Hassani S. Mathematical Methods: For Students of Physics and Related Fields

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1.2 Coordinate Systems 11

In 1827, while still an undergraduate, he was appointed Professor of Astronomy

at Trinity College in which capacity he had to manage the astronomical observations

and teach science. He did not do much of the former, but he was a ﬁne lecturer.

Hamilton had very good intuition, and knew how to use analogy to reason from

the known to the unknown. Although he lacked great ﬂashes of insight, he worked

very hard and very long on special problems to see what generalizations they would

lead to. He was patient and systematic in working on speciﬁc problems and was

willing to go through detailed and laborious calculations to check or prove a point.

After mastering and clarifying the concept of complex numbers and their relation

to planar vectors (see Problem 18.11 for the connection between complex multiplica-

tion on the one hand, and dot and cross products on the other), Hamilton was able

to think more clearly about the three-dimensional generalization. His eﬀorts led

unfortunately to frustration because the vectors (a) required four components, and

(b) deﬁed commutativity! Both features were revolutionary and set the standard

for algebra. He called these new numbers quaternions.

In retrospect, one can see that the new three-dimensional complex numbers had

to contain four components. Each “number,” when acting on a vector, rotates the

latter about an axis and stretches (or contracts) it. Two angles are required to

specify the axis of rotation, one angle to specify the amount of rotation, and a

fourth number to specify the amount of stretch (or contraction).

Hamilton announced the invention of quaternions in 1843 at a meeting of the

Royal Irish Academy, and spent the rest of his life developing the subject.

1.2 Coordinate Systems

Coordinates are “functions” that specify points of a space. The smallest

number of these functions necessary to specify a point is called the dimension

of that space. For instance, a point of a plane is speciﬁed by two numbers, and

as the point moves in the plane the two numbers change, i.e., the coordinates

are functions of the position of the point. If we designate the point as P,we

may write the coordinate functions of P as (f(P ),g(P )).

Each pair of such coordinate

systems as

functions.

functions is called a coordinate system.

There are two coordinate systems used for a plane, Cartesian, denoted

by (x(P ),y(P )), and polar, denoted by (r(P ),θ(P )). As shown in Figure 1.7,

y(P)

x(P)

θ(P)

r(P)

Figure 1.7: Cartesian and polar coordinates of a point P in two dimensions.

Think of f (or g) as a rule by which a unique number is assigned to each point P .

12 Coordinate Systems and Vectors

the “function” x is deﬁned as giving the distance from P to the vertical axis,

while θ is the function which gives the angle that the line OP makes with a

given ﬁducial (usually horizontal) line. The origin O and the ﬁducial line are

completely arbitrary. Similarly, the functions r and y give distances from the

origin and to the horizontal axis, respectively.

Box 1.2.1. In practice, one drops the argument P and writes (x, y) and

(r, θ).

We can generalize the above concepts to three dimensions. There are three

coordinate functions now. So for a point P inspacewewrite

the three common

coordinate

systems:

Cartesian,

cylindrical and

spherical

(f(P ),g(P ),h(P )),

where f , g,andh are functions on the three-dimensional space. There are

three widely used coordinate systems, Cartesian (x(P),y(P ),z(P )), cylin-

drical (ρ(P ),ϕ(P ),z(P )), and spherical (r(P ),θ(P ),ϕ(P )). ϕ(P ) is called

the azimuth or the azimuthal angle of P , while θ(P ) is called its polar

angle. To ﬁnd the spherical coordinates of P, one chooses an arbitrary point

as the origin O and an arbitrary line through O called the polar axis.One

measures

OP and calls it r(P ); θ(P ) is the angle between OP and the polar

axis. To ﬁnd the third coordinate, we construct the plane through O and per-

pendicular to the polar axis, drop a projection from P to the plane meeting

the latter at H, draw an arbitrary ﬁducial line through O in this plane, and

measure the angle between this line and

OH. This angle is ϕ(P ). Cartesian

and cylindrical coordinate systems can be described similarly. The three co-

ordinate systems are shown in Figure 1.8. As indicated in the ﬁgure, the polar

axis is usually taken to be the z-axis, and the ﬁducial line from which ϕ(P )

is measured is chosen to be the x-axis. Although there are other coordinate

systems, the three mentioned above are by far the most widely used.

