An invariant is a variant of order zero, i.e. a single number or single vector, that has the
same value in all coordinate systems. Since an invariant is independent of coordinates, it must
represent a meaningful geometric quantity. It is for this reason that producing an invariant is the
ultimate goal of any analysis. Crucial to constructing invariants is the concept of a tensor
which we will introduce over the course of this and the next Chapters. Tensors are variants of any
order whose values in different coordinate systems are related by a special set of relationships
known as the tensor transformation rules. Tensors are closed under addition, multiplication,
and contraction. In other words, a sum of tensors is also a tensor, a product of tensors is also a
tensor, and a contraction of a tensor is also a tensor. Crucially, a tensor of order zero is
guaranteed to be an invariant. Thus, although tensors have different values in different coordinate
systems, and therefore contain within them artifacts of the coordinate system, they may be thought
of as preserving the geometric meaning of the objects that they represent in the coordinate space.
This Chapter is devoted to the analysis of the transformation rules, i.e. the relationships between
the values of a variant in two different coordinate systems, for the most fundamental variants in a
Euclidean space. This will lay the necessary groundwork for the exploration of tensors in the next
Chapter.
13.1Alternative coordinate systems
In a three-dimensional Euclidean space, simultaneously introduce two coordinate systems
and
or, collectively, and
.
We will refer to these coordinates as unprimed and primed. You may be surprised to
see the prime placed on the index rather than the letter . However, you will soon see the
tremendous utility of this style, as it will allow us to use every letter twice -- once for the
unprimed coordinate system and once for the primed. It will also enable us to treat the two
coordinate systems in a more parallel fashion.
Although the two coordinate systems play equal roles, we will tend to think of the unprimed
coordinates as
the "original" coordinate system and the primed coordinates
as the "new" coordinate system. We will therefore tend to highlight the transformation from the
unprimed to the primed coordinates even though the inverse transformation is equally relevant.
With each point in the Euclidean space we have
associated two sets of coordinates: and . This correspondence creates a natural
analytical relationship between the two coordinate systems: with each set of unprimed coordinates
we can associate the set of primed coordinates
that correspond to the same point . This relationship can be described
by a set of three functions ,
,
and ,
i.e.
The inverse relationship can likewise be described by a set of three functions ,
, and
, i.e.
Collectively, we refer to these relationships as the equations of the coordinate
transformation.
For an example of equations of a coordinate transformation, let be
Cartesian coordinates and
be spherical coordinates aligned with the Cartesian coordinates in the usual way. Then the
functions ,
,
and
are
while the functions ,
, and
are
It may be helpful to see these functions in terms of the familiar variables and . We have
and
In the spirit of the tensor notation, let us switch to a more economical way of recording the
relationships between the coordinate systems. Switch from the symbols ,
,
and ,
,
to
,
,
and ,
,
to
denote the functions and enumerate them with indices. Thus, the transformation from the unprimed to
the primed coordinates reads
while the inverse transformation
reads
Furthermore, collapse the arguments
of the functions resulting in the compact equations
and
It must be noted that the last two
equations exhibit such economy of notation, that they are nearly cryptic out of context. The reader
is therefore encouraged to apply the technique of unpacking until they become comfortable with this
exceedingly concise notation.
13.2The transformation rule of the covariant basis
Recall that in a Euclidean space referred to a coordinate system , the
covariant basis is
constructed by differentiating the position vector as a function of the coordinates with
respect to each of the coordinates, i.e.
Similarly, in the primed coordinates
,
the covariant basis is
constructed by differentiating the position vector as a function of the coordinates
with respect to each of the coordinates:
Also recall that the fact that the
two algorithms in the two coordinate systems are identical is what makes the covariant basis a
variant.
Our present goal is to determine the transformation rule for the covariant basis , i.e.
the relationship between and
. To
this end, we must first relate the functions and . Clearly, is obtained by substituting the
equations of the coordinate transformation
into . This relationship is expressed by the
identity
which, in the fully unpacked form,
reads:
It is intuitively obvious why this identity holds. The function on the right translates the primed
coordinates
to the equivalent unprimed coordinates by
the equations of the transformation and then uses the function to map the resulting unprimed coordinates to the
corresponding position vector -- and that is precisely the mapping that the function accomplishes in a single step.
