Appendix A: Properties of Spherical Coordinates in n Dimensions

The purpose of this appendix is to present in an essentially "self-contained" manner the important properties of a set of spherical coordinates in n dimensions. In particular, a derivation of the Jacobian of the transformation is provided.

In the determination of the probability distribution of the Euclidean length between a fixed point and a second point picked at random from the surface of a sphere in n dimensions the following spherical coordinates were introduced. [These can be found stated explicitly for n = 4 in 17 and, in general for any n, in 4, pp. 593-595, 10, 13, pp. 435-437, 14 p. 11]

This particular choice of  coordinates is but one example of a family of  "polyspherical coordinates" which can be used to specify points in n dimensional space. A graphical scheme or tree for developing such coordinates was given by Vilenkin, Kuznetsov and Smorodinskii in the 1960's [15], though G. N. Watson made implicit use of such coordinates [16, pp 420-421] long before. Polyspherical coordinates seem to be used primarily to solve problems that arise in mathematical physics [2, 3, 10]. It might therefore be desirable to introduce this transformation, which basically generalizes a topic from elementary calculus, to a wider mathematical audience.
 

For  and  let  be the transformation on  in  to  in  defined by

. (1)

Theorem A1: For  . (2)

Proof: so the identity is satisfied when .

Suppose that  for  Now consider

.

Hence by an inductive argument equation (2) is established.

Theorem A2:  (3)

Proof: 
.

From equations (1) and (2), this simplifies to

.

From theorem A2, it follows that  maps elements of  to points in  which lie on the  dimensional "surface" of a  hypersphere of radius centered at the origin.

Let and  be subsets of  and a subset of  defined as follows:
 

,
,
,
.

Theorem A3: For  the map   is a bijection (i.e., one to one and onto).

Proof: Let , then by theorem A2  However, 
. Hence,  and so   .

Now, let  , then  . Define the unique angle,  ; since  , and  .

Now for  . Hence, , so  is a unique angle in  with  , and  .

Now suppose that for  there are k unique angles  , in  with . Then it follows that for each such m.

Now,  . But theorem A1 implies that  Hence, , so  is a unique angle in with , and  So by an inductive argument there exist  unique angles  in  with  and  for  . Let r be the positive number given by  . Then , but by theorem A1 this reduces to So the ordered pair lies on the unit circle and there is a unique angle  in with  and  Thus, there exists a unique with  for  . This completes the proof.

For  the transformation from  to is obviously many to one since any choice of angles in  gives   for . However, even for the transformation is many to one as the following theorem makes explicit.

Theorem A4: For  and  there are uncountably many  with  for .

Proof: Let  , then  and  is a unique angle in .

Case 1a: If , then  for  and  or  . Hence,  and therefore any choice of the angles for  gives  for .

If , this is the only possible case.
 

Case 1b: If , then at least one  for  and with  . However,  , so  is a unique angle in .

Case 2a: If , then  and  for  and  or . Hence,  and therefore any choice of the angles  for  gives  for .

Case 2b: If  , then and at least one  for  and  with . However,  , so  is a unique angle in .

Proceeding in this manner, suppose for  it has been established that there are m unique angles  for and with

.

Now , so by theorem A1  . Therefore,  is a unique angle in .

Case ma: If , then by theorem A1  .
Therefore,  for  and  or . Hence,  and any choice of the angles  for  gives  for .

Case mb: If  , then by theorem A1  and so at least one  for  and  with  . However, again by theorem A1  , so  is a unique angle in  . This case is only possible for or  . Therefore, the maximum number of uniquely defined angles in  is the maximum of  , i.e., .

Hence, for any element of  one of the following two cases must occur.

Case a: All for   for some l with and there are uncountably many  in  with  for .

Case b: There are  unique angles with
.
 .
In particular,  , but by theorem A1 . So  and  is either 0 or  with . Thus, any angle  in gives the correct results that

.

Therefore, there are uncountably many  in  with  for .

