Jacobian for N-Dimensional Spherical Coordinates in This Article We Will

Jacobian for n-Dimensional Spherical Coordinates

In this article we will derive the general formula for the Jacobian of the transformation from the Cartesian coordinates to the spherical coordinates in n dimensions without the use of determinants. In general, the equation for the sphere of radius R in integer n dimensions is 2 2 2 2 x1 + x2 + ... + xn = R (1) where x1, x2, . . . , xn are Cartesian coordinates. The n-dimesnsional sphere is often called n-hypersphere. For n = 2 we have just the equation of a circle, and for n = 3 the equation of a three-dimensional sphere. To compute the area of a circle or the volume of a three-dimensional sphere it is convenient to carry out the appropriate integrations in azimuthal and spherical coordinates, respectively. The computation of the volume of the n-dimensional sphere would require integration in n-dimensional spherical coordinates. The deriva- tion of the transformation from the Cartesian coordinates x1, x2, . . . , xn to the n-dimensional spherical coordinates r, θ, φ1, . . . φn−2 has been presented in [1]. For example the transformation for ﬁve dimensions is given by equations (19) and for n dimensions is

1 : x1 = r cos φ1

2 : x2 = r sin φ1 cos φ2

3 : x3 = r sin φ1 sin φ2 cos φ3 ...

i : xi = r sin φ1 sin φ2 ... sin φi−1 cos φi (2) ...

n − 2 : xn−2 = r sin φ1 sin φ2 ... sin φn−3 cos φn−2

n − 1 : xn−1 = r sin φ1 sin φ2 ... sin φn−2 cos θ

n : xn = r sin φ1 sin φ2 ... sin φn−2 sin θ where 0 ≤ φi ≤ π, i = 1, . . . , n − 2 and 0 ≤ θ ≤ 2π. The n-dimensional spherical coordinates are created in such way that they are orthogonal what means that the scalar product of their any two basis vectors, which are ˆ ˆ ˆ sometimes called versors, ir, iθ, iφi for i = 1, 2, . . . , n − 2 is equal zero. The n-dimensional Cartesian coordinates are also orthogonal.

1 The transformation from one set of coordinates to another one involves the change of the inﬁnitesimally small volume element. In Cartesian coordinates the volume element is simply

(x1,x2,...,xn) dV = dx1dx1 . . . dxn (3) and the change from the Cartesian to the spherical coordinates involves the

Jacobian J(r, θ, φ1, φ2, . . . , φn−2) of the transformation, so we must write the formula for the volume element in the n-dimensional spherical coordinates as

(r,θ,φ1,φ2,...,φn−2) dV = J(r, θ, φ1, φ2, . . . , φn−2)drdθdφ1dφ2 . . . dφn−2 (4)

The Jacobian is a determinant of the n by n matrix of partial derivatives

 ∂x1 ∂x1 ∂x1 ··· ∂x1  ∂r ∂θ ∂φ1 ∂φn−2  ∂x2 ∂x2 ∂x2 ··· ∂x2   ∂r ∂θ ∂φ1 ∂φn−2   . . . . .  (5)  ......   . . . .  ∂xn ∂xn ∂xn ··· ∂xn ∂r ∂θ ∂φ1 ∂φn−2 For how Jacobian determinant emerges in the transformation of variables we will point the reader to [2]. We will analyze the Jacobians of transformations from the Cartesian to the spherical coordinates for dimensions n = 1, 2, 3, 4, 5 without actually computing any determinants, and we will develop the general formula for the Jacobian of the transformation of coordinates for any dimension n > 2. Computing the Jacobian determinants even for a three-dimensional spherical coordinates transformation is cumbersome. We will employ another method which is based on the deﬁnition of the angle measure in radians and on the orthogonality of the spherical coordinates. The radian measure dα of a central angle of a circle is deﬁned as the ratio of the length dlα of the arc the angle subtends divided by the radius r of the circle dl dα = α (6) r We may express the value of the volume element dV (r,θ,φ1,φ2,...,φn−2) as

(r,θ,φ1,φ2,...,φn−2) dV = drdlθdlφ1 dlφ2 . . . dlφn−2 (7)