(

)

(

)

(

)

(a) (b)

(

)

(

)

(

)

(c)

(

)

(

)

(

)

Figure 1.8: (a) Cartesian, (b) cylindrical, and (c) spherical coordinates of a point P in

three dimensions.

1.2 Coordinate Systems 13

Which one of the three systems of coordinates to use in a given physi-

cal problem is dictated mainly by the geometry of that problem. As a rule,

spherical coordinates are best suited for spheres and spherically symmetric

problems. Spherical symmetry describes situations in which quantities of in-

terest are functions only of the distance from a ﬁxed point and not on the

orientation of that distance. Similarly, cylindrical coordinates ease calcula-

tions when cylinders or cylindrical symmetries are involved. Finally, Cartesian

coordinates are used in rectangular geometries.

Of the three coordinate systems, Cartesian is the most complete in the

following sense: A point in space can have only one triplet as its coordinates.

This property is not shared by the other two systems. For example, a point

limitations of

non-Cartesian

coordinates

P locatedonthez-axis of a cylindrical coordinate system does not have a

well-deﬁned ϕ(P ). In practice, such imperfections are not of dire consequence

and we shall ignore them.

Once we have three coordinate systems to work with, we need to know

how to translate from one to another. First we give the transformation rule

from spherical to cylindrical. It is clear from Figure 1.9 that

transformation

from spherical to

cylindrical

coordinates

ρ = r sin θ, ϕ

cyl

= ϕ

sph

,z= r cos θ. (1.7)

Thus, given (r, θ, ϕ)ofapointP , we can obtain (ρ, ϕ, z)ofthesamepointby

substituting in the RHS.

Next we give the transformation rule from cylindrical to Cartesian. Again

transformation

from cylindrical to

Cartesian

coordinates

Figure 1.9 gives the result:

x = ρ cos ϕ, y = ρ sin ϕ, z

car

= z

cyl

. (1.8)

We can combine (1.7) and (1.8) to connect Cartesian and spherical coordi-

transformation

from spherical to

Cartesian

coordinates

nates:

x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ. (1.9)

Figure 1.9: The relation between the cylindrical and spherical coordinates of a point

P can be obtained using this diagram.

14 Coordinate Systems and Vectors

Box 1.2.2. Equations (1.7)–(1.9) are extremely important and worth be-

ing committed to memory. The reader is advised to study Figure 1.9

carefully and learn to reproduce (1.7)–(1.9) from the ﬁgure!

The transformations given are in their standard form. We can turn them

around and give the inverse transformations. For instance, squaring the ﬁrst

and third equations of (1.7) and adding gives ρ

+ z

= r

or r =



+ z

Similarly, dividing the ﬁrst and third equation yields tan θ = ρ/z,which

implies that θ =tan

−1

(ρ/z), or equivalently,

=cosθ ⇒ θ =cos

−1





=cos

−1





+ z



Thus, the inverse of (1.7) is

transformation

from cylindrical to

spherical

coordinates

r =



+ z

,θ=tan

−1





=cos

−1





+ z



,ϕ

sph

= ϕ

cyl

(1.10)

Similarly, the inverse of (1.8) is

ρ =



+ y

ϕ =tan

−1





=cos

−1





+ y



=sin

−1





+ y



, (1.11)

cyl

= z

car

and that of (1.9) is

transformation

from Cartesian to

spherical

coordinates

r =



+ y

+ z

θ =tan

−1





+ y



=cos

−1





+ y

+ z



=sin

−1





+ y



+ y

+ z



, (1.12)

ϕ =tan

−1





=cos

−1





+ y



=sin

−1





+ y



An important question concerns the range of these quantities. In other

words: In what range should we allow these quantities to vary in order to cover

the whole space? For Cartesian coordinates all three variables vary between

−∞ and +∞.Thus,

range of

coordinate

variables

−∞ <x<+∞, −∞ <y<+∞, −∞ <z<+∞.

The ranges of cylindrical coordinates are

0 ≤ ρ<∞, 0 ≤ ϕ ≤ 2π, −∞ <z<∞.

