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A Brief Note on Calculus and Its Principles

Paul Williams*

Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK

*Corresponding Author:
Paul Williams
Department of Applied Mathematics and Theoretical Physics,
University of Cambridge,
Cambridge,
UK
E-mail: [email protected]

Received: 13-Jan-2022, Manuscript No. JSMS-22-52315; Editor assigned: 17- Jan-2022, Pre QC No. JSMS-22-52315 (PQ); Reviewed: 31- Jan-2022, QC No. JSMS-22-52315; Accepted: 02-Feb-2022, Manuscript No. JSMS-22-52315 (A); Published: 09-Feb-2022, DOI: 10.4172/ J Stats Math Sci.8.1.e001.

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Editorial Note

Calculus

Calculus, often known as infinitesimal calculus or "calculus of infinitesimals," is a branch of mathematics that studies continuous change in the same manner that geometry studies shape and algebra studies arithmetic operations in general.

Differential calculus and integral calculus are the two main disciplines; the former deals with instantaneous rates of change and curve slopes, while the latter deals with accumulation of quantities and areas under or between curves. The fundamental theorem of calculus connects these two branches, and they make use of the fundamental principles of infinite sequences and infinite series converge to a well-defined limit [1]. Isaac Newton and Gottfried Wilhelm Leibniz independently created infinitesimal calculus in the late 17th century. Calculus is used widely in science, engineering, and economics today.

Calculus is a term used in mathematics education to describe courses in elementary mathematical analysis that focus on the study of functions and limits. Calculus (plural calculi) is a Latin term that originally meant "little stone" (the meaning of which is preserved in medicine – see Calculus (medical). Because such stones were employed in ancient Rome to count (or measure) the distance travelled by transportation machines, the definition of the term has changed, and it now commonly refers to a technique of computing [2-4]. As a result, it's commonly used to refer to particular calculating methods and theories such propositional calculus, Ricci calculus, calculus of variations, lambda calculus, and process calculus. Calculus is the study of change in mathematics, similar to how geometry is the study of shape and algebra is the study of operations and how they are used to solve problems. Calculus is a term used in mathematics education to describe courses in elementary mathematical analysis that focus on the study of functions and limits [5-7]. The word calculus comes from the Latin word calculus, which means "little pebble" (the diminutive of calx, meaning "stone"). The name evolved to denote a technique of calculating since such stones were used for counting distances, tallying votes, and conducting abacus arithmetic. It was used in English in this meaning as early as 1672, several years before Leibniz and Newton published their works. The word is also used to refer to particular methods of computation and associated theories, such as propositional calculus, Ricci calculus, calculus of variations, lambda calculus, and process calculus, in addition to differential and integral calculus.

Principles

Limits and infinitesimals

Calculus is generally learned by dealing with tiny amounts of data. Infinitesimals were historically the first technique of doing. These are things that can be regarded like real numbers but are "infinitely tiny" in some ways. An infinitesimal number, for example, might be higher than 0 but smaller than any number in the series 1, 1/2, 1/3,.., and hence smaller than any positive real number. Calculus, in this sense, is a collection of methods for managing infinitesimals. The derivative {\displaystyle dx}dx and {\displaystyle dy}dy was simply their ratio, and the symbols {\displaystyle dy/dx}dy/dx were believed to be infinitesimal.

Because it was impossible to make the concept of an infinitesimal accurate, the infinitesimal method fell out of favour in the nineteenth century. However, in the twentieth century, the notion was revitalised by the emergence of non-standard analysis and smooth infinitesimal analysis, which gave strong foundations for manipulating infinitesimals [8].

In academia, infinitesimals were supplanted by the epsilon, delta approach to limits in the late nineteenth century. Limits characterise a function's value at a given input in terms of its values at other inputs. In the context of the real number system, they represent small-scale dynamics. Calculus is defined as a set of procedures for altering particular limitations in this context.

Differential calculus

The study of the definition, characteristics, and applications of a function's derivative is known as differential calculus. Differentiation is the process of determining the derivative. The derivative at a point in the domain is a technique of storing the function's small-scale behaviour at that point given a function and a point in the domain [9]. It is possible to create a new function, termed the derivative function or simply the derivative of the original function, by determining the derivative of a function at each point in its domain [10]. In mathematical language, the derivative is a linear operator that takes a function as input and outputs another function.

This is more abstract than many of the processes covered in elementary algebra, where functions typically take in one number and output another. If you provide the doubling function three as an input, it will produce six, and if you give the squaring function three as an input, it will output nine. The squaring function, on the other hand, may be used as an input to the derivative [11-12]. This implies that the derivative takes all of the information from the squaring function-such as the fact that two is sent to four, three to nine, four to sixteen, and so on-and applies it to create a new function. The doubling function is the function obtained by deriving the squaring function.

References

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