Scalars, Vectors, Tensors

Appendix B
Scalars, Vectors, Tensors
The fundamental requirement for the mathematical expression of each physical law
is that it is written in a way that is independent of the particular coordinate system
that is being used. For example, in Newtonian dynamics, the second law is expressed
using vector notation as
m d2 ⃗r
dt = ⃗ F,
(B.1)
where m and ⃗r are the mass and position vector of a particle that is moving under
the influence of a force ⃗ F .If,moreover,theforceispotential,i.e.,itcanbewritten
as the gradient of a scalar potential function Φ, then Newton’s second law takes the
form
m d2 ⃗r
d 2 t = −⃗ ∇Φ .
(B.2)
This is a symbolic representation of the vectorial form of Newton’s second law, Manifestly
covariant forms
but it is not particularly useful for computations. What we would like to have is
aformofNewton’ssecondlawintermsofthe components of the various vectors,
but written in a way that is invariant under coordinate transformations. We would
call such an expression the manifestly covariant form of a physical law.
In an orthonormal Cartesian coordinate system with unit vectors ˆx i (i=1,2,3),
we can write the position vector in component form as ⃗r = x i ˆx i ,andthegradient
operator as ⃗ ∇ =(∂/∂x i )ˆx i . In this case, Newton’s second law in Cartesian
coordinates takes the form
m d2 x i
dt 2
= − ∂Φ
∂x i .
(B.3)
This is, however, not a manifestly covariant form of the physical law, as we can
easily prove. Consider a transformation from the Cartesian coordinates x i to a set
of general coordinates ξ i that may be neither orthogonal nor normal. We understand
this transformation to imply the existence of one-to-one functions of the form
x i = x i (ξ 1 ,ξ 2 ,ξ 3 )
(B.4)
as well as of their inverse
ξ i = ξ i (x 1 ,x 2 ,x 3 )
(B.5)
that are well defined in all but a small number of spatial points in space that we
315

316 APPENDIX B. SCALARS, VECTORS, TENSORS
will call the poles. Then, we can use the chain rule of differentiation to write
and
d 2 x i 3∑
dt 2 = ∂x i d 2 ξ j
∂ξ j dt 2
∂Φ
∂x i =
j=1
3∑
j=1
∂Φ ∂ξ j
∂ξ j ∂x i .
Inserting these two expressions in equation (B.3) we obtain
3∑
j=1
( ) ∂x
i
∂ξ j m d2 ξ j
dt 2
= − 3∑
j=1
( ∂ξ
j
∂x i ) ∂Φ
∂ξ j .
(B.6)
(B.7)
(B.8)
where
This last expression can be put in the form of equation (B.3) if and only if
δ i j ≡
∂x i
∂ξ j = δi j ,
{ 1, if i = j
0, if i ≠ j
(B.9)
(B.10)
is Kronecker’s delta. As a result, Newton’s second law in the coordinate form of
equation (B.3) is not manifestly covariant.
What went wrong in this derivation
As we will see below, in writing the
components of the two vectors as ⃗r = x i ˆx i and ⃗ ∇ =(∂/∂x i )ˆx i ,weactuallyusedtwo
different basis vectors, even though we denoted them both by ˆx i . We, therefore,
need to start the discussion from the beginning by defining properly coordinate
systems and basis vectors.
B.1 Coordinate and Dual Basis Vectors
We consider a flat space with N dimensions and define a coordinate system as an
one-to-one map between an ordered set of N real numbers ξ 1 , ξ 2 ,...,ξ N and each
individual point in space. The position of each point in space can, therefore, be
written in the form
⃗r = ⃗r(ξ 1 ,ξ 2 ,...,ξ N ) .
(B.11)
Coordinate Lines We define coordinate lines as the curves along which only one of the coordinates
and Surfaces changes, whereas the other remain constant. This is illustrated in Figure B.1 for two
sample coordinate systems. At the same time, we also define coordinate surfaces as
the surfaces on which only one of the coordinates remains constant. It follows from
their definition that, e.g., in a three dimensional space, the intersection between
two coordinate surfaces is a coordinate line.
Using this definition of coordinate lines and surfaces, we have an infinite number
of options of defining basis vectors, three of which are particularly useful in
describing physical laws.
We define the coordinate basis vectors at each point in space as the ordered set
of vectors ê i (i =1, 2,...,N), with the property that each of them is tangent to the
corresponding coordinate line. Formally, we define them by relations of the form