1.2 Coordinate Systems 15

Note that ρ, being a distance, cannot have negative values.

Similarly, the

ranges of spherical coordinates are

0 ≤ r<∞, 0 ≤ θ ≤ π, 0 ≤ ϕ ≤ 2π.

Again, r is never negative for similar reasons as above. Also note that the

range of θ excludes values larger than π. This is because the range of ϕ takes

care of points where θ “appears” to have been increased by π.

Historical Notes

One of the greatest achievements in the development of mathematics since Euclid

was the introduction of coordinates. Two men take credit for this development: Fer-

mat and Descartes. These two great French mathematicians were interested in the

uniﬁcation of geometry and algebra, which resulted in the creation of a most fruitful

branch of mathematics now called analytic geometry. Fermat and Descartes who

were heavily involved in physics, were keenly aware of both the need for quantitative

methods and the capacity of algebra to deliver that method.

Fermat’s interest in the uniﬁcation of geometry and algebra arose because of his

involvement in optics. His interest in the attainment of maxima and minima—thus

Pierre de Fermat

1601–1665

his contribution to calculus—stemmed from the investigation of the passage of light

rays through media of diﬀerent indices of refraction, which resulted in Fermat’s

principle in optics and the law of refraction. With the introduction of coordinates,

Fermat was able to quantify the study of optics and set a trend to which all physicists

of posterity would adhere. It is safe to say that without analytic geometry the

progress of science, and in particular physics, would have been next to impossible.

Born into a family of tradespeople, Pierre de Fermat was trained as a lawyer

and made his living in this profession becoming a councillor of the parliament of

the city of Toulouse. Although mathematics was but a hobby for him and he could

devote only spare time to it, he made great contributions to number theory, to

calculus, and, together with Pascal, initiated work on probability theory.

The coordinate system introduced by Fermat was not a convenient one. For one

thing, the coordinate axes were not at right angles to one another. Furthermore,

the use of negative coordinates was not considered. Nevertheless, he was able to

translate geometric curves into algebraic equations.

Ren´e Descartes was a great philosopher, a founder of modern biology, and a

superb physicist and mathematician. His interest in mathematics stemmed from his

desire to understand nature. He wrote:

...I have resolved to quit onlyabstract geometry, thatis to say, the

consideration of questions which serveonlytoexercisethemind,and

this, in order to study another kind of geometry, which has for its object

the explanation of the phenomena of nature.

His father, a relatively wealthy lawyer, sent him to a Jesuit school at the age

Ren´e Descartes

1596–1650

of eight where, due to his delicate health, he was allowed to spend the mornings in

bed, during which time he worked. He followed this habit during his entire life. At

twenty he graduated from the University of Poitier as a lawyer and went to Paris

where he studied mathematics with a Jesuit priest. After one year he decided to

In some calculus books ρ is allowed to have negative values to account for points on the

opposite side of the origin. However, in physics literature ρ is assumed to be positive.To go

to “the other side” of the origin along ρ, we change ϕ by π, keeping ρ positive at all times.

16 Coordinate Systems and Vectors

join the army of Prince Maurice of Orange in 1617. During the next nine years he

vacillated between various armies while studying mathematics.

He eventually returned to Paris, where he devoted his eﬀorts to the study of

optical instruments motivated by the newly discovered power of the telescope. In

1628 he moved to Holland to a quieter and freer intellectual environment. There he

lived for the next twenty years and wrote his famous works. In 1649 Queen Christina

of Sweden persuaded Descartes to go to Stockholm as her private tutor. However

the Queen had an uncompromising desire to draw curves and tangents at 5 a.m.,

causing Descartes to break the lifelong habit of getting up at 11 o’clock! After only

a few months in the cold northern climate, walking to the palace for the 5 o’clock

appointment with the queen, he died of pneumonia in 1650.

Descartes described his algebraic approach to geometry in his monumental work

La G´eom´etrie. It is in this work that he solves geometrical problems using algebra

by introducing coordinates. These coordinates, as in Fermat’s case, were not lengths

along perpendicular axes. Nevertheless they paved the way for the later generations

of scientists such as Newton to build on Descartes’ and Fermat’s ideas and improve

on them.

