Notation for differentiation
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In differential calculus, there is no single uniform notation for differentiation. Instead, several different notations for the derivative of a function or variable have been proposed by different mathematicians. The usefulness of each notation varies with the context, and it is sometimes advantageous to use more than one notation in a given context. The most common notations for differentiation (and its opposite operation, the antidifferentiation or indefinite integration) are listed below.
The original notation employed by Gottfried Leibniz is used throughout mathematics. It is particularly common when the equation y = f(x) is regarded as a functional relationship between dependent and independent variables y and x. Leibniz's notation makes this relationship explicit by writing the derivative as
The function whose value at x is the derivative of f at x is therefore written
Higher derivatives are written as
This is a suggestive notational device that comes from formal manipulations of symbols, as in,
Logically speaking, these equalities are not theorems. Instead, they are simply definitions of notation. Indeed, evaluating above using the quotient rule and using dd to distinguish from d2 in the above notation gives
The value of the derivative of y at a point x = a may be expressed in two ways using Leibniz's notation:
Leibniz's notation allows one to specify the variable for differentiation (in the denominator). This is especially helpful when considering partial derivatives. It also makes the chain rule easy to remember and recognize:
Leibniz's notation for differentiation does not require assigning a meaning to symbols such as dx or dy on their own, and some authors do not attempt to assign these symbols meaning. Leibniz treated these symbols as infinitesimals. Later authors have assigned them other meanings, such as infinitesimals in non-standard analysis or exterior derivatives.
Leibniz's notation for antidifferentiation
∫∫ y dx2
One of the most common modern notations for differentiation is named after Joseph Louis Lagrange, even though it was actually invented by Euler and just popularized by the former. In Lagrange's notation, a prime mark denotes a derivative. If f is a function, then its derivative evaluated at x is written
It first appeared in print in 1749.
Higher derivatives are indicated using additional prime marks, as in for the second derivative and for the third derivative. The use of repeated prime marks eventually becomes unwieldy. Some authors continue by employing Roman numerals, usually in lower case, as in
to denote fourth, fifth, sixth, and higher order derivatives. Other authors use Arabic numerals in parentheses, as in
This notation also makes it possible to describe the nth derivative, where n is a variable. This is written
Unicode characters related to Lagrange's notation include
- U+2032 ◌′ PRIME (derivative)
- U+2033 ◌″ DOUBLE PRIME (double derivative)
- U+2034 ◌‴ TRIPLE PRIME (third derivative)
- U+2057 ◌⁗ QUADRUPLE PRIME (fourth derivative)
When there are two independent variables for a function f(x, y), the following convention may be followed:
Lagrange's notation for antidifferentiation
When taking the antiderivative, Lagrange followed Leibniz's notation:
However, because integration is the inverse operation of differentiation, Lagrange's notation for higher order derivatives extends to integrals as well. Repeated integrals of f may be written as
- for the first integral (this is easily confused with the inverse function ),
- for the second integral,
- for the third integral, and
- for the nth integral.
Leonhard Euler's notation uses a differential operator suggested by Louis François Antoine Arbogast, denoted as D (D operator) or D̃ (Newton–Leibniz operator). When applied to a function f(x), it is defined by
- for the second derivative,
- for the third derivative, and
- for the nth derivative.
Euler's notation leaves implicit the variable with respect to which differentiation is being done. However, this variable can also be notated explicitly. When f is a function of a variable x, this is done by writing
- for the first derivative,
- for the second derivative,
- for the third derivative, and
- for the nth derivative.
When f is a function of several variables, it's common to use a "∂" rather than D. As above, the subscripts denote the derivatives that are being taken. For example, the second partial derivatives of a function f(x, y) are:
Euler's notation is useful for stating and solving linear differential equations, as it simplifies presentation of the differential equation, which can make seeing the essential elements of the problem easier.
Euler's notation for antidifferentiation
- for a first antiderivative,
- for a second antiderivative, and
- for an nth antiderivative.
