608 VIII. ON THE STRUCTURE OF MATHEMATICS |
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Not so, however, with temperature, or density, or many other derived magnitudes, as we call them. If we have a body of temperature of one degree and combine it with another body of equal temperature the synthesis will not have a temperature of two degrees, (as in the case of weight), but of one degree. This applies to density . , two bodies of density one each will not give us a body of density two, but of density one.
Before further analysis of the problems of linearity and additivity, it will be well to consider a few definitions.
If an entity
u is changed into an entity v by some process, the change may be regarded as the result of an operation performed upon u, the operand, which has converted it into v. If we denote the operation by/, then the result might be written as v =fu. The symbol of the operation / is called the operator. We are familiar with many such; indeed the symbols for all mathematical operations may be treated as operators. So for instance the symbol indicates the operation of extracting the square root. If we deal with a range of values for a variable x, what we have defined as the function symbol,< may be treated as an operator whose operation on may be indicated by the symbol. The operation of differentiation may be symbolized by D, the result of whose operation on the variable «, Du is the derivative of u. The sign ofthe definite integral may be taken as indicating an operation which converts a function into a number , .
It is important to know that many of the rules of algebra and arithmetic when defined in this way, give rise to a calculus of operations. The fundamental notion in such a calculus is that of a product. If « is operated upon by / the result
v is indicated by, or symbolically, If t; in turn is operatedupon by
g the result w is indicated by gv, or symbolically,whencethe operation
gf which converts w directly into w is called the product of / and g. If this operation is repeated several times in succession the usual notation of powers is used, for instance. Not applying the operator atall, which we would denote by
f°, leaves w unchanged, which we indicate symbolically by the equation. The operatoris equivalent to multi-plication by , whencemay be called the
idem operator. We see alsothat the law of indices holds; namely, the
For our purpose we will analyse only one special case; namely, where we have
u, v and u +v as operands, and such an operator, /, that Expressed in words, this means that the operator applied to the sum of the two operands gives a result equal to the sum of the results of operating upon each operand separately. Such a special operator is called a linear, or distributive, operator,In terms of functions we would havewhich may be
called a functional equation.
It has been proved that such a functional equation has one type of solutions; namely, when / is equivalent to a multiplication by a constant, or This fact is of great importance for us. Many problemsin science are stated in terms of variation. For purposes of analysis a statement |
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