Matrices, Vectors, and Vector Calculus

In this chapter, we will focus on the mathematical tools required for the course. The main
concepts that will be covered are:
• Coordinate transformations
• Matrix operations
• Scalars and vectors
• Vector calculus
• Differentiation and integration

Coordinate transformation
In order to be able to specify the position of a point P we first must specify the coordinate
system that will be used. The coordinates of point P will be a function of the coordinate system
being used, and coordinate transformations al low us to define the relation between the
coordinates of point P in different coordinate systems.
Different types of coordinate systems are used for different applications. The most
commonly used coordinate systems are:

• Cartesian coordinate systems. These systems consist out of three perpendicular
coordinate axes, called the x, y, and z (or x₁, x₂, and x₃) axes. The coordinates of a point P
are usually specified by three coordinates (x, y, z) or (x₁, x₂, x₃). Note: the coordinate axes
in a Cartesian coordinate system are usually independent of time.

• Spherical coordinate systems. Spherical coordinate systems are most often used when
the system under conside ration has spherical symmetry. The origin of the coordinate
system is chosen to coincide with the point of spherical symmetry. The position of a
point P is de termined by specifying the distance r from the origin of the coordinate
system and the polar and azimuthal angles θ and Ø. Note: we still need to define a
Cartesian coordinate system to define the origin and the two angles . Note 2: the unit
vectors associated with the position vector and the angles will be a function of time if the
position described is time dependent.

• Cylindrical coordinate systems. Cylindrical coordinate systems are most often used
when the system under consideration has cylindrical symmetry. The coordinate system is
characterized by the axis of cylindrical symmetry, which is usually called the z axis. The
position of a point P is determine by specifying the distance r to the z axis, the azimuthal
angle φ, and the z coordinate. Note: we still need to define a Cartesian coordinate system
to define the origin and the two angles. Note 2: the unit vectors associated with the
position vector and the azimuthal angle will be a function of time if the position
described is time dependent.

A coordinate system is not uniquely defined. For example, the coordinates of a point P in a
Cartesian coordinate system depend on the choice of coordinate axes. Different choices will
result in different coordinates. Coordinate transformations are used to transform the
coordinates between coordinate systems.
There are a number of different types of coordinate transformations we will encounter in this
course: translation, rotation, and the standard Lorentz transformation. In this chapter we will
only focus on the rotational transformation.
Consider two coordinate systems, related to each other via a transformation around the x₃
axis, as shown in Figure 1.

Figure 1. Two different coordinate systems used to re present the position of P.

The relation between the coordinates of P in the two coordinate systems can be written as



where



is the cosine of the angle between the x_i’-axis and the x_j-axis (also called the direction cosine).
The coordinate transformation is most often written in matrix notation:

Consider the two coordinate systems shown in Figure 1. Since the two coordinate systems are
related to each other via a rotation around the x₃-axis we conclude that:

• The direction cosine between the x₃’-axis and the x₁- and x₂-axes will be zero (angle is
90°). Thus: .
• The direction cosine between the x₃’-axis and the x₃-axis will be one (angle is 0°). Thus
= 1.
• The direction cosine between the x₁’- and x₂’-axes and the x₃-axis will be zero (angle is
90°). Thus: .
• The direction cosines between the x₁’- and the x₂-axis is cos(π/2-θ) and the direction
cosine between the x₂’- and the x₁-axis is cos(π/2+θ). Thus: = cos(π/2-θ) and =
cos(π/2+θ).
• The direction cosines between the x₁’- and the x₁-axes and between the x₂’- and the x₂-
axes is cos(θ). Thus: .
The coordinate transformation for this particular transformation is thus given by

Although each rotation matrix has 9 parameters, only 3 are truly independent. This can be seen
by realizing that in order to specify a rotation we need to specify the rotation axis (which
requires the specification of a polar and azimuthal angle) and the rotation angle. Consider some
of the relations we can determine between the parameters of the rotation matrix:
• The rotation preserves the length of a vector. Rotating a unit vector will produce another
unit vector. This requires that

• The rotation preserves the angle between two vectors. Since the angle between the
coordinate axes is 90°, the angle between these axes after transformation must also be
90°. This requires that

These six equations can be combined in the following manner

This equation is called the orthogonal condition, and is only satisfied if the coordinate axes are
mutually perpendicular.

Matrix operations

When we are use the rotation matrix to carry out coordinate transformations we carry out a
matrix operation. There are several important facts to remember about matrix operations:
• The unit matrix is defined as a matrix for which the diagonal values are 1 and the nondiagonal
values are 0. When the unit vector operates on a vector, the result is the same
vector.
• The inverse of a matrix is defined such that when it operates on the original matrix, the
result is the unit matrix.
• A transposed matrix is derived from the original matrix by interchanging the rows and
columns.
• When we combine coordinate transformations, we can obtain the resulting transformation
by multiplying the rotation matrices.
• When we multiply rotation matrices we must realize that the order of the multiplication
matters. See for example Figure 2.

Figure 2. The order of transformation matters.

• A matrix inversion is a transformation that results in the reflection through the origin of
all axes. We can not find any series of rotations that result in an inversion. Matrix
inversions are examples of the so-called improper rotations, and are characterized by a
matrix with a determinant equal to -1. Proper rotations are those rotations that are
characterized by a matrix with a determinant equal to +1.

Vectors and Scalars
Consider the following coordinate transformation

where

Vectors and scalars are defined based on what happens as a result of such coordinate
transformations:
• If a quantity is unaffected by this transformation, it is called a scalar.
• If a set of three quantities transforms in the same manner as the coordinates of a point P,
these quantities are the comp onents of what we call a vector.

Note:

• These definitions define scalars and vectors in a very different way from what you may
have been used to. For example, the current definition of a vector makes no reference to
the geometrical interpretation the vector.
• You will encounter parameters in Physics that look like vectors, but do not transform like
vectors under certain operations (a pseudo vector is an example). Just having a
magnitude and a direction is not sufficient to define a vector!
Vector and scalar addition and multiplication have a number of properties in common :
• Both satisfy the commutative law: the order of addition does not change the final result.
• Both satisfy the associative law: when we determine the sum of more than two vectors or
scalars, the final results will not depend on which pair of vectors or scalars we add first.
• The product of a scalar with a scalar transforms like a scalar (and is thus a scalar).
• The product of a vector with a scalar transforms like a vector (and is thus a vector).

The following operations are unique to vectors and do not have equivalents for scalars :