Newton's notation for differentiation (also called the dot notation, or sometimes, rudely, the flyspeck notation for differentiation) places a dot over the dependent variable. That is, if y is a function of t, then the derivative of y with respect to t is
Higher derivatives are represented using multiple dots, as in
Newton extended this idea quite far:
Unicode characters related to Newton's notation include:
- U+0307 ◌̇ COMBINING DOT ABOVE (derivative)
- U+0308 ◌̈ COMBINING DIAERESIS (double derivative)
- U+20DB ◌⃛ COMBINING THREE DOTS ABOVE (third derivative) ← replaced by "combining diaeresis" + "combining dot above".
- U+20DC ◌⃜ COMBINING FOUR DOTS ABOVE (fourth derivative) ← replaced by "combining diaeresis" twice.
- U+030D ◌̍ COMBINING VERTICAL LINE ABOVE (integral)
- U+030E ◌̎ COMBINING DOUBLE VERTICAL LINE ABOVE (second integral)
- U+25AD ▭ WHITE RECTANGLE (integral)
- U+20DE ◌⃞ COMBINING ENCLOSING SQUARE (integral)
- U+1DE0 ◌ᷠ COMBINING LATIN SMALL LETTER N (nth derivative)
Newton's notation is generally used when the independent variable denotes time. If location y is a function of t, then denotes velocity and denotes acceleration. This notation is popular in physics and mathematical physics. It also appears in areas of mathematics connected with physics such as differential equations. It is only popular for first and second derivatives, but in applications these are usually the only derivatives that are necessary.
When taking the derivative of a dependent variable y = f(x), an alternative notation exists:
Newton's notation for integration
Newton developed many different notations for integration in his Quadratura curvarum (1704) and later works: he wrote a small vertical bar or prime above the dependent variable (y̍ ), a prefixing rectangle (▭y), or the inclosure of the term in a rectangle ( ) to denote the fluent or time integral (absement).
To denote multiple integrals, Newton used two small vertical bars or primes (y̎), or a combination of previous symbols ▭y̍ , to denote the second time integral (absity).
Higher order time integrals were as follows:
For a function f of an independent variable x, we can express the derivative using subscripts of the independent variable:
This type of notation is especially useful for taking partial derivatives of a function of several variables.
Partial derivatives are generally distinguished from ordinary derivatives by replacing the differential operator d with a "∂" symbol. For example, we can indicate the partial derivative of f(x, y, z) with respect to x, but not to y or z in several ways:
What makes this distinction important is that a non-partial derivative such as may, depending on the context, be interpreted as a rate of change in relative to when all variables are allowed to vary simultaneously, whereas with a partial derivative such as it is explicit that only one variable should vary.
Other notations can be found in various subfields of mathematics, physics, and engineering; see for example the Maxwell relations of thermodynamics. The symbol is the derivative of the temperature T with respect to the volume V while keeping constant the entropy (subscript) S, while is the derivative of the temperature with respect to the volume while keeping constant the pressure P. This becomes necessary in situations where the number of variables exceeds the degrees of freedom, so that one has to choose which other variables are to be kept fixed.
Higher-order partial derivatives with respect to one variable are expressed as
and so on. Mixed partial derivatives can be expressed as
In this last case the variables are written in inverse order between the two notations, explained as follows:
Notation in vector calculus
where the terminology symbolically reflects that the operator ∇ will also be treated as an ordinary vector.
- Gradient: The gradient of the scalar field is a vector, which is symbolically expressed by the multiplication of ∇ and scalar field ,
- Divergence: The divergence of the vector field A is a scalar, which is symbolically expressed by the dot product of ∇ and the vector A,
- Laplacian: The Laplacian of the scalar field is a scalar, which is symbolically expressed by the scalar multiplication of ∇2 and the scalar field φ,
- Rotation: The rotation , or , of the vector field A is a vector, which is symbolically expressed by the cross product of ∇ and the vector A,
Many symbolic operations of derivatives can be generalized in a straightforward manner by the gradient operator in Cartesian coordinates. For example, the single-variable product rule has a direct analogue in the multiplication of scalar fields by applying the gradient operator, as in
Many other rules from single variable calculus have vector calculus analogues for the gradient, divergence, curl, and Laplacian.
Further notations have been developed for more exotic types of spaces. For calculations in Minkowski space, the d'Alembert operator, also called the d'Alembertian, wave operator, or box operator is represented as , or as when not in conflict with the symbol for the Laplacian.
- Morris, Carla C. (2015-07-28). Fundamentals of calculus. Stark, Robert M., 1930-2017. Hoboken, New Jersey. ISBN 9781119015314. OCLC 893974565.
