Page 103 - Intro to Tensor Calculus

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Note in the definition of a bilinear form that the scalar function ϕ is linear in both the arguments ~x and
~y. An example of a bilinear form is the dot product relation

ϕ(~x, ~y)= ~x · ~y (1.3.69)

where both ~x and ~y belong to the same vector space V.
The definition of a bilinear form suggests how multilinear forms can be defined.

Definition: (Multilinear forms) A multilinear form of degree M or a M degree
linear form in the vector arguments

~x 1 ,~x 2 ,..., ~x M

is a scalar function
ϕ(~x 1 ,~x 2 ,..., ~x M )
of M vector arguments which satisfies the property that it is a linear form in each of its
arguments. That is, ϕ must satisfy for each j =1, 2,... ,M the properties:

(i) ϕ(~x 1 ,..., ~x j1 + ~x j2 ,... ~x M )= ϕ(~x 1 ,..., ~x j1 ,..., ~x M )+ ϕ(~x 1 ,..., ~x j2 ,..., ~x M )

(ii) ϕ(~x 1 ,... ,µ~x j ,... , ~x M )= µϕ(~x 1 ,..., ~x j ,..., ~x M )
(1.3.70)
for all arbitrary vectors ~x 1 ,... , ~x M in the vector space V and all real numbers µ.

An example of a third degree multilinear form or trilinear form is the triple scalar product

ϕ(~x, ~y,~)= ~x · (~y × ~). (1.3.71)
z
z
Note that multilinear forms are independent of the coordinate system selected and depend only upon the
vector arguments. In a three dimensional vector space we select the basis vectors ( b e 1 , b e 2 , b e 3 ) and represent
all vectors with respect to this basis set. For example, if ~x, ~y,~ are three vectors we can represent these
z
vectors in the component forms
j
i
~x = x b e i , ~y = y b e j , z k (1.3.72)
~ = z b e k
where we have employed the summation convention on the repeated indices i, j and k. Substituting equations
(1.3.72) into equation (1.3.71) we obtain

i j k
i
k
j
ϕ(x b e i ,y b e j ,z b e k )= x y z ϕ( b e i , b e j , b e k ), (1.3.73)
since ϕ is linear in all its arguments. By defining the tensor quantity

ϕ( b e i , b e j , b e k )= e ijk (1.3.74)