ñòð. 6 |

Î¸

y-axis

Ï•

.

x-axis

Figure 3.1: De nition of the angles used in the spherical coordinates.

Problem a: Use gure (3.1) to show that the Cartesian coordinates are given by:

x = r sin cos '

y = r sin sin ' (3.3)

z = r cos

Problem b: Use these expressions to derive the following expression for the spherical

coordinates in terms of the Cartesian coordinates:

p

r = x2 + y2 + z 2p2 2 2

= arccos z= x + y + z (3.4)

' = arctan (y=x)

We now have obtained the relation between the Cartesian coordinates (x y z) and the

spherical coordinates (r '). We want to express the vector u of equation (3.2) also in

spherical coordinates:

u =ur^+u ^ + u''

r ^ (3.5)

and we want to know the relation between the components (ux uy uz ) in Cartesian co-

ordinates and the components (ur u u' ) of the same vector expressed in spherical coor-

dinates. In order to do this we rst need to determine the unit vectors ^, ^ and '. In

r ^

Cartesian coordinates, the unit vector x points along the x-axis. This is a di erent way

^

of saying that it is a unit vector pointing in the direction of increasing values of x for

constant values of y and z in other words, x can be written as: x =@r=@x.

^ ^

3.1. INTRODUCING SPHERICAL COORDINATES 19

Problem c: Verify this by carrying out the di erentiation that the de nition x =@r=@x

^

01

B 1 C.

leads to the correct unit vector in the x-direction: x = @ 0 A

^

0

Now consider the unit vector ^. Using the same argument as for the unit vector x we ^

know that ^ is directed towards increasing values of for constant values of r and '. This

means that ^ can be written as ^ = C@r=@ . The constant C follows from the requirement

that ^ is of unit length.

Problem d: Use this reasoning for all the unit vectors ^, ^ and ' and expression (3.3)

r ^

to show that:

^ = @r ^ = 1 @r 1 @r

r @r ^ r sin @'

'= (3.6)

r@

and that this result can also be written as

0 1 0 1 0 1

' ' B cos '' C : (3.7)

B sin cos ' C B cos cos ' C sin

^ = @ cos sin A

;

^ = @ sin sin A

r ^

'=@ A

cos sin 0

;

These equations give the unit vectors ^, ^ and ' in Cartesian coordinates.

r ^

In the right hand side of (3.6) the derivatives of the position vector are divided by 1, r

and r sin respectively. These factors are usually shown in the following notation:

hr = 1 h =r h' = r sin : (3.8)

These scale factors play a very important role in the general theory of curvilinear coordi-

nate systems, see Butkov 14] for details. The material presented in the remainder of this

chapter as well as the derivation of vector calculus in spherical coordinates can be based

on the scale factors given in (3.8). However, this approach will not be taken here.

Problem e: Verify explicitly that the vectors ^, ^ and ' de ned in this way form an

r ^

orthonormal basis, i.e. they are of unit length and perpendicular to each other:

(^ ^) = ^ ^ = (^ ') = 1

rr '^ (3.9)

^ ^ = (^ ') = ^ ' = 0 :

r r^ ^ (3.10)

Problem f: Using the expressions (3.7) for the unit vectors ^, ^ and ' show by calculating

r ^

the cross product explicitly that

^ ^=' ^ '= r ^=^:

r ^ ^ ^ r (3.11)

'

;^

The Cartesian basis vectors x, y and ^ point in the same direction at every point in space.

^^ z

This is not true for the spherical basis vectors ^, ^ and ' for di erent values of the angles

r ^

and ' these vectors point in di erent directions. This implies that these unit vectors

are functions of both and '. For several applications it is necessary to know how the

basis vectors change with and '. This change is described by the derivative of the unit

vectors with respect to the angles and '.

CHAPTER 3. SPHERICAL AND CYLINDRICAL COORDINATES

20

Problem g: Show by direct di erentiation of the expressions (3.7) that the derivatives

of the unit vectors with respect to the angles and ' are given by:

@^=@ = ^

r r ^

@^=@' = sin '

@ ^=@ = r @ ^=@' = cos ' ^ (3.12)

;^

@^ =@' = sin ^ cos ^

r

@^ =@ = 0

' ' ; ;

3.2 Changing coordinate systems

Now that we have derived the properties of the unit vectors ^, ^ and ' we are in the

r ^

position to derive how the components (ur u u' ) of the vector u de ned in equation

(3.5) are related to the usual Cartesian coordinates (ux uy uz ). This can most easily be

achieved by writing the expressions (3.7) in the following form:

^ = sin cos ' x + sin sin ' y + cos ^

r ^ ^ z

^ = cos cos ' x + cos sin ' y sin ^

^ ^ z (3.13)

;

' = sin ' x + cos ' y

^ ^ ^

;

Problem a: Convince yourself that this expression can also be written in a symbolic form

01 01

as

^C

r x

^

ñòð. 6 |