume of a paraboloidKerala school in India, Taylor seriesCavalieri, integrals as sums of infinitesimal slicesFermat, infinitesimal approximationsBarrow and Gregory, definite integrals from any antiderivative (second fundamental theorem of calculus)Newton, fluxions, including derivatives of powers and dot notation for time derivatives; Taylor series; analytical functions given by convergent power seriesLeibniz, d-notation, product rule, chain ruleIt was Newton and Leibniz who demonstrated how to take the derivative of any function expressed as a combination of the standard set of functions, including exponential, logarithm, trigonometric and hyperbolic functions, and many elementary integrals. The problem of which such functions can be integrated explicitly was not solved until three centuries later, in the Risch algorithm.Many other major contributions followed after Newton and Leibniz, including partial derivatives, differential and integral equations, the epsilon-delta method for limits, differential geometry, differential topology, complex variables, Fourier analysis, vector calculus, measure theory, operator theory, and Abraham Robinson's Non-Standard Analysis, with a consistent version of infinitesimals.We are not done with this subject.
Is it possible to discover calculus or trigonometry without formal schooling just by accumulating the necessary precursors?
Not quite sure what you mean. You could pick up a book on calculus or trig and work through it. Teach yourself Calculus by Paul Abbott for instance, I think it is pretty much self contained, so if you can understand the basics, theoretically you could work through that. Similarly for Trig - there’s a bunch of cross fertilization between the two subjects so you can’t really study one without the other.If you are sufficiently intelligent, disciplined and organized, its ‘theoretically possible’ to study anything by yourself, I don’t know you, so I can’t say if you will succeed.
When should the Frobenius method be used?
In traditional method of solving linear differential equation what find as solution? we get linear combination of some elementary functions like x^2, lnx, e^ax, sin(ax), cos(ax) etc as general & particular solution.From Power series(or can say Taylor Series expansion)or Polynomial expansion we can find any elementary functions as a sum of power series. Thus can’t we consider that solutions of linear differential equation is expressed as linear combination of some power series? Yes we can.Now , In general aspect we don’t need to solve any linear differential equation in series solution method.BUT there is some linear differential equation(Bessel differential equation,Legendre differential equation, Hermite & Lagurre differential equation)in which we may fail to find its solution by traditional method. In such situation we must seek series solution method for the solution.Frobenius method is special method to find solution of linear differential equation at regular singular point(such a point which is not analytic or diverges )
What are some topics i could do for calculus research paper?
You didn't say how long the paper should be or really what level you're aiming at, so giving appropriate suggestions is hard. I'll try to give a bunch of varied ones. * Series for pi, e, etc., proven using, say, Taylor's theorem. (Eg. Leibniz series.) * Basics of complex analysis. Maybe discuss complex power series expansions and use them to prove Gauss' mean value theorem in that context. * Root finding methods, eg. Newton's method and generalizations, based on calculus. The error analysis also uses calculus. * Basic Fourier analysis (non-rigorous, most likely) * Solving differential equations using Laplace transforms * Computing difficult integrals with residue calculus (proving the residue theorem takes a lot of effort, but understanding it enough to understand example applications should be much easier) * The history of calculus is fascinating; eg. the Leibniz/Newton priority dispute, ancient Indian and Greek "glimmers" of Calculus * Proof of the irrationality of pi (and e) using clever integral tricks; for a much harder project, write up a proof of Lindemann-Weierstrass * Exploration of badly-behaved functions, motivating the Lebesgue integral and generalizations of the Riemann integral * Exploration of introductory real analysis--how do you make all the statements you probably glossed over in class rigorous? (As a starting point, try to tell me what the real numbers are, assuming I don't know about them. Now try to convince me that every continuous function from a closed interval [a, b] to the reals attains its maximum.) * Basic differential geometry, eg. the Frenet frame * The main intro multivariable calculus theorems, Green's, Stokes', Gauss'