The resulting relationship
is an identity in the primed
coordinates
and can therefore be differentiated with respect to each coordinate. An application of the chain
rule yields
On the left, we immediately
recognize
as . On
the right, is
. Thus,
the relationship between and
reads
This identity is precisely the transformation rule for the covariant basis that
we have sought. It tells us how the elements of the
covariant basis in the "new" coordinate system can be obtained from the elements of the
covariant basis in the "original" coordinate system. Looking ahead to the concept of a
tensor, this is precisely the special kind of transformation described by that term.
The inverse relationship
can be obtained by applying the same
approach to the inverse relationship
between the functions and .
13.3The Jacobian of a coordinate transformation
13.3.1The definition
The collection of the partial derivatives
is known as the Jacobian of
the coordinate transformation
It is denoted by the symbol , i.e.
The Jacobian of the inverse
transformation
is denoted by , i.e.
In terms of the newly defined
symbols
and ,
the transformation rules for the covariant basis read
In Chapter 3, we discovered that two bases have the
same orientation when the determinant of the matrix that relates them is positive, and
opposite orientations when the determinant is negative. Therefore, the coordinate
transformation preserves the orientation of the basis if the determinant of the Jacobian is
positive and reverses the orientation otherwise.
13.3.2The Jacobians for the transformation between Cartesian and polar coordinates
Since this example uses two of the most common coordinate systems, let us use the standard names
and for the variables and the
corresponding functions representing the coordinate transformations. The "forward" and the
"inverse" equations of the coordinate transformation read
Adopting the convention that the
superscript is the first index and the subscript is the second, we find that
and
Note that we have already
encountered these matrices in Section 8.6 when discussing
the partial derivatives of inverse sets of functions.
Also note that each Jacobian has its own natural variables. For example, is
naturally a function of and and is
naturally a function of and . In general, is
naturally a function of the unprimed coordinates and
is
naturally a function of the primed coordinates .
However, nothing is preventing us from converting either Jacobian to the other set of variables by
substituting the equations of the coordinate transformation. In the case of the interplay between
Cartesian and polar coordinates, we may state that
and
This observation will take on
significant relevance often, including in the next Section as well as later in this Chapter when we
are faced with the need to differentiate the Jacobians with respect to both natural and "unnatural"
variables.
Interestingly, we have already discussed the conversion of the matrix corresponding to to
the variables and in Section 8.6. However, at that point, the substitution
and was a formal analytical matter dictated by
the chain rule. Specifically, in the chain rule
the derivative is
evaluated at , i.e. is substituted into . In
the present context, on the other hand, the substitution is much more intuitive: since all
analytical objects are associated with the physical point at which they are being evaluated,
we are free to express them in any coordinates we wish. And, if a particular object happens to be
expressed in terms of and , we can re-express it in terms of
and by utilizing the equations of the
coordinate transformation
since these questions translate the polar coordinates of a point to its Cartesian coordinates.
13.3.3Confirming the identity
As a sanity check, let us confirm that the covariant bases and
for
Cartesian and polar coordinates are indeed related by these matrices. Denote the covariant basis in
Cartesian coordinates by the familiar symbols and , and in polar coordinates by the symbols and
. The
identity
in matrix form reads
In other words
An inspection of the geometric arrangement of the vectors , , , and
confirms the correctness of the above identities:
(13.46)
13.3.4The inverse relationship between and
Consider the matrices associated with the systems and
arranged in such a way that the superscripts refers to the row and the subscripts to the column. As
we demonstrated in Section 8.3, matrices that represent
opposite transformations between two bases are the inverses of each other, i.e.
and
Note that these relationships also
follow from the discussion in Section 8.6, where we showed
that the matrices of partial derivatives of inverse sets of functions are inverses of each other.
Looking ahead, the inverse relationship between the Jacobians
and
will prove key to achieving invariance. In a nutshell, when a variant that transforms by is
contracted with another that transforms by ,
the two transformations cancel each other leading to invariance.
In order to confirm the above identities for the interplay between Cartesian and polar coordinates,
the two Jacobians need to be expressed in the same variables. When expressed in the same
coordinates, the two matrices can be multiplied to confirm they are the inverses of each other. For
example, in polar coordinates, the identity
reads
while in Cartesian coordinates, it
reads
13.3.5The operation of inverting the Jacobian
Note that we derived the transformation rules
for the covariant basis independently from each other. But, of course, given our experience with
inverting the metric tensor, first introduced in Section 9.5.2 and subsequently used on a number of occasions, we
recognize that the two identities above are equivalent and can be derived from each other on the
basis of the matrix inverse relationship between the two Jacobians. As a refresher, let us give the
details of the analytical tactic that will show the equivalence of the above identities. Naturally,
it follows the Linear Algebra logic of multiplying both sides of by to
yield .