B.1. COORDINATE AND DUAL BASIS VECTORS 317
Figure B.1: Coordinate lines and coordinate basis vectors for two different coordinate systems.
The left panel depicts a Cartesian orthonormal coordinate system. In both cases, a third coordinate
is assumed to extend from each point, perpendicular to the plane of the paper.
ê i ≡ ∂⃗r
∂ξ i (B.12) Coordinate
Basis Vectors
and use a subscript notation to denote the coordinate along which this vector is
tangent. Figure B.1 shows the coordinate basis vectors for two sample coordinate
systems.
We can also define the dual basis vectors at each point in space in terms of the
coordinate surfaces. Given than each coordinate surface can be represented by an
equation of the form ξ i (⃗r) =constant,wedefinethedualbasisvectorsas
ê i ≡ ∇ξ ⃗ i . (B.13) Dual Basis
Vectors
Note that we use a subscript notation for the dual basis vectors, in order to distinguish
them from the coordinate basis vectors. Figure B.2 shows the dual basis
vectors for two sample coordinate systems. Note that for an orthonormal Cartesian
coordinate system
ê ∗ i =ê∗i ,
where we have used the star to denote a Cartesian system.
(B.14)
In general, the coordinate and dual basis vectors depend on position in space.
However, they always satisfy
ê i ê j = δ j i .
(B.15)
In order to prove this property, we will use an orthonormal Cartesian coordinate
system (x 1 ,x 2 ,...,x N )andmakeacoordinatetransformationtosomeunspecified
coordinate system (ξ 1 ,ξ 2 ,...,ξ N ). The position vector of any point in space can be
written in terms of the Cartesian basis vectors as
N∑
⃗r = x k ê ∗ k .
k=1
(B.16)

318 APPENDIX B. SCALARS, VECTORS, TENSORS
Figure B.2: Coordinates lines and dual basis vectors for the coordinate systems shown in Figure
B.1.
Figure B.3: Aexamplesetofcoordinatebasisvectors(ê 1 , ê 2 )andofthecorrespondingdual
basis vectors (ê 1 , ê 2 ). Equation (B.15) requires that ê 1 ⊥ ê 2 and ê 2 ⊥ ê 1 but not necessarily that
ê 1 ↑↑ ê 1 or ê 2 ↑↑ ê 2 .
We use this together with the definition of the coordinate basis vectors to obtain
ê i = ∂⃗r
∂ξ i = N ∑
k=1
∂x k
∂ξ i ê∗ k
We also write explicitly the definition of the dual basis vectors as
ê j = ⃗ ∇ξ j =
N∑
k=1
∂ξ j
∂x k ê∗ k .
(B.17)
(B.18)
(Don’t worry for the moment about the apparent asymmetry in the k−index in the
last expression; this is a Cartesian system for which ê ∗ k =ê∗k .) Taking the product
of the two vectors we finally obtain
ê i ê j =
N∑
k=1
∂ξ j ∂x k
∂x k ∂ξ i
= ∂ξj
∂ξ i = δj i .
(B.19)
Relation (B.15) leads to a number of important results regarding the two sets
of basis vectors. In general, it implies that each coordinate basis vector ê i is perpendicular
to all the dual basis vectors ê j with j ≠ i, butmaynotbeparallelto

B.2. COVARIANT AND CONTRAVARIANT COMPONENTS 319
the dual basis vector ê i . The same is also true for each dual basis vector ê i : it is
perpendicular to all the coordinate basis vectors ê j with j ≠ i, butmaynotbe
parallel to the coordinate basis vector ê i .ThisisillustratedinFigureB.3.
In an orthogonal coordinate system, i.e., one in which the coordinate lines are
intersecting at right angles, the coordinate basis vectors and the dual basis vectors
are mutually orthogonal and, hence, in this case
ê i ↑↑ ê i . (B.20) Orthogonal Coordinate
System
B.2 Covariant and Contravariant Components
Having defined different basis vectors, we can express the components of any vector
⃗A with respect to either of them. When we use the coordinate basis vectors, i.e.,
⃗A = A i ê i (B.21) Contravariant
Components
we use a superscript (or upstairs) notation for the components, which we call the
contravariant components of the vector. On the other hand, when we use the dual
basis vector, i.e.,
⃗A = A i ê i (B.22) Covariant
Components
we use a subscript (or downstairs) notation for the components, which we call the
covariant components of the vector ⃗ A.
The contravariant and covariant components of a vector transform in different
ways from one coordinate system to another. In order to study their transformations,
we will consider two coordinate systems ξ i and ξ ′j and use unprimed and
primed quantities, respectively, to denote the various vector components in the two
systems.
We will start by inserting the definition of the coordinate basis (B.12) into
equation (B.21), i.e.,
⃗A = A i ê i = A i ∂⃗r
∂ξ i
(B.23)
We will then perform a change of coordinates in the derivatives using the chain rule,
⃗A = A i ∂⃗r ∂ξ ′j
∂ξ ′j ∂ξ i
= A i ∂ξ′j
∂ξ i ê′i .
(B.24)
In the last equation we have used again the definition (B.12) but for the primed
coordinate frame. Comparing the last term with the definition (B.21) we finally
find
A ′j = A i ∂ξ′j
∂ξ i . (B.25) Transformation of
Contravariant Components
In order to derive the transformation rule for the covariant components of a
vector, we well use an auxiliary Cartesian coordinate system x i with basis vectors

