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Introduction
Published in Roberts Charles, Elementary Differential Equations, 2018
By 1670 the idea of a limit had been conceived; integration had been defined; many integrals had been calculated to find the areas under curves, the volumes of solids, and the arc lengths of curves; differentiation had been defined; tangents to many curves had been effected; many minima and maxima problems had been solved; and the relationship between integration and differentiation had been discovered and proved. What remained to be done? And why should Isaac Newton and Gottfried Wilhelm Leibniz be given credit for inventing the calculus? The answers to these question are: A general symbolism for integration and differentiation needed to be invented and strictly formal rules, independent of geometric meaning, for analytic operations needed to be discovered. Working independently of each other, Newton and Leibniz both developed the required symbolism and rules for operation. Newton’s “fluxional calculus” was invented as early as 1665, but he did not publish his work until 1687. Leibniz, on the other hand, formulated his “differential calculus” about 1676, ten years later than Newton, but published his results in 1684, thus provoking a bitter priority dispute. It is noteworthy that Leibniz’s notation is superior to Newton’s, and it is Leibniz’s notation which we use today.
Solving First-Order Equations
Published in Steven G. Krantz, Differential Equations, 2022
The usual Leibniz notation dy/dx implies that x is the independent variable and y is the dependent variable. In solving a differential equation, it is sometimes useful to reverse the roles of the two variables. Treat each of the following equations by reversing the roles of y and x: (a)(ey−2xy)y′=y2(c)xy′+2=x3(y−1)y′(b)y−xy′=y′y2ey(d)f(y)2dxdy+3f(y)f′(y)x=f′(y)
Online integral calculators and the concept of indefinite integral
Published in International Journal of Mathematical Education in Science and Technology, 2023
The issues discussed in our paper can be attributed to the topic ‘Symbolism in mathematics’, which, in particular, places great importance on the heuristic potential of mathematical symbols. The Leibniz notation for the indefinite integral indicates unambiguously the operation performed, the integrand, and the variable of integration, which is necessary in the case when the integrand depends on parameters. This notation is convenient for performing substitutions and integrating by parts. However, it does not contain at least minimal information about what is considered the result of calculating the indefinite integral. This leads to mismatches in the results of calculating the same integral by different online calculators and the resulting misunderstandings that students experience as well as to errors when applying the results of calculation to definite integrals.