Start with the equation
and contract both sides with ,
i.e.
Since ,
we have
and therefore
Renaming into , we arrive at the inverse identity
as we set out to show.
Of course, the same approach applies to any variant , i.e.
Naturally, the operation can be
reversed leading to the conclusion that
We will refer to this type of
operation as inverting the Jacobian.
One of the consequences of our ability to invert the Jacobians is that for some variants -- namely,
tensors -- we will only need to formulate the transformation rules from the original to the new
coordinates since the inverse transformation follows easily.
13.4The transformation rule for the covariant metric tensor
From the transformation rule for the covariant basis , we
can easily derive the transformation rule for the metric tensor .
In the primed coordinates, the expression for
reads
Dotting the identities
we find
The dot product on the left yields
and the one on the right yields .
Therefore, we arrive at the following rule for the transformation of the metric tensor
The inverse transformation rule, of
course, reads
In these identities, we begin to see the tendency of each index to be contracted with its own
Jacobian in the coordinate transformation rule. We also see that the placements of indices, which
were determined by somewhat informal (albeit intuitive) rules in our earlier explorations
effectively predict the manner in which the corresponding variants transform. Generally speaking,
we will observe that variants with subscripts change by
while variants with superscripts change by .
Of course, this is by design: the rules for placing indices were specifically formulated so that
those placements signal the correct transformation rule.
To close out this Section, let us confirm the rule
for the Cartesian (unprimed) to
polar (primed) transformation. Recall that
Thus, the identity
corresponds to
which is easily confirmed to be
true.
13.5The transformation rule for the contravariant components of a vector
The derivation of the transformation rule for the contravariant coordinates of a
vector is particularly satisfying. By definition, the components
are
the decomposition coefficients of with respect to , i.e.
In the primed coordinate system, the
components are,
similarly, the coefficients of with respect to , i.e.
The key to the relationship between
and
is, of course, that is the same vector in both equations. Therefore, we can
equate the two right sides of those equations, i.e.
Next, substitute the transformation
rule
for on the
right:
Both sides of this identity
represent linear decomposition with respect to the basis .
Matching up the coefficients, we arrive at the desired transformation rule for :
Note, crucially, that the components
transform by an opposite rule compared to the basis .
Indeed,
transforms by ,
while
transforms by ,
as we observe by inspecting the two transformation rules side by side:
These two modes of transformation are called covariant -- i.e. changing in the same
manner as the basis -- and contravariant -- i.e. changing in the opposite way. When
we combine the two types of objects in a contraction, the inverse matrices meet and cancel each
other, leading to invariance.
To illustrate this point further, consider the combination
In primed coordinates, the
corresponding variant is
Using the transformation rules
above, we find that
Note that we had to use the letter
in the contraction
since was already used in .
Now, observe that
and
participate in a contraction on .
Recalling that ,
we have
Since , we
have
We therefore conclude that and
are the same vector:
In other words, the combination
is an
invariant.
13.6The transformation rule for the contravariant basis
Given the transformation rules that we have discovered so far, we fully expect that
transforms according to the equation
This is indeed the case. However,
demonstrating this relationship requires more inventiveness than the earlier ones. What makes this
derivation more challenging is the fact that all concise definitions for the contravariant basis
are
implicit. According to one definition, is
obtained by contracting the covariant basis with
the contravariant metric tensor :
This definition appears explicit,
but recall
is defined implicitly as the matrix inverse of
and we will not have an explicit
expression for the matrix inverse until Chapter 16. Thus, if we wanted to establish the transformation rule for
first and, from it, the rule for , we
would be faced with the same lack of an explicit expression for .
We will instead base our argument on the definition of the contravariant basis contained in the
identity
To discover the transformation rule
for ,
recall the corresponding rule for the covariant basis
expressed in the form
Substituting this relationship into
the identity
yields
In order to eliminate on
the left, contract both sides with its matrix inverse .
