When you make a definition in the form f[args]=rhs or f[args]:=rhs, the Wolfram Language 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.
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 g, and to correspond to an upvalue of g.
In simple cases, you will get the same answers to calculations whether you give a definition for f[g[x]] 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 f[g[x]] 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[g[x]], Exp is a built‐in Wolfram Language function, while g is presumably a function you have added. In such a case, you will typically think of definitions for Exp[g[x]] 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.
Since the full form of the pattern g[x_]+g[y_] is Plus[g[x_],g[y_]], 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 g.
In general, whenever the Wolfram Language encounters a particular function, it tries all the definitions you have given for that function. If you had made the definition for g[x_]+g[y_] a downvalue for Plus, then the Wolfram Language would have tried this definition whenever Plus occurs. The definition would thus be tested every time the Wolfram Language added expressions together, making this very common operation slower in all cases.
However, by giving a definition for g[x_]+g[y_] as an upvalue for g, you associate the definition with g. In this case, the Wolfram Language only tries the definition when it finds a g inside a function such as Plus. Since g 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,…]^=value||make assignments associated with the heads of all the argi|
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.
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 f[args], 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.
|f[…]:=rhs||downvalue for f|
|f/:f[g[…]][…]:=rhs||downvalue for f|
|g/:f[…,g,…]:=rhs||upvalue for g|
|g/:f[…,g[…],…]:=rhs||upvalue for g|
As discussed in "The Meaning of Expressions", you can use Wolfram Language symbols as "tags", to indicate the "type" of an expression. For example, complex numbers in the Wolfram Language are represented internally in the form Complex[x,y], 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 quat. You can represent each object of this type by a Wolfram Language expression of the form quat[data].
In a typical case, you might want quat 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 quat with respect to Plus and Times.
When you define an upvalue for quat with respect to an operation like Plus, what you are effectively doing is to extend the domain of the Plus operation to include quat objects. You are telling the Wolfram Language to use special rules for addition in the case where the things to be added together are quat objects.
In defining addition for quat objects, you could always have a special addition operation, say quatPlus, to which you assign an appropriate downvalue. It is usually much more convenient, however, to use the standard Wolfram Language Plus operation to represent addition, but then to "overload" this operation by specifying special behavior when quat objects are encountered.
You can think of upvalues as a way to implement certain aspects of object‐oriented programming. A symbol like quat represents a particular type of object. Then the various upvalues for quat specify "methods" that define how quat objects should behave under certain operations, or on receipt of certain "messages".