# The Leibniz Differential and the Perron–Stieltjes Integral

## Abstract

We implement Leibniz’s idea about the differential as the length of an infinitesimally small elementary interval (a monad) in a form satisfying modern standards of rigor. The concept of sequential differential introduced in this paper is shown to be in good alignment with the standard convention of the integral calculus. As an application of this concept we simplify and generalize the construction of the Perron–Stieltjes integral.

- Authors:

- Steklov Mathematical Institute of the Russian Academy of Sciences (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 22773908

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Mathematical Sciences

- Additional Journal Information:
- Journal Volume: 233; Journal Issue: 1; Other Information: Copyright (c) 2018 Springer Science+Business Media, LLC, part of Springer Nature; http://www.springer-ny.com; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1072-3374

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; AGREEMENTS; ALIGNMENT; INTEGRAL CALCULUS; INTEGRALS; LENGTH

### Citation Formats

```
Shchepin, E. V., E-mail: scepin@mi.ras.su.
```*The Leibniz Differential and the Perron–Stieltjes Integral*. United States: N. p., 2018.
Web. doi:10.1007/S10958-018-3932-8.

```
Shchepin, E. V., E-mail: scepin@mi.ras.su.
```*The Leibniz Differential and the Perron–Stieltjes Integral*. United States. doi:10.1007/S10958-018-3932-8.

```
Shchepin, E. V., E-mail: scepin@mi.ras.su. Wed .
"The Leibniz Differential and the Perron–Stieltjes Integral". United States. doi:10.1007/S10958-018-3932-8.
```

```
@article{osti_22773908,
```

title = {The Leibniz Differential and the Perron–Stieltjes Integral},

author = {Shchepin, E. V., E-mail: scepin@mi.ras.su},

abstractNote = {We implement Leibniz’s idea about the differential as the length of an infinitesimally small elementary interval (a monad) in a form satisfying modern standards of rigor. The concept of sequential differential introduced in this paper is shown to be in good alignment with the standard convention of the integral calculus. As an application of this concept we simplify and generalize the construction of the Perron–Stieltjes integral.},

doi = {10.1007/S10958-018-3932-8},

journal = {Journal of Mathematical Sciences},

issn = {1072-3374},

number = 1,

volume = 233,

place = {United States},

year = {2018},

month = {8}

}

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