- Osborne, George A. (1908). Differential and Integral Calculus. Boston: D. C. Heath and co. pp. 63-65.
- The Differential and Integral Calculus (Augustus De Morgan, 1842). pp. 267-268
- Lagrange, Nouvelle méthode pour résoudre les équations littérales par le moyen des séries (1770), p. 25-26. http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN308900308%7CLOG_0017&physid=PHYS_0031
- "The D operator - Differential - Calculus - Maths Reference with Worked Examples". www.codecogs.com. Archived from the original on 2016-01-19.
- Weisstein, Eric W. "Differential Operator." From MathWorld--A Wolfram Web Resource. "Archived copy". Archived from the original on 2016-01-21. Retrieved 2016-02-07.CS1 maint: archived copy as title (link)
- Weisstein, Eric W. "Repeated Integral." From MathWorld--A Wolfram Web Resource. "Archived copy". Archived from the original on 2016-02-01. Retrieved 2016-02-07.CS1 maint: archived copy as title (link)
- Zill, Dennis G. (2009). "1.1". A First Course in Differential Equations (9th ed.). Belmont, CA: Brooks/Cole. p. 3. ISBN 978-0-495-10824-5.
- Newton's notation reproduced from:
- 1st to 5th derivatives: Quadratura curvarum (Newton, 1704), p. 7 (p. 5r in original MS: "Archived copy". Archived from the original on 2016-02-28. Retrieved 2016-02-05.CS1 maint: archived copy as title (link)).
- 1st to 7th, nth and (n+1)th derivatives: Method of Fluxions (Newton, 1736), pp. 313-318 and p. 265 (p. 163 in original MS: "Archived copy". Archived from the original on 2017-04-06. Retrieved 2016-02-05.CS1 maint: archived copy as title (link))
- 1st to 5th derivatives : A Treatise of Fluxions (Colin MacLaurin, 1742), p. 613
- 1st to 4th and nth derivatives: Articles "Differential" and "Fluxion", Dictionary of Pure and Mixed Mathematics (Peter Barlow, 1814)
- 1st to 4th, 10th and nth derivatives: Articles 622, 580 and 579 in A History of Mathematical Notations (F .Cajori, 1929)
- 1st to 6th and nth derivatives: The Mathematical Papers of Isaac Newton Vol. 7 1691-1695 (D. T. Whiteside, 1976), pp.88 and 17
- 1st to 3rd and nth derivatives: A History of Analysis (Hans Niels Jahnke, 2000), pp. 84-85
- Weisstein, Eric W. "Overdot." From MathWorld--A Wolfram Web Resource. "Archived copy". Archived from the original on 2015-09-05. Retrieved 2016-02-05.CS1 maint: archived copy as title (link)
- Weisstein, Eric W. "Double Dot." From MathWorld--A Wolfram Web Resource. "Archived copy". Archived from the original on 2016-03-03. Retrieved 2016-02-05.CS1 maint: archived copy as title (link)
- Article 580 in Florian Cajori, A History of Mathematical Notations (1929), Dover Publications, Inc. New York. ISBN 0-486-67766-4
- "Patterns of Mathematical Thought in the Later Seventeenth Century", Archive for History of Exact Sciences Vol. 1, No. 3 (D. T. Whiteside, 1961), pp. 361-362,378
- S.B. Engelsman has given more strict definitions in Families of Curves and the Origins of Partial Differentiation (2000), pp. 223-226
- Newton's notation for integration reproduced from:
- 1st to 3rd integrals: Quadratura curvarum (Newton, 1704), p. 7 (p. 5r in original MS: "Archived copy". Archived from the original on 2016-02-28. Retrieved 2016-02-05.CS1 maint: archived copy as title (link))
- 1st to 3rd integrals: Method of Fluxions (Newton, 1736), pp. 265-266 (p. 163 in original MS: "Archived copy". Archived from the original on 2017-04-06. Retrieved 2016-02-05.CS1 maint: archived copy as title (link))
- 4th integrals: The Doctrine of Fluxions (James Hodgson, 1736), pp. 54 and 72
- 1st to 2nd integrals: Articles 622 and 365 in A History of Mathematical Notations (F .Cajori, 1929)