Throughout the seventeenth century, mathematicians used one axis with the y

values drawn at an oblique or right angle onto that axis. Newton, however, in a bookNewton uses polar

coordinates for the

ﬁrst time

called The Method of Fluxions and Inﬁnite Series written in 1671, and translated

much later into English in 1736, describes a coordinate system in which points are

located in reference to a ﬁxed point and a ﬁxed line through that point. This was

the ﬁrst introduction of essentially the polar coordinates we use today.

1.3 Vectors in Diﬀerent Coordinate Systems

Many physical situations require the study of vectors in diﬀerent coordinate

systems. For example, the study of the solar system is best done in spherical

coordinates because of the nature of the gravitational force. Similarly calcu-

lation of electromagnetic ﬁelds in a cylindrical cavity will be easier if we use

cylindrical coordinates. This requires not only writing functions in terms of

these coordinate variables, but also expressing vectors in terms of unit vectors

suitable for these coordinate systems. It turns out that, for the three coordi-

nate systems described above, the most natural construction of such vectors

renders them mutually perpendicular.

Any set of three (two) mutually perpendicular unit vectors in space (in the

plane) is called an orthonormal basis.

Basis vectors have the propertyorthonormal basis

that any vector can be written in terms of them.

Let us start with the plane in which the coordinate system could be Carte-

sian or polar. In general, we construct an orthonormal basis at a point and

note that

The word “orthonormal” comes from orthogonal meaning “perpendicular,” and normal

meaning “of unit length.”

1.3 Vectors in Diﬀerent Coordinate Systems 17

(a)

(b)

Figure 1.10: The unit vectors in (a) Cartesian coordinates and (b) polar coordinates.

The unit vectors at P and Q are the same for Cartesian coordinates, but diﬀerent in

polar coordinates.

Box 1.3.1. The orthonormal basis, generally speaking, depends on the

point at which it is constructed.

The vectors of a basis are constructed as follows. To ﬁnd the unit vector

corresponding to a coordinate at a point P , hold the other coordinate ﬁxed

and increase the coordinate in question. The initial direction of motion of P

is the direction of the unit vector sought. Thus, we obtain the Cartesian unit

vectors at point P of Figure 1.10(a):

is obtained by holding y ﬁxed and

letting x vary in the increasing direction; and

is obtained by holding x ﬁxed

at P and letting y increase. In each case, the unit vectors show the initial

direction of the motion of P. It should be clear that one obtains the same set

general rule for

constructing a

basis at a point

of unit vectors regardless of the location of P . However, the reader should

take note that this is true only for coordinates that are deﬁned in terms of

axes whose directions are ﬁxed, such as Cartesian coordinates.

If we use polar coordinates for P , then holding θ ﬁxed at P gives the

direction of

as shown in Figure 1.10(b), because for ﬁxed θ, that is the

direction of increase for r. Similarly, if r is ﬁxed at P , the initial direction

of motion of P when θ is increased is that of

shown in the ﬁgure. If we

choose another point such as Q shown in the ﬁgure, then a new set of unit

vectors will be obtained which are diﬀerent form those of P .Thisisbecause

polar coordinates are not deﬁned in terms of any ﬁxed axes.

Since {

} and {

} form a basis in the plane, any vector a in the

plane can be expressed in terms of either basis as shown in Figure 1.11. Thus,

we can write

a = a

+ a

= a

+ a

= a

+ a

, (1.13)

where the coordinates are subscripted to emphasize their dependence on the

points at which the unit vectors are erected. In the case of Cartesian coor-

dinates, this, of course, is not necessary because the unit vectors happen to

be independent of the point. In the case of polar coordinates, although this

18 Coordinate Systems and Vectors

(a) (b)

Figure 1.11: (a) The vector a has the same components along unit vectors at P and Q

in Cartesian coordinates. (b) The vector a has diﬀerent components along unit vectors

at diﬀerent points for a polar coordinate system.

dependence exists, we normally do not write the points as subscripts, being

aware of this dependence every time we use polar coordinates.