Since ,
, and
,
we have
Rename into
:
This intermediate identity is of
interest in its own right as it relates elements from two different coordinate systems. It also
gives an elegant expression for the Jacobian :
Returning to the form
we next wish to "attach" the
Jacobian to
in
anticipation that the combination
corresponds to .
To this end, contract both sides of the above identity with
Since ,
we have
This identity is sufficient to
conclude that
equals .
Recall that
is defined by the identity
Therefore, a comparison of the last
two identities immediately yields
as we set out to prove.
13.7The transformation rule for the contravariant metric tensor
As was the case for the covariant metric tensor ,
the transformation rule for the contravariant metric tensor
follows readily from that for the contravariant basis .
Recall that
is given by
Thus,
in the primed coordinates is given by
Since
while ,
we have
Finally, we recognize the dot
product
on the right as ,
and therefore arrive at the following transformation rule for :
An alternative derivation of this identity can be constructed directly on the basis of the original
definition of the contravariant metric tensor
In the alternative coordinate
system,
is given by the analogous identity
Utilizing the transformation rule
for the covariant basis, we find
Our goal is, of course, to isolate
on the left. This can be accomplished by successive inversions of the two Jacobians and the
covariant metric tensor on the left, although, importantly, not in that order. Inverting
yields
or
Next, inverting
yields
and subsequently inverting
yields
Finally, renaming into on the right and taking advantage of the symmetry of the
metric tensor yields
This derivation of the transformation rule for the contravariant metric tensor has great
significance. Note that it is conducted strictly in the coordinate space -- or, more accurately, in
two coordinate spaces -- and does not make any references to geometric objects in the
Euclidean space. This insight will play an important role in the development of Riemannian spaces.
13.8A preview of the definition of a tensor
The tensor property of variants is the subject of the next Chapter. However, it is important to
give the definition of a tensor at this time in order to illustrate the importance of the
relationships we have derived in this Chapter.
A tensor is a variant whose transformation rule consists precisely of contractions with the
appropriate Jacobian for each index. For example, a variant of
order one is a covariant tensor if
and a variant of
order one is a contravariant tensor if
Since the covariant basis
transforms according to the rule
it is, indeed a covariant tensor and
thus the term covariant in its name is finally justified. Since the contravariant basis
transforms according to the rule
it is, indeed, a contravariant
tensor and thus the term contravariant in its name is finally justified. Similar statements
can be made of the contravariant components and
the covariant components of a
vector .
A variant of of
order two is a tensor if
It may be described as a tensor
of one contravariant and one covariant order or, if the superscript is treated as the first
index, a contravariant-covariant tensor. Similarly a variant is
a tensor if
and may be described as a
covariant tensor, a doubly-covariant tensor, or a covariant-covariant tensor.
Finally, a variant
is a tensor if
and may be described as a
contravariant tensor, a doubly-contravariant tensor, or a
contravariant-contravariant tensor. Clearly, these descriptions are redundant when the
indicial signature of the variant is available. In this, most common, situation, it is easier to
describe the variant as a tensor of the type indicated by the indicial signature or a
tensor of the type indicated by the flavors of the indices.
Since the covariant metric tensor
transforms according to the rule
and the contravariant metric tensor
transforms according to the rule
the terms covariant and
contravariant in the names of these central objects are finally justified.
It should now be clear how the definition of a tensor can be extended to a variant with an
arbitrary indicial signature. For example, a variant
is a tensor of the type indicated by the flavors of its indices if
Interestingly, all variants analyzed so far in this Chapter are tensors. It would, thus, be quite
insightful to see an example of a variant that is not a tensor. An essential example of a
non-tensor is the Christoffel symbol .
Therefore, we will devote the remainder of this Chapter to studying the transformation rule of the
Christoffel symbol.
13.9The second-order Jacobians
Recall that the Jacobians and
represent the collections of first-order derivatives
and
of the coordinate transformations.
In order to establish the transformation rule for the Christoffel symbol, we must consider the
collections, denoted by symbols
and ,
of the second-order derivatives, i.e.
and
Note that
and
are third-order systems. Therefore, in an -dimensional space, they represent collections of
elements. Since, generally speaking, partial derivatives commute, the systems
and
are symmetric in their subscripts, i.e.
Thus, in actuality,
and
have degrees of freedom.
For the Cartesian to polar transformation, we have
Similar to the identities
relating the first-order Jacobians,
there exists identities relating the second-order Jacobians. The latter can be obtained by
differentiating the first-order identities. For example, consider the identity
In order to differentiate it, we
must refer all of the elements in this identity to a consistent set of coordinates, either the
unprimed coordinates or
the primed coordinates .
