Page 101 - Intro to Tensor Calculus

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are tensor quantities. These quantities are called the components of the generalized velocity. The coordinates
2
1
3
x ,x ,x are generalized coordinates. This means we can select any set of three independent variables for
the representation of the motion. The variables selected might not have the same dimensions. For example,
3
1
1
2
3
in cylindrical coordinates we let (x = r, x = θ, x = z). Here x and x have dimensions of distance but x 2
has dimensions of angular displacement. The generalized velocities are
dx 1 dr dx 2 dθ dx 3 dz
1 2 3
v = = , v = = , v = = .
dt dt dt dt dt dt
2
1
3
Here v and v have units of length divided by time while v has the units of angular velocity or angular
change divided by time. Clearly, these dimensions are not all the same. Let us examine the physical
components of the generalized velocities. We find in cylindrical coordinates h 1 =1,h 2 = r, h 3 =1 and the
physical components of the velocity have the forms:
dr 2 dθ 3 dz
1
v r = v(1) = v h 1 = , v θ = v(2) = v h 2 = r , v z = v(3) = v h 3 = .
dt dt dt
Now the physical components of the velocity all have the same units of length divided by time.

Additional examples of the use of physical components are considered later. For the time being, just
remember that when tensor equations are derived, the equations are valid in any generalized coordinate
system. In particular, we are interested in the representation of physical laws which are to be invariant and
independent of the coordinate system used to represent these laws. Once a tensor equation is derived, we
can chose any type of generalized coordinates and expand the tensor equations. Before using any expanded
tensor equations we must replace all the tensor components by their corresponding physical components in
order that the equations are dimensionally homogeneous. It is these expanded equations, expressed in terms
of the physical components, which are used to solve applied problems.

Tensors and Multilinear Forms
Tensors can be thought of as being created by multilinear forms defined on some vector space V. Let
us define on a vector space V a linear form, a bilinear form and a general multilinear form. We can then
illustrate how tensors are created from these forms.

Definition: (Linear form) Let V denote a vector space which
contains vectors ~x, ~x 1 ,~x 2 ,.... A linear form in ~x is a scalar function
ϕ(~x) having a single vector argument ~x which satisfies the linearity
properties:

(i) ϕ(~x 1 + ~x 2 )= ϕ(~x 1 )+ ϕ(~x 2 )
(1.3.66)
(ii) ϕ(µ~x 1 )= µϕ(~x 1 )

for all arbitrary vectors ~x 1 ,~x 2 in V and all real numbers µ.