From theorems A3 and A4 the map  is onto but many to one. This degeneracy of the  transformation is already seen in three dimensions when  and  for then the azimuthal angle  is completely undetermined.

The dimensional volume (or content) of  is 0 since are held fixed. Therefore, the dimensional volume of  equals the dimensional volume of . This "volume" (the "surface area" of a hypersphere of radius) equals the dimensional integral of the Jacobian of the  transformation from  to over the domain  [11 p.186]. The n dimensional integral of this same Jacobian over the domain  gives the volume in n dimensions of a hypersphere of radius a.
 
 

The direction in associated with a change of a given coordinate u is along the vector whose k' th rectangular component is given by  with  [5, pp. 212-215, 12, pp. 141-142]. For the  transformation the k' th rectangular component of a vector which "points" in the direction of increasing  is  . Similarly, the k'th rectangular component of a vector in the direction of increasing is  .

The transformation equations of (1) yield the following explicit expressions.

.(4)

.         (5)

For ,

.(6)

.                                      (7)

For  let  be an  matrix defined as follows:

(8)

and for 

.(9)

Thus,  and  .

While in general, 

.

From equations (4) and (8)

.(10)

From equations (5) and (9)

.(11)

From equations (6), (7), and (9) for ,

.(12)

To explore the properties of the matrix, , the following trigonometric identity will prove useful.
 

Lemma 1: For  and  , .

Proof: This will be demonstrated by induction on n .

For the base case,  and 

.

Now assume that  .

Consider  . In the sum over k make the substitution  . This gives

, which is 1 by the induction hypothesis.
 
 

An  matrix B is orthogonal if and only if , where is the transpose of B [7, pp. 291-292, 312-313,
8, p. 93]. Thus, if B is orthogonal,  where  is the identity matrix.

These two products are expressed in the following two equations.

(13)

(14)

So an equivalent definition of an orthogonal matrix is an  matrix whose n columns (or rows) form an orthonormal basis for the space of n dimensional column (or row) vectors. Finally, from the product rule for determinants, if B is an orthogonal matrix, [7, pp. 206-207]

,

 so  .
 

Theorem A5: For  is an orthogonal matrix.

Proof: It is sufficient to establish that equation (13) is satisfied. This is directly verified for  or . So assume
.

Part I. Demonstrate that for 
 

Case 1j = 1.

From equation (8) and theorem A2 
 

Case 2 .

From equation (9) 

,

but by lemma 1 this reduces to .
 

Case 3 .

.
 

Case 4.

.
 

Part II. Demonstrate that for  and  and ,

.

From the definition of in equations (8) and (9) this reduces to considering the following nine cases.
 

Case 1 and  for  .


 

Case 2 and  for  .

.

By lemma 1 this is simply  .
 

Case 3 and  for  .


 

Case 4 and  for  .

from lemma 1.
 

Case 5 and  for  .


 

Case 6 and  for  .


 

Case 7 and  for  .


 
 

Case 8 and  for  .

from lemma 1.
 

Case 9 and  for  and  .

   from lemma 1.
 

The n columns of  are the n orthogonal unit vectors associated with the n variables  . This means that  constitute a system of orthogonal curvilinear coordinates
[6 pp. 21-27, 8, p. 219, 9 pp. 279-284, 18]. From equations (10) through (12) the "scale" factors of this curvilinear system are given as follows: [4, p 594, 13, p.436]
 

. (15)
for 

Since  is an orthogonal matrix its determinant must be either 1 or  . A more precise statement is made in the next theorem.

Theorem A6: For   is  raised to the power .

Proof: Let 

For  .

Assume  .

Suppose  , then the bottom row is . Expanding the determinant about the bottom row gives  , where  is an  minor determinant. The matrix M is defined as .

From the substitution  for  and equations (8) and (9) it follows tha  t Thus, .

Suppose  . Multiply the first column of by  and add this product to the second column of  to generate a new column, B .

.