320 APPENDIX B. SCALARS, VECTORS, TENSORS
ê ∗i . We will start again by inserting the definition of the dual basis (B.13) into
equation (B.22) and use the chain rule to perform a change of coordinates in the
derivatives. In detail
⃗A = A i ê i = A i
∂ξ i
∂ξ i ∂ξ ′k ∂ξ i
∂x j ê∗j = A i
∂ξ ′k ∂x j ê∗j = A i
∂ξ ′k ê′k .
(B.26)
Comparing the last term with the definition (B.22) and simply changing the dummy
index for k to j, weobtain
Transformation of
Covariant Components
A ′ j = A i
∂ξ i
∂ξ ′j .
(B.27)
Note in these transformations how useful the notation of subscripts and superscripts
has been.
Until this point, we have assumed that the space on which we have defined coordinates,
basis vectors, and vector components is flat. Extending these definitions
to a general curved space will require a different understanding of the various quantities
involved. For example, in a curved space,directedlinesegmentscanonly
be infinitesimally short and, therefore, the position vector ⃗r that we used in the
definition of the coordinate basis is not well defined. Even though there is a way
of extending all these geometric interpretations in curved spaces, it is sufficient for
the purposes of this class to take a somewhat backward approach.
We will define as the contravariant components of a vector in an N-dimensional
space, and denote them with superscript notation, an ordered set of N physical
quantities (i.e., components of velocities, momenta, fields) that transform between
coordinate systems according to equation (B.25).
Similarly, we will define as the covariant components of a vector in an N-
dimensional space, and denote them with subscript notation, an ordered set of N
physical quantities (i.e., components of velocities, momenta, fields) that transform
between coordinate systems according to equation (B.27).
In general, we will define as a tensor and denote by
T ijkl... αβγδ...
(B.28)
an ordered set of physical quantities, some of which transform according to the rules
for contravariant components (and we will use superscript notation) and some of
which transform according to the rule for covariant components, i.e.,
Transformation of
Tensor Components
T ′ijk... αβγ... = ∂ξ′i ∂ξ ′j ∂ξA ∂ξ B
∂ξ I ...
∂ξJ ∂ξ ′α ∂ξ ′β ...T IJK... ABΓ... .
(B.29)
The total number of indices is called the rank of the tensor. A vector is a tensor of
rank one. A scalar is a tensor of rank zero.
Example: The inner product of two vectors
In this example, we will derive some useful expressions for the inner product
between two vectors and show that it is a scalar quantity, i.e., that it is invariant
between coordinate transformations.