2 ˆ ˆ ˆ by virtue of orthogonality of the versors ir, iθ, iφi for i = 1, 2, . . . , n − 2 along the radius r and tangent to the coordinate lines θ, φi for i = 1, 2, . . . , n − 2, respectively at the point (r, θ, φ1, . . . , φn−2). The versors for threedimesional ˆ ˆ ˆ spherical coordinates which are denoted in this article by ir, iθ, iφ1 are illus- trated in [3]. θ is azimuthal angle coordinate, and φi is called i-th polar angle coordinate. For n = 1

1 : x1 = r (8) we just make a variable substitution and

dV (r) = J(r)dr = dr (9) what gives J1 = J(r) = 1. For n = 2 we add azimuthal angle θ as the second coordinate

1 : x = r cos θ 1 (10) 2 : x2 = r sin θ and we have for

θ : dθ = dlθ/r (11)

The volume element for n = 2 is

(r,θ) dV = J(r, θ)drdθ = drdlθ = rdrdθ (12) and J2 = J(r, θ) = r.

For n = 3 we need to add to the coordinates the polar angle φ1

1 : x1 = r cos φ1

2 : x2 = r sin φ1 cos θ (13)

3 : x3 = r sin φ1 sin θ and we have for

φ : dφ = dl /r 1 1 φ1 (14) θ : dθ = dlθ/(r sin φ1)

3 We come to the above formulas just by taking into account that the angle dφ1 subtends the arc of length dlφ1 of the radius r, and that the angle dθ subtends the arc of length dlθ of the radius r sin φ1. The volume element in 3 dimensions is

(r,θ,φ1) 2 dV = J(r, θ, φ1)drdθdφ1 = drdlθdlφ1 = r sin φ1drdθdφ1 (15)

2 and the Jacobian J3 = J(r, θ, φ1) = r sin φ1.

For n = 4 we add to the coordinates the polar angle φ2

1 : x1 = r cos φ1 2 : x = r sin φ cos φ 2 1 2 (16) 3 : x3 = r sin φ1 sin φ2 cos θ

4 : x4 = r sin φ1 sin φ2 sin θ and it gives for

φ1 : dφ1 = dlφ1 /r

φ2 : dφ2 = dlφ2 /(r sin φ1) (17)

θ : dθ = dlθ/(r sin φ1 sin φ2)

Now the situation is that the angle dφ1 subtends the arc of length dlφ1 of the radius r as before, the angle dφ2 subtends the arc of length dlφ2 of the radius r sin φ1, which is the new radius as given in (16) in the formula for x2, and that the angle dθ subtends the arc of length dlθ of the radius r sin φ1 sin φ2 also by analogy to the situation for 3 dimensions. In other words we develop the above relations as a consequence of the deﬁnition of the spherical coordinates in 3 dimensions in equations (13) and by following the relations in (14). The volume element in 4 dimensions is

(r,θ,φ1,φ2) dV = J(r, θ, φ1, φ2)drdθdφ1dφ2 = drdlθdlφ1 dlφ2 (18) 3 2 = r sin φ1 sin φ2drdθdφ1dφ2

3 2 and the Jacobian J4 = J(r, θ, φ1, φ2) = r sin φ1 sin φ2. For n = 5 we have

1 : x1 = r cos φ1

2 : x2 = r sin φ1 cos φ2

3 : x3 = r sin φ1 sin φ2 cos φ3 (19)

4 : x4 = r sin φ1 sin φ2 sin φ3 cos θ

5 : x5 = r sin φ1 sin φ2 sin φ3 sin θ

4 and for

φ1 : dφ1 = dlφ1 /r φ : dφ = dl /(r sin φ ) 2 2 φ2 1 (20) φ3 : dφ3 = dlφ3 /(r sin φ1 sin φ2)

θ : dθ = dlθ/(r sin φ1 sin φ2 sin φ3) This gives the volume element for 5 dimensions

(r,θ,φ1,φ2,φ3) dV = J(r, θ, φ1, φ2, φ3)drdθdφ1dφ2dφ3 = drdlθdlφ1 dlφ2 dlφ3 (21) 4 3 2 = r sin φ1 sin φ2 sin φ3drdθdφ1dφ2dφ3

