Leibniz's notation

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In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, was originally the use of expressions such as dx and dy and to represent "infinitely small" (or infinitesimal) increments of quantities x and y, just as Δx and Δy represent finite increments of x and y respectively. So for y being a function of x, or

y=f(x) \!\ ,

then the derivative of y with respect to x, which later came to be viewed as

\lim_{\Delta x\rightarrow 0}\frac{\Delta y}{\Delta x} = \lim_{\Delta x\rightarrow 0}\frac{f(x + \Delta x)-f(x)}{\Delta x},

was, according to Leibniz, the quotient of an infinitesimal increment of y by an infinitesimal increment of x, or

\frac{\mathrm dy}{\mathrm dx}=f'(x),

where the right hand side is Lagrange's notation for the derivative of f at x.

Similarly, although mathematicians usually now view an integral

\int f(x)\,\mathrm dx

as a limit

\lim_{\Delta x\rightarrow 0}\sum_{i} f(x_i)\,\Delta x,

where Δx is an interval containing xi, Leibniz viewed it as the sum (the integral sign denoting summation) of infinitely many infinitesimal quantities f(x) dx.

One advantage of Leibniz's point of view is that it is compatible with dimensional analysis. For, example, in Leibniz's notation, the second derivative (using implicit differentiation) is:

\frac{\mathrm d^2 y}{\mathrm dx^2}=f''(x)

and has the same dimensional units as \frac{y}{x^2}.[1]

Contents

[edit] History

The Newton-Leibniz approach to calculus was introduced in the 17th century. In the 19th century, mathematicians ceased to take Leibniz's notation for derivatives and integrals literally. That is, mathematicians saw that the concept of infinitesimals contained logical contradictions in the development. A number of 19th century mathematicians (Cauchy, Weierstrass and others) found logically rigorous ways to treat derivatives and integrals without infinitesimals using limits as shown above. Nonetheless, Leibniz's notation is still in general use. Although the notation need not be taken literally, it is usually simpler than alternatives when the technique of separation of variables is used in the solution of differential equations. In physical applications, one may for example regard f(x) as measured in meters per second, and dx in seconds, so that f(x) dx is in meters, and so is the value of its definite integral. In that way the Leibniz notation is in harmony with dimensional analysis.

However, in the 1950s and 1960s, Abraham Robinson introduced ways of treating infinitesimals both literally and logically rigorously, and so rewriting calculus from that point of view. But Robinson's methods are not used by most mathematicians. (One mathematician, Jerome Keisler, has gone so far as to write a first-year-calculus textbook according to Robinson's point of view.)

[edit] Leibniz's notation for differentiation

In Leibniz's notation for differentiation, the derivative of the function f(x) is written:

\frac{\mathrm{d}\bigl(f(x)\bigr)}{\mathrm{d}x}

If we have a variable representing a function, for example if we set

y=f(x) \!\ ,

then we can write the derivative as:

\frac{\mathrm{d}y}{\mathrm{d}x}

Using Lagrange's notation, we can write:

\frac{\mathrm{d}\bigl(f(x)\bigr)}{\mathrm{d}x} = f'(x).

Using Newton's notation, we can write:

\frac{\mathrm{d}x}{\mathrm{d}t} = \dot{x}.

For higher derivatives, we express them as follows:

\frac{\mathrm{d}^n\bigl(f(x)\bigr)}{\mathrm{d}x^n} or \frac{\mathrm{d}^ny}{\mathrm{d}x^n}

denotes the nth derivative of f(x) or y respectively. Historically, this came from the fact that, for example, the third derivative is:

\frac{\mathrm{d} \left(\frac{\mathrm{d} \left( \frac{\mathrm{d} \left(f(x)\right)} {\mathrm{d}x}\right)} {\mathrm{d}x}\right)} {\mathrm{d}x}

which we can loosely write as:

\left(\frac{\mathrm{d}}{\mathrm{d}x}\right)^3 \bigl(f(x)\bigr) = \frac{\mathrm{d}^3}{\left(\mathrm{d}x\right)^3} \bigl(f(x)\bigr)

Now drop the brackets and we have:

\frac{\mathrm{d}^3}{\mathrm{d}x^3}\bigl(f(x)\bigr)\ \mbox{or}\ \frac{\mathrm{d}^3y}{\mathrm{d}x^3}

The chain rule and integration by substitution rules are especially easy to express here, because the "d" terms appear to cancel:

\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{\mathrm{d}y}{\mathrm{d}u} \cdot \frac{\mathrm{d}u}{\mathrm{d}v} \cdot \frac{\mathrm{d}v}{\mathrm{d}w} \cdot \frac{\mathrm{d}w}{\mathrm{d}x} etc.

and:

\int y \, \mathrm{d}x = \int y \frac{\mathrm{d}x}{\mathrm{d}u} \, \mathrm{d}u.

[edit] Notes

  1. ^ Note that \frac{\mathrm d^2 y}{\mathrm d x^2} is shorthand for \frac{\mathrm d{\frac{\mathrm dy}{\mathrm dx}}}{\mathrm dx}, or in other words the second differential of y over the square of the first differential of x. The denominator is not the differential of x2, nor is it the second differential of x.

[edit] See also