As output, Maple can display the partial derivative  as ; that is, subscript notation can be used to display partial derivatives, and it can be done with two completely different mechanisms. We describe these two techniques, and then investigates the extent to which partial derivatives can be calculated by subscript notation.

The prime as the symbol of ordinary differentiation is very compact (and generally convenient), especially when the derivative is to be evaluated at some specific value of the independent variable. Thus, for example, we have  to indicate that the derivative of  is to be evaluated at the argument .  Of course, the notation

is perhaps a bit less ambiguous, but if  is declared to be a function of , and prime to mean differentiation with respect to , then the prime notation is pretty clear. Moreover, for typesetting in a manuscript, the prime notation is a definite space-saver.

The parallel for partial derivatives is the literal subscript. There are texts such as [1] that indicate partial derivatives with numeric subscripts so that for the function , partial differentiation with respect to  would be indicated by . If it is clear that  is a function of  and , then, like the case for primes with ordinary derivatives, the notation  is unambiguous.

Given the function , for evaluating its partial with respect to  at , the notation  is more compact and just as clear as the operator form

Many texts use the literal subscript for partial differentiation; this notation appears, for example, in [2, 3], two of the classics I still have on my shelf.

The first mechanism in Maple that allowed partial derivatives to be displayed with subscripts is the declare command in the PDEtools package. This command predates the introduction of the Typesetting package that allows commands to be entered in math mode, or in what Maplesoft calls "2D math". Thus, declare would function in math mode, but it is best understood as an artifact of the "1D math" environment.

Within the confines of PDEtools package, executing the declare command changes the display of the partial derivative of  with respect to  to . So, the derivative is expressed with a literal subscript, and the display of the independent variables is suppressed. Table 1 illustrates the basic functionality of this command.

The "Nothing declared" echo in the fourth line of the table indicates that no specific function has been "declared;" hence, all unspecified functions will display with independent variables suppressed, and all partial derivatives will display as literal subscripts. There is a "quiet" option for the declare command that suppresses any output it would otherwise provide. There are undeclare, ON, and OFF commands to remove declarations, and to turn on and off the functionality of the display without having to invoke undeclare. These details are available in the help page for ?PDEtools/declare . where one can find the important excerpt we provide below.

This paragraph clarifies that the changes induced by declare affect only the display (i.e., printing) of derivatives and functions. Neither input nor output is changed in any essential way. Just the display changes.

Maple does not make directly available the subscript as a partial differentiation operator. The experiment that led to this blog consists of the display in Table 3.

The desire to use subscripts as partial differentiation operators is at the heart of this blog. That Maple fails as per Table 3 was a surprise, even after 23 years of using Maple as the mathematical tool of daily usage. Fortunately, the Maple programmers explained that this is a feature of the software, namely, that both  and  evaluate to the same function. In fact, this is the feature that allows  to point to the natural log function, and  to point to the base  logarithm. The secret is in the coding, an example of which is given in Table 4.

Proof that the code in Table 4 defines the function  in such a way as to permit literal subscripts to act as partial differentiation operators is in the calculations in Table 5.

Of course, the coding of the function in Table 4 is beyond tedious. Although the code illustrates what is possible, it is not a practical solution to the problem of using subscripts operationally for partial differentiation.

Table 6 summarizes a practical and workable method for defining subscripts as partial differentiation operators. The method hinges on the Atomic Identifier construct available in math mode. Loosely speaking, an Atomic Identifier is a collection of symbols that have been "locked" together to form a single name. In Maple, an unassigned letter or string of letters is the name of a variable. Some strings of symbols - for example,  - do not form a Maple name because of the space between the two letters. Assignments cannot be made to such collections of characters. However, when frozen into the construction called an Atomic Identifier, they form a valid name.

There are several ways to form an Atomic Identifier. For complex collections of symbols, it is best to assemble the object, select it, and use Context Menu: 2D Math_Convert To_Atomic Identifier. The alternative to the Context Menu is the Format Menu's "Convert To" option, ending at "Atomic Identifier." The keyboard equivalent in the Windows®, and UNIX® operating systems is Control+Shift+A. For the Macintosh® operating system, it's Command+Shift+A.

For a literal subscript, it's easier. Instead of setting the subscript with the underscore key, add the Control key to the combination. Thus, it's Control+Shift+minus in the first two systems, and Command+Shift+minus on the last. Since this adds only one key to the normal combination for setting a subscript, setting the literal subscript and thereby forming an Atomic Identifier is relatively painless. Finally, there is the  template in the Expression palette. This also creates a subscripted name as an Atomic Identifier.

Table 6 summarizes how to use subscripts, formed as Atomic Identifiers, as partial differentiation operators.

The astute reader will realize that the partial derivative has been defined as a function, just as  has been. The Atomic Identifier device has allowed the symbols  and  to be seen as unrelated by Maple, thus bypassing the built-in rule that Maple functions do not distinguish between a function's name and that name subscripted. This may not be completely satisfactory, since each partial derivative has to be defined as a separate function. But the notational clarity that this device can bring to a manuscript makes it worth considering.