Triple scalar product
The triple scalar product is the scalar product of the first vector with the vector product of the other two vectors. It is denoted as (
abc ) or [
abc ].
Recall that
b ×
c has a magnitude equal to the area of a parallelogram with sides
b and
c and a direction normal to the plane of
b and
c . The scalar product of this normal vector and the vector
a is equal to the altitude of the parallelepiped with a common origin and sides
a ,
b , and
c . The triple scalar product has a magnitude equal to the volume of a parallelepiped with a common origin and sides
a ,
b , and
c . The sign of the triple scalar product can be either positive or negative. If
a ,
b , and
c are coplanar, then the altitude of the parallelepiped is zero and thus the triple scalar product is zero.
The triple scalar product can be expressed in terms of the components by using the earlier definitions of the vector product and scalar product.
From the definition of the permutation symbol, the triple scalar product is unchanged by even permutations of
a ,
b , and
c but have the opposite algebraic sign for odd permutations. Also, if any two of
a ,
b , and
c are identical, then permutation of the two identical vectors results in a triple scalar products that are identical and also opposite in sign. This implies that the triple scalar product is zero if two of the vectors are identical.
Triple vector product
The triple vector product of vectors
a ,
b , and
c results from the repeated application of the vector product, i.e.,
a ×(
b ×
c ). Since
b ×
c is normal to the plane of
b and
c and
a ×(
b ×
c ) is normal to
b ×
c, a ×(
b ×
c ) must be in the plane of
b and
c . It is left as an exercise to show that
Second order tensors
A second order tensor can be written as a 3×3 matrix.
A tensor is a physical entity that is the same quantity in different coordinate systems. Thus a second order tensor is defined as an entity whose components transform on rotation of the Cartesian frame of reference as follows.
If
A
ij
=A
ji
the tensor is said to be
symmetric and a symmetric tensor has only six distinct components. If
A
ij
=-A
ij
the tensor is said to be
antisymmetric and such a tensor is characterized by only three nonzero components for the diagonal terms,
A
i
i , are zero. The tensor whose
ij
th element is
A
ji is called the transpose
A’ of
A . The determinant of a tensor is the determinant of the matrix of its components.
Examples of second order tensors
A second order tensor we have already encountered is the
Kronecker delta
. Of its nine components, the six off-diagonal components vanish and the three diagonal components are equal to unity. It transforms as a tensor upon transforming its components to a rotated frame of reference.
because of the orthogonality relation between the directional cosines
l
ij . In fact, the components of
δ
ij in all coordinates remain the same.
δ
ij is called the isotropic tensor for that reason. The transport coefficients (e.g., thermal conductivity) of an
isotropic medium can be expressed as a scalar quantity multiplying
δ
ij
.