B.3. THE METRIC TENSOR 321
We will start with two vectors,
⃗A = A i ê i = A i ê i (B.30)
⃗B = B j ê j = B j ê j (B.31)
and calculate their product as
⃗A · ⃗B =(A i ê i ) · (B j ê j )=A i B j (ê i · ê j )=A i B j δ i j = A i B i .
(B.32)
Here we used the fact that ê i · ê j = δ i j . We can follow the exact same procedure
using the covariant components of vector ⃗ A and the contravariant components of
vector ⃗ B.Thefinalsetofexpressionsfortheinnerproductoftwovectorsis
⃗A · ⃗B = A i B i = A i B i (B.33) Inner product
of Vectors
We will now consider a change of coordinates from a system ξ i to another system
ξ ′i and evaluate the inner product of the two vectors in that system:
⃗A· ⃗B = A ′i B i ′ = ∂ξ′i ∂ξk
Aj
∂ξj ∂ξ ′i B k =( ∂ξk ∂ξ ′i
∂ξ ′i ∂ξ j )Aj B k = ∂ξk
∂ξ j Aj B k = δ k jA j B k = A j B j .
(B.34)
This last expression proves that the inner product of two vectors is a scalar quantity.
Example: Projecting onto basis vectors
In an orthonormal coordinate system, e.g., in a Cartesian system (ê x , ê y , ê z ), we
can calculate the component of a vector A ⃗ along one of the coordinate lines using
inner products of the form A x = A ⃗ · ê x .Thisisnot,ofcourse,thecaseifthesystem
is non-orthogonal. However, the definitions of the coordinate and dual basis provide
us with a very useful tool in calculating contravariant and covariant components of
vectors, independent of whether the coordinate system is orthogonal or not.
Starting with the definition of the contravariant components of a vector A ⃗ and
multiplying both sides of the equation with a dual basis vector ê i we obtain
⃗A = A j ê j ⇒ ⃗ A · ê i = A j (ê j · ê i )=A j δ j
i
(B.35)
and, therefore,
A i = ⃗ A · ê i
(B.36)
Similarly, we can also prove that
A i = ⃗ A · ê i
(B.37)
B.3 The Metric Tensor
We will use the dot product of two vectors, and in particular of two basis vectors,
in order to specify the geometry of a general curved space. Starting from
⃗A · ⃗B =(A i ê i ) · (B j ê j )=A i B j (ê i · ê j ) ,
(B.38)

322 APPENDIX B. SCALARS, VECTORS, TENSORS
we only need to specify the elements of the rank-2 covariant tensor
Metric
Tensor
Line
Element
g ij =ê i · ê j
(B.39)
which we will call the metric tensor.
If, instead of the product of two vectors, we calculate the product of an infinitesimal
translational vector to itself, i.e.,
ds 2 = d⃗x · d⃗x = g ij dx i dx j
(B.40)
we call the result the line element of the space.
We can also define the contravariant components of the metric tensor as
and the mixed components as
g ij =ê i · ê j
g i j =ê i · ê j .
(B.41)
(B.42)
Because of the orthogonality of the coordinate and dual basis vectors, g i j = δ i j.
The metric tensor has very many uses in problem solving, one of which is to help
us transform the components of a tensor between the coordinate and dual basis (i.e.,
to raise on lower indices). For example, we show earlier that the inner product of
two vectors is equal to
⃗A · ⃗B = A i B i .
(B.43)
However, we can write the same inner product using the definition of the metric
tensor as
⃗A · ⃗B = A i B j g ij .
(B.44)
Comparing these two equations we obtain
Lowering
an Index
Similarly, we can prove that
B i = g ij B j .
(B.45)
Raising
an Index
B i = g ij B j .
(B.46)
Finally, we can use this last property of the metric tensor to prove that g ij is
the inverse of g ij .Wewillstartfromthedotproductofacoordinatetoadualbasis
vector,
ê i · ê k = δ i k ⇒ (g ij ê j ) · ê k = δ i k ⇒ g ij (ê j · ê k )=δ i k (B.47)
from which we obtain
g ij g jk = δ i k .
(B.48)

B.3. THE METRIC TENSOR 323
Useful Expressions
Coordinate Basis Vectors
Dual Basis Vectors
Orthogonality of Basis Vectors
ê i ≡ ∂⃗r
∂ξ i
ê i ≡ ⃗ ∇ξ i .
ê i ê j = δ j i
(B.49)
(B.50)
(B.51)
Contravariant Components of Vector
⃗A = A i ê i
(B.52)
Covariant Components of Vector
⃗A = A i ê i
(B.53)
Transformation of Contravariant Components
Transformation of Covariant Components
A ′j = A i ∂ξ′j
∂ξ i . (B.54)
A ′ j = A ∂ξ i
i
∂ξ ′j .
(B.55)
Transformation of General Tensor Components
T ′ijk... αβγ... = ∂ξ′i ∂ξ ′j ∂ξA ∂ξ B
∂ξ I ...
∂ξJ ∂ξ ′α ∂ξ ′β ...T IJK... ABΓ... .
(B.56)
Inner Product of Vectors
Vector Components
The Metric Tensor
⃗A · ⃗B = A i B i = A i B i
A i = A ⃗ · ê i
A i = A ⃗ · ê i
g ij =ê i · ê j
(B.57)
(B.58)
(B.59)
(B.60)
Other components of the metric tensor
g ij g jk = δ i
k
g i j = δ i j
(B.61)
(B.62)
Lowering and Raising an Index
B i = g ij B j B i = g ij B j (B.63)