4 3 2 and the Jacobian J5 = J(r, θ, φ1, φ2, φ3) = r sin φ1 sin φ2 sin φ3. We notice that our Jacobian for 5 dimensions is just the product of the denominators from the equations (20). The pattern for the Jacobian of the transformation from n Cartesian coordinate system to the system of n-dimensional spherical coordinates clearly reveals itself. For n > 2 n−2 n−1 Y n−1−k Jn = J(r, θ, φ1, φ2, . . . , φn−2) = r sin φk (22) k=1

The Jacobian we derived may be used in computing the volume Vn(c) or the surface Sn(r) of a n-dimensional sphere of radius c or r, respectively. Z c Z 2π Z π Z π Z π Vn(c) = ··· Jn drdθdφ1dφ2 . . . dφn−2 (23) r=0 θ=0 φ1=0 φ2=0 φn−2=0 Z c Z 2π n−2 Z π n−1 Y n−1−k = r dr dθ sin φk dφk r=0 θ=0 k=1 φk=0

Z 2π Z π Z π Z π Sn(r) = ··· Jn dθdφ1dφ2 . . . dφn−2 (24) θ=0 φ1=0 φ2=0 φn−2=0 Z 2π n−2 Z π n−1 Y n−1−k = r dθ sin φk dφk θ=0 k=1 φk=0 The further computation is an exercise in applying the formula for the integral of the type ! Z π/2 1 1 1 sinn x cosm dx = B (n + 1), (m + 1) (25) 0 2 2 2

5 where B is the Beta function, which is deﬁned in [4] and [5] to compute the integral of powers of sine, and then the application of the Euler gamma function Γ which is described in [4], [6] and [7] and which is related to the function Beta Γ(x)Γ(y) B(x, y) = (26) Γ(x + y) Properties of Euler Gamma function used in this article are presented also in [8]. For natural values of x the Euler Gamma function has the property

Γ(x) = (x − 1)! (27)

If we substitute x + 1 for x in the above equation, then we obtain

Γ(x + 1) = x! = x(x − 1)! = xΓ(x) (28)

The relation Γ(x + 1) = xΓ(x) (29) is valid also for real values of x. Also 1 √ Γ = π (30) 2 Then we have

n−2 1 1 2 n n−1 n−1 Γ ( 2 ) 2π r Sn(r) = r 2π 1 = 1 (31) Γ( 2 n) Γ( 2 n)

R 1 n n Z 2π 2 R Vn(R) = Sn(r) dr = 1 (32) r=0 nΓ( 2 n) In particular for n = 3, i.e. for three dimensions we can obtain the formulas for the surface area and for the volume of a sphere. For the sphere surface area from equation (31) we have

3 2 √ 3 2 2π 2 r 2( π) r S3(r) = 3 = 3 (33) Γ( 2 ) Γ( 2 )

3 From equations (29) and (30) we can compute the value of Γ 2 3 1 1 1 1√ Γ = Γ + 1 = Γ = π (34) 2 2 2 2 2 6 and substitute it into equation (33) √ 2( π)3r2 S3(r) = 1 √ (35) 2 π obtaining the well known formula for the surface area S3(r) of a three- dimensional sphere of radius r

2 S3(r) = 4πr (36)

For the sphere volume V3(R) of a three-dimensional sphere from equation (32) we obtain √ √ 2( π)3R3 2( π)3R3 V3(R) = 3 = 3 √ (37) 3Γ( 2 ) 2 π and we receive 4 V (R) = πR3 (38) 3 3 what is also a familiar formula for the volume of a sphere of radius R.

References

[1] K.S. Miller, Multidimensional Gaussian Distributions, John Wiley & Sons, Inc., New York, London, Sydney, 1964.

[2] R.A. Hunt, Calculus, 2nd ed., HarperCollins College Publishers, 1994.

[3] http://mathworld.wolfram.com/SphericalCoordinates.html

[4] N.N. Lebedev, Special Functions and Their Applications, Translation by R.A. Silverman, Dover Publications, Inc., New York, 1972.

[5] http://mathworld.wolfram.com/BetaFunction.html

[6] Emil Artin, The Gamma Function, Translation by Michael Butler, Dover Publications, Inc., Mineola, New York, 2015.

[7] http://mathworld.wolfram.com/GammaFunction.html

[8] Donald A. McQuarrie, Mathematical Methods for Scientists and Engi- neers, University Science Books, Sausalito, California 2003.

7 Pawel Jan Piskorz ([email protected]) 426 Tynan Ct Erie, CO 80516-7208 USA