Let us choose .
As we discussed above, the natural variables for the Jacobian are
the primed coordinates .
Thus, the natural function is ready for differentiation.
However, for the Jacobian ,
the natural coordinates are the unprimed coordinates .
Thus, in order to refer to
the primed coordinates ,
we must substitute the coordinate transformation
into the natural function , i.e.
The Kronecker delta
has constant values and therefore its derivatives will equal zero regardless of the independent
variables that we choose. However, for the sake of consistency, let us discuss its functional
dependence on the coordinates. Since, in our present analysis, the Kronecker delta
appears with unprimed indices, it is more natural to think of it as a function of the unprimed coordinates. Thus, the
function of the primed coordinates can be
obtained by the composition
Combining all the elements, we find that, when referred to the primed coordinates ,
the identity
reads
Differentiating with respect to ,
we find
An application of the chain rule on
the second term on the left and the term on the right yields
Since
we have
Renaming the dummy index into
in
the first term and changing the order of the Jacobians in the second term yields a more elegant
form of the same relationship
This is one of many equivalent forms that this relationship can take. Other forms can be obtained
by inverting the first-order Jacobians present in the identity. For example, inverting
yields
The advantage of this form is that
it gives the Jacobian
explicitly in terms of .
Alternatively, inverting
in the penultimate identity yields
The advantage of this form is that
all live indices are unprimed and thus correspond to the same coordinate system. Note that there
are equivalent forms of this relationship depending on which
second-order Jacobian each first-order Jacobian is contracted with.
13.10The transformation rule for the Christoffel symbol
Let us now establish the transformation rule for the Christoffel symbol .
Our present analysis will be based on the equation
Let us first determine the transformation rule for the vector variant
Consider the primed version of the
same variant
We will relate to
by
differentiating the identity
In order to differentiate this
identity, refer all elements in it to the primed coordinates ,
i.e.
Now, differentiate both sides with
respect to .
By an application of the product rule, we have
A subsequent application of the
chain rule to the first term on the right yields
Since
the transformation rule for
reads
We must now make the observation that is
not a tensor, which would have required the above identity to read ,
which it does not. This is our first example of a variant that is not a tensor. It is, therefore,
not surprising that the Christoffel symbol ,
so closely related to ,
is also not a tensor. We will now show this.
The Christoffel symbol
in the primed coordinate system is
given by
Since
and, as we just derived,
we have
Multiplying out the expression on
the right yields
The first term equals .
Since ,
the second term equals .
Thus, the transformation rule for the Christoffel symbol
reads
Therefore, the Christoffel symbol
is
not a tensor.
The above formula provides a convenient way of calculating the elements of the Christoffel symbol
by relating them to the elements of the Christoffel symbol in another coordinate system. In
particular, if the latter system is affine where, if you recall, the Christoffel symbol vanishes,
then is
given by
It is left as an exercise to
calculate the elements of the Christoffel symbol in polar and spherical coordinates using this
formula.
13.11Exercises
Exercise 13.1On the basis of the equation
show that the covariant components of a vector transform according to the rule
Exercise 13.2Calculate the elements of for a transformation from one affine coordinate system to another.
Exercise 13.3Confirm that
for the transformation between Cartesian and polar coordinates.
Exercise 13.4Let denote the third-order Jacobian
Show that
Exercise 13.5Consider a circular change of variables: from to , from to , and from back to . Show that
Exercise 13.6Consider the transformation between the Cartesian and spherical coordinates in the three-dimensional Euclidean space:
and
Show that
and
Also show that in terms of ,
and use this form to confirm that the matrices corresponding to and are the inverses of each other. Repeat the task by instead converting to Cartesian coordinates.
Exercise 13.7 Derive the following rule for the transformation of the Christoffel symbol :
This is one of the few relationships where the metric tensor appears in a contraction and does not get absorbed by index juggling. Of course, we could introduce a new symbol for the combination but there is not a sufficiently compelling reason for doing so.
Exercise 13.8Use the equation
for the transformation of the Christoffel symbol from affine coordinates to calculate the elements of the Christoffel symbol in polar coordinates and cylindrical coordinates.
Exercise 13.9Repeat the previous exercise for spherical coordinates.