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Problem c: Use the expressions (3.3) and (3.31) to show that

J = r2 sin : (3.33)

Note that the Jacobian J in (3.33) is the product of the scale factors de ned in equation

(3.8): J = hr h h' . This is not a coincidence in general the scale factors contain all

the information needed to compute the Jacobian for a curvilinear coordinate system, see

Butkov 14] for details.

Problem d: A volume element dV is in spherical coordinates thus given by dV = r2 sin drd d'.

Consider the volume element dV in gure (3.3) that is de ned by in nitesimal incre-

ments dr, d and d'. Give an alternative derivation of this expression for dV that

is based on geometric arguments only.

In some applications one wants to integrate over the surface of a sphere rather than

integrating over a volume. For example, if one wants to compute the cooling of the Earth,

one needs to integrate the heat ow over the Earth's surface. The treatment used for

deriving the volume integral in spherical coordinates can also be used to derive the surface

integral. A key element in the analysis is that the surface spanned by two vectors a and

b is given by bj. Again, an increment d of the angle corresponds to a change

ja

@r=@ d of the position vector. A similar result holds when the angle ' is changed.

Problem e: Use these results to show that the surface element dS corresponding to

in nitesimal changes d and d' is given by

@r @r

dS = @ @' d d' : (3.34)

CHAPTER 3. SPHERICAL AND CYLINDRICAL COORDINATES

26

z-axis

dÏ• dr

dÎ¸ r

y-axis

x-axis

Figure 3.3: De nition of the geometric variables for an in nitesimal volume element dV.

Problem f: Use expression (3.3) to compute the vectors in the cross product and use this

to derive that

dS = r2 sin d d' : (3.35)

Problem g: Using the geometric variables in gure (3.3) give an alternative derivation

of this expression for a surface element that is based on geometric arguments only.

Problem h: Compute the volume of a sphere with radius R using spherical coordinates.

Pay special attention to the range of integration for the angles and ', see section

(3.1).

3.5 Cylinder coordinates

Cylinder coordinates are useful in problems that exhibit cylinder symmetry rather than

spherical symmetry. An example is the generation of water waves when a stone is thrown

in a pond, or more importantly when an earthquake excites a tsunami in the ocean. In

p

cylinder coordinates a point is speci ed by giving its distance r = x2 + y2 to the z-

axis, the angle ' and the z-coordinate, see gure (3.4) for the de nition of variables. All

the results we need could be derived using an analysis as shown in the previous sections.

However, in such an approach we would do a large amount of unnecessary work. The key is

to realize that at the equator of a spherical coordinate system (i.e. at the locations where

= =2) the spherical coordinate system and the cylinder coordinate system are identical,

3.5. CYLINDER COORDINATES 27

z-axis

z

. (x, y, z)

y-axis

Ï•

r

.

x-axis

Figure 3.4: De nition of the geometric variables used in cylinder coordinates.

see gure (3.5). An inspection of this gure shows that all results obtained for spherical

coordinates can be used for cylinder coordinates by making the following substitutions:

p2 2 2 p2 2

r = x +y +z x +y!

=2 (3.36)

!

^z ! ;^

rd ! ;dz

Problem a: Convince yourself of this. To derive the third line consider the unit vectors

pointing in the direction of increasing values of and z at the equator.

Problem b: Use the results of the previous sections and the substitutions (3.36) to show

the following properties for a system of cylinder coordinates:

x = r cos '

y = r sin ' (3.37)

z=z

0 1 0 1 01

cos ' sin '

B0C

^ = B sin ' C ' = B cos ' C

;

r@ ^@ ^ =@ 0 A

z

A A (3.38)

0 0 1

dV = rdrd'dz (3.39)

dS = rdzd' : (3.40)

CHAPTER 3. SPHERICAL AND CYLINDRICAL COORDINATES

28

z-axis

x 2+ y 2+ z 2= constant

x 2 + y 2 = constant

Î¸=Ï€

y-axis

2

x-axis

Figure 3.5: At the equator the spherical coordinate system has the same properties as a

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