Page 48 - Intro to Tensor Calculus

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Definition: ( Absolute scalar field) Assume there exists a coordinate
transformation of the type (1.2.30) with Jacobian J different from zero. Let
the scalar function
2
1
N
f = f(x ,x ,... ,x ) (1.2.36)
i
be a function of the coordinates x ,i =1,... ,N in a space V N . Whenever
there exists a function
1 2 N
f = f(x , x ,... , x ) (1.2.37)
i
which is a function of the coordinates x ,i =1,... ,N such that f = J W f,
then f is called a tensor of rank or order zero of weight W in the space V N .
Whenever W = 0, the scalar f is called the component of an absolute scalar
field and is referred to as an absolute tensor of rank or order zero.

That is, an absolute scalar field is an invariant object in the space V N with respect to the group of
coordinate transformations. It has a single component in each coordinate system. For any scalar function
of the type defined by equation (1.2.36), we can substitute the transformation equations (1.2.30) and obtain

1
1
1
N
N
N
f = f(x ,... ,x )= f(x (x),...,x (x)) = f(x ,... , x ). (1.2.38)
Vector Transformation, Contravariant Components
In V N consider a curve C defined by the set of parametric equations

i
i
C : x = x (t), i =1,... ,N
where t is a parameter. The tangent vector to the curve C is the vector

1 2 N
dx dx dx
~
T = , ,..., .
dt dt dt
In index notation, which focuses attention on the components, this tangent vector is denoted
dx i
i
T = , i =1,... ,N.
dt
For a coordinate transformation of the type defined by equation (1.2.30) with its inverse transformation
defined by equation (1.2.32), the curve C is represented in the barred space by

i
N
i
1
2
i
x = x (x (t),x (t),...,x (t)) = x (t), i =1,... ,N,
with t unchanged. The tangent to the curve in the barred system of coordinates is represented by
i
dx i ∂x dx j
= , i =1,... ,N. (1.2.39)
j
dt ∂x dt