The determinant  is not changed by replacing the second column of by B . In this new matrix the only non-zero entry in the bottom row is  in the  position. Evaluating  by expanding about this row gives

,

where  is an  minor determinant. The matrix N is defined as ,

but as before the substitution for  and the definition of the matrix  results in the simplified form:

 .
Hence,  . This verifies that the recursion  is valid for any angle in  . Using this recursion  times results in the following expression.

.

However, this is just  to the power .

Theorem A7: For the transformation has the Jacobian matrix  with the property that

. (16)

Proof: From equations (10) through (12)  is given in terms of by the following equations.
 

for 

Now by factoring out the scale factor from each column of  ,
 
 

So,  .

An equivalent argument would be to use the result thatis an orthogonal matrix, so that from the scale factors the volume element in n dimensions is given by  .
 
 

For and  . The inverse function theorem [ 1 pp. 256-257, 5, pp. 150-151, 8, p. 205 ] then provides an additional argument that the map  is injective (i.e., one to one).

The author thanks Professor Charles Goebel of the University of Wisconsin-Madison Physics Department for the reference to the work by Vilenkin, Kuznetsov and  Smorodinskii. This proved invaluable in finding other relevant references.

References:

1. R. Bartle, The Elements of Real Analysis, John Wiley, 1964.

2. X. Chapuisat and C. Iung, Vector parametrization of the N-body problem in quantum mechanics: polyspherical
     coordinates, Phys. Rev. A 45, 6217 (1992).

3. Ye.M.Hakobyan, G.S.Pogosyan, and A.N.Sissakian, On a Generalized D-Dimensional Oscillator: Interbasis Expansions,
      Phys. Atom. Nucl. 61, 1762-1767 (1998).

4. S. Hassani, Mathematical Physics: A Modern Introduction to its Foundations, Springer, 1999.

5. W. Kaplan, Advanced Calculus, 2nd ed., Addison-Wesley, 1973.

6. P. Morse and H. Feschbach, Methods of Theoretical Physics, Part I, McGraw Hill, 1953.

7. B. Noble, Applied Linear Algebra, Prentice-Hall, 1969.

8. L. Rade aqnd B. Westergren, Beta Mathematics Handbook, 2nd ed., CRC Press, 1990.

9. K. Riley, M. Hobson, and S. Bence, Mathematical Methods for Physics and Engineering, Cambridge University Press,
    1997.

10. M.A. Rodriguez and P.Winternitz, Quantum superintegrability and exact solvability in n dimensions,
      J. Math. Phys., 43, 1309-1322, (2002).

11. W. Rudin, Real and Complex Analysis, 2nd ed., McGraw-Hill, 1974.

12. M. Speigel, Schawm's Outline of Theory and Problems of Advanced Calculus, Schawm Publishing Company, 1963.

13. N. J. Vilenkin, Special Functions and the Theory of Group Representations: Translations of Mathematical
      Monographs, Vol 22, American Mathematical Society, 1968.

14. N. J. Vilenkin and A. U. Klimyk, Representation of Lie Groups and Special Functions, Vol 2: Class I
      Representations, Special Functions, and Integral Transforms, Kluwer Academic Press, 1993.

15.  N. J. Vilenkin, G. I. Kuznetsov and Y. A. Smorodinskii, Eigenfunctions of the Laplace operator realizing
       representations of  the groups U(2), SU(2), SO(3), U(3) and the symbolic method, Sov. J. Nucl. Phys 2, 645-655
       (1966).
16. G. N. Watson, Theory of Bessel Functions, 2nd edition, Cambridge University Press, 1944.
17. Wolfram Research: The hypersphere entry at the website http://mathworld.wolfram.com/Hypersphere.html .

18. Wolfram Research: The curvilinear coordinates entry at the website
       http://mathworld.wolfram.com/CurvilinearCoordinates.html .
 
 
 
 

Al Lehnen
Mathematics Department
Madison Area Technical College 
3550 Anderson Street
Madison, WI 53704
(608) 246-6567
alehnen@madisoncollege.edu
http://my.execpc.com/~aplehnen/al

Document first posted 6/19/2002
Document last revised 9/27/2002