The Standard Notation of the Differential d in Derivatives and Integrals

In mathematical notation, the differential symbol d can appear in both roman upright and italic slanted forms. The standard convention, particularly in formal writing and typesetting, is to use the roman upright form for the differential, especially in calculus where it denotes the differential in derivatives and integrals. This article explores the standard notations for the differential symbol in mathematical expressions.

Usage of Roman Upright d

The roman upright form of the differential symbol d is the preferred style in most mathematical texts. It is widely used to denote the differential in derivatives and integrals. For example, in a derivative, we might see:

Derivative: frac{dy}{dx}

And in an integral, the differential would be represented as:

Integral: int fx dx

Usage of Italic Slanted d

While the roman upright form is the standard, the italic slanted form of the differential d can be seen in some informal contexts or specific typesetting styles. However, it is not the standard practice in formal mathematics.

Historical Context

Historically, the differential symbol d has been written in italic, dating back to its origins by Leibniz in the 1600s. The standard at that time was to use italic for mathematical symbols and roman for abbreviated trigonometric functions. For instance, in the expression int 4x sin x dx, the trigonometric function sin is written in roman while x and dx are in italic.

Modern Notations

In recent years, there has been a movement to make constants roman and variables italic, based on a misinterpretation of historical reasons. However, this practice is not widely accepted in textbooks, which continue to adhere to the traditional rule.

ISO 31/XI Recommendations

According to the ISO 31/XI standard, the d in derivatives and integrals should be in roman. However, it is important to note that this standard also writes the variable e in roman, which is not the typical usage in mathematical notation. While the ISO standard exists, it does not force mathematicians to follow it, as it is ultimately a matter of personal preference.

Many mathematicians prefer the italic form due to it being more common and easier to read. The preference for italics is rooted in tradition and personal preference rather than a strict standard.

Conclusion

While the ISO 31/XI standard recommends the roman form for the differential d, the most common and traditional usage in mathematical texts is the italic slanted form. This preference has been upheld for centuries and continues to be the standard in most formal mathematical writing and typesetting.