So far we have used parentheses to designate the (components of) a vector.

angle brackets

denote vector

components

Since, parentheses—as a universal notation—are used for coordinates of points,

we shall write components of a vector in angle brackets. So Equation (1.13)

can also be written as

a = a



= a



= a



where again the subscript indicating the point at which the unit vectors are

deﬁned is normally deleted. However, we need to keep in mind that although

a

 is independent of the point in question, a

 is very much point-

dependent. Caution should be exercised when using this notation as to the

location of the unit vectors.

The unit vectors in the coordinate systems of space are deﬁned the same

way. We follow the rule given before:

Box 1.3.2. (Rule for Finding Coordinate Unit Vectors). To ﬁnd

the unit vector corresponding to a coordinate at a point P , hold the other

coordinates ﬁxed and increase the coordinate in question. The initial di-

rection of motion of P is the direction of the unit vector sought.

It should be clear that the Cartesian basis {

} is the same for all

points, and usually they are drawn at the origin along the three axes. An

arbitrary vector a can be written as

a = a

+ a

or a = a

, (1.14)

where we used angle brackets to denote components of the vector, reserving

the parentheses for coordinates of points in space.

1.3 Vectors in Diﬀerent Coordinate Systems 19

(ρ, ϕ, z)

Figure 1.12: Unit vectors of cylindrical coordinates.

The unit vectors at a point P in the other coordinate systems are obtained

similarly. In cylindrical coordinates,

lies along and points in the direction

of increasing ρ at P ;

is perpendicular to the plane formed by P and the

z-axis and points in the direction of increasing ϕ;

points in the direction of

positive z (see Figure 1.12). We note that only

is independent of the point

at which the unit vectors are deﬁned because z is a ﬁxed axis in cylindrical

coordinates. Given any vector a,wecanwriteitas

a = a

+ a

or a = a

. (1.15)

The unit vectors in spherical coordinates are deﬁned similarly:

is taken

along r andpointsinthedirectionofincreasingr; this direction is called

radial direction

radial;

is taken to lie in the plane formed by P and the z-axis, is per-

pendicular to r, and points in the direction of increasing θ;

is as in the

cylindrical case (Figure 1.13). An arbitrary vector in space can be expressed

in terms of the spherical unit vectors at P :

a = a

+ a

or a = a

. (1.16)

It should be emphasized that

Box 1.3.3. The cylindrical and spherical unit vectors

,and

are dependent on the position of P .

Once an origin O is designated, every point P in space will deﬁne a vector,

called a position vector and denoted by r. This is simply the vector drawn

position vector

from O to P . In Cartesian coordinates this vector has components x, y, z,

thus one can write

r = x

+ y

+ z

. (1.17)

20 Coordinate Systems and Vectors

, θ, ϕ

)

Figure 1.13: Unit vectors of spherical coordinates. Note that the intersection of the

shaded plane with the xy-plane is a line along the cylindrical coordinate ρ.

But (x, y, z)arealsothecoordinates of the point P . This can be a source ofdiﬀerence between

coordinates and

components

explained

confusion when other coordinate systems are used. For example, in spherical

coordinates, the components of the vector r at P are r, 0, 0 because r has

only a component along

and none along

.Onewrites

r = r

. (1.18)

However, the coordinates of P are still (r, θ, ϕ)! Similarly, the coordinates of

P are (ρ, ϕ, z) in a cylindrical system, while

r = ρ

+ z

, (1.19)

because r lies in the ρz-plane and has no component along

. Therefore,

Box 1.3.4. Make a clear distinction between the components of the

vector r and the coordinates of the point P .

A common symptom of confusing components with coordinates is as fol-

lows. Point P

has position vector r

with spherical components r

, 0, 0

at P

. The position vector of a second point P

is r

with spherical compo-

nents r

, 0, 0 at P

. It is easy to fall into the trap of thinking that r

− r

has spherical components r

− r

, 0, 0! This is, of course, not true, because

the spherical unit vectors at P

are completely diﬀerent from those at P

and, therefore, contrary to the Cartesian case, we cannot simply subtract

components.

We should really label everything with P . But, as usual, we assume this labeling to be

implied.