Mathematical Tools for Physics

12—Tensors 355

in this cased~will be mostly in thexdirection (k 1 ) and is not aligned withF~. In any case there is a relation

betweend~andF~,

d~=f(F~). (2)

The functionfis a tensor.

In both of these examples, the functions involved werevector valued functions of vector variables. They
have the further property that they are linear functions,i.e.ifαandβare real numbers,

I(α~ω 1 +β~ω 2 ) =αI(~ω 1 ) +βI(~ω 2 ), f

(

αF~ 1 +βF~ 2

)

=αf

(~

F 1

)

+βf

(~

F 2

)

,

These two properties are the firstdefinitionof a tensor. (A generalization will come later.) There’s a point

here that will probably cause some confusion. Notice that in the equation~L=I(~ω), the tensor is the function
I. I didn’t refer to “the functionI(~ω)” as you commonly hear in casual discussions. The reason is thatI(~ω),

which equalsL~, is a vector, not a tensor. It is the output of the functionI after the independent variable~ω
has been fed into it. For an analogy, retreat to the case of a real valued function of a real variable. In common
language, you would look at the equationy=f(x)and say thatf(x)is a function, but it’s better to say that
fis a function, and thatf(x)is the single number obtained by feeding the numberxtofin order to obtain
the numberf(x). In this language,fis regarded as containing a vast amount of information, all the relations
betweenxandy.f(x)however is just a single number. Think offas the whole graph of the function andf(x)
as telling you one point on the graph. This apparently trivial distinction will often make no difference, but there
are a number of cases (particularly here) where a misunderstanding of this point will cause confusion.

Definition of “Function”
An abstract definition of a function is useful: X and Y are sets (possibly the same set) andxandyare elements
of these sets (x∈X,y∈Y). A new set F is formed consisting of some collection of ordered pairs of elements,
one from X and one from Y. That is, a typical element of the set F would be(x 1 , y 1 )wherex 1 ∈X andy 1 ∈Y.
Such a set is called a “relation” between X and Y.
This relation is not yet a function. One additional item is needed. We now require of F that if(x,y 1 )∈F
and(x,y 2 )∈F theny 1 =y 2. This is the statement that the function is single-valued. The ordinary notation for
a function isy=F(x); in the language of sets we say(x,y)∈F. The set F is the function. You can picture it
as a graph, containing all the information about the function; it is by definition single-valued.