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Associating Definitions with Different Symbols

When you make a definition in the form or , Mathematica associates your definition with the object f. This means, for example, that such definitions are displayed when you type ?f. In general, definitions for expressions in which the symbol f appears as the head are termed downvalues of f.
Mathematica however also supports upvalues, which allow definitions to be associated with symbols that do not appear directly as their head.
Consider for example a definition like Exp[g[x_]]:=rhs. One possibility is that this definition could be associated with the symbol Exp, and considered as a downvalue of Exp. This is however probably not the best thing either from the point of view of organization or efficiency.
Better is to consider Exp[g[x_]]:=rhs to be associated with , and to correspond to an upvalue of .
f[args]:=rhsdefine a downvalue for f
f[g[args],...]^:=rhsdefine an upvalue for g

Associating definitions with different symbols.

This is taken to define a downvalue for .
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You can see the definition when you ask about .
This defines an upvalue for .
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The definition is associated with .
It is not associated with Exp.
The definition is used to evaluate this expression.
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In simple cases, you will get the same answers to calculations whether you give a definition for as a downvalue for f or an upvalue for g. However, one of the two choices is usually much more natural and efficient than the other.
A good rule of thumb is that a definition for should be given as an upvalue for g in cases where the function f is more common than g. Thus, for example, in the case of Exp, Exp is a built-in Mathematica function, while g is presumably a function you have added. In such a case, you will typically think of definitions for Exp as giving relations satisfied by g. As a result, it is more natural to treat the definitions as upvalues for g than as downvalues for Exp.
This gives the definition as an upvalue for .
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Here are the definitions for so far.
The definition for a sum of 's is used whenever possible.
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Since the full form of the pattern is Plus, a definition for this pattern could be given as a downvalue for Plus. It is almost always better, however, to give the definition as an upvalue for .
In general, whenever Mathematica encounters a particular function, it tries all the definitions you have given for that function. If you had made the definition for a downvalue for Plus, then Mathematica would have tried this definition whenever Plus occurs. The definition would thus be tested every time Mathematica added expressions together, making this very common operation slower in all cases.
However, by giving a definition for as an upvalue for , you associate the definition with . In this case, Mathematica only tries the definition when it finds a inside a function such as Plus. Since presumably occurs much less frequently than Plus, this is a much more efficient procedure.
f[g]^=value or f[g[args]]^=value
make assignments to be associated with g, rather than f
f[g]^:=value or f[g[args]]^:=value
make delayed assignments associated with g
f[arg1,arg2,...]^=valuemake assignments associated with the heads of all the

Shorter ways to define upvalues.

A typical use of upvalues is in setting up a "database" of properties of a particular object. With upvalues, you can associate each definition you make with the object that it concerns, rather than with the property you are specifying.
This defines an upvalue for which gives its area.
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This adds a definition for the perimeter.
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Both definitions are now associated with the object .
In general, you can associate definitions for an expression with any symbol that occurs at a sufficiently high level in the expression. With an expression of the form , you can define an upvalue for a symbol g so long as either g itself, or an object with head g, occurs in args. If g occurs at a lower level in an expression, however, you cannot associate definitions with it.
occurs as the head of an argument, so you can associate a definition with it.
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Here appears too deep in the left-hand side for you to associate a definition with it.
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f[...]:=rhsdownvalue for f
f/:f[g[...]][...]:=rhsdownvalue for f
g/:f[...,g,...]:=rhsupvalue for g
g/:f[...,g[...],...]:=rhsupvalue for g

Possible positions for symbols in definitions.

As discussed in "The Meaning of Expressions", you can use Mathematica symbols as "tags", to indicate the "type" of an expression. For example, complex numbers in Mathematica are represented internally in the form Complex, where the symbol Complex serves as a tag to indicate that the object is a complex number.
Upvalues provide a convenient mechanism for specifying how operations act on objects that are tagged to have a certain type. For example, you might want to introduce a class of abstract mathematical objects of type . You can represent each object of this type by a Mathematica expression of the form quat[data].
In a typical case, you might want objects to have special properties with respect to arithmetic operations such as addition and multiplication. You can set up such properties by defining upvalues for with respect to Plus and Times.
This defines an upvalue for with respect to Plus.
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The upvalue you have defined is used to simplify this expression.
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When you define an upvalue for with respect to an operation like Plus, what you are effectively doing is to extend the domain of the Plus operation to include objects. You are telling Mathematica to use special rules for addition in the case where the things to be added together are objects.
In defining addition for objects, you could always have a special addition operation, say , to which you assign an appropriate downvalue. It is usually much more convenient, however, to use the standard Mathematica Plus operation to represent addition, but then to "overload" this operation by specifying special behavior when objects are encountered.
You can think of upvalues as a way to implement certain aspects of object-oriented programming. A symbol like represents a particular type of object. Then the various upvalues for specify "methods" that define how objects should behave under certain operations, or on receipt of certain "messages".