THE CHANGE OF VARIABLES FORMULA USING MATRIX VOLUME

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1 SIAM J. Matrix Analysis 21(1999), THE CHANGE OF ARIABLES FORMLA SING MATRIX OLME ADI BEN-ISRAEL Abstract. The matrix volume is a generalization, to rectangular matrices, of the absolute value of the determinant. In particular, the matrix volume can be used in change-of-variables formulæ, instead of the determinant (if the Jacobi matrix of the underlying transformation is rectangular). This result is applicable to integration on surfaces, illustrated here by several examples. 1. Introduction The change-of-variables formula in the title is f(v) dv = (f φ)(u) det J φ (u) du (1) where, are sets in R n, φ is a sufficiently well-behaved function :, and f is integrable on. Here dx denotes the volume element dx 1 dx 2 dx n, and J φ is the Jacobi matrix (or Jacobian) ( ) φi (v 1, v 2,, v n ) J φ :=, also denoted u j (u 1, u 2,, u n ), representing the derivative of φ. An advantage of (1) is that integration on is translated to (perhaps simpler) integration on. This formula was given in 1841 by Jacobi [8], following Euler (the case n = 2) and Lagrange (n = 3). It gave prominence to functional (or symbolic) determinants, i.e. (non numerical) determinants of matrices including functions or operators as elements. If and are in spaces of different dimensions, say R n and R m with n > m, then the Jacobian J φ is a rectangular matrix, and (1) cannot be used in its present form. However, if J φ is of full column rank throughout, we can replace det J φ in (1) by the volume vol J φ of the Jacobian to get f(v) dv = (f φ)(u) vol J φ (u) du. (2) Recall that the volume of an m n matrix of rank r is vol A := det 2 A IJ (3) (I,J) N where A IJ is the submatrix of A with rows I and columns J, and N is the index set of r r nonsingular submatrices of A, see e.g. [1]. Alternatively, vol A is the product of the singular values of A. If A is of full column rank, its volume is simply vol A = det A T A (4) Date: February 25, Revised May 10, Mathematics Subject Classification. Primary 15A15, 26B15; Secondary 53A05, 42B10, 65R10. Key words and phrases. Determinants, Jacobians, matrix volume, change-of-variables in integration, surface integrals, Radon transform, Fourier transform, generalized Pythagorean theorem. Research done at the Indian Statistical Institute, New Delhi , India, and supported by a Fulbright Grant and the S Educational Foundation in India. 1

2 2 ADI BEN-ISRAEL If m = n then vol J φ = det J φ, and (2) reduces to the classical result (1). The formula (2) is well known in differential geometry, see e.g. [2, Proposition 6.6.1] and [6, 3.2.3]. Although there are elementary accounts of this formula (see e.g. [3, ol. II, Ch. I, 4], [7, 8.1] and [13, 3.4]), it is seldom used in applications. The purposes of this note are: (i) to establish the usefulness of (2) for various surface integrals, (ii) to simplify the computation of the Radon and Fourier Transforms, and general integrals in R n (see Examples 7 9 and Appendix A below), and (iii) to introduce the functional matrix volume, in analogy with the functional determinant. We illustrate (2) for an elementary calculus example. Let S be a subset of a surface in R 3 represented by z = g(x, y), (5) and let f(x, y, z) be a function integrable on S. Let A be the projection of S on the xy plane. Then S is the image of A under a mapping φ S = φ(a), or x y = x ( ( y x x = φ, A. (6) y) y) z g(x, y) The Jacobi matrix of φ is the 3 2 matrix J φ (x, y) = (x, y, z) (x, y) = , (7) where g x = g, g x y = g. The volume of (7) is, by (4), y vol J φ (x, y) = 1 + gx 2 + gy 2. (8) Substituting (8) in (2) we get the well-known formula f(x, y, z) ds = f(x, y, g(x, y)) 1 + gx 2 + gy 2 dx dy, (9) S A giving an integral over S as an integral over its projection in the xy plane. The simplicity of this approach is not lost in higher dimensions, or with different coordinate systems, as demonstrated below by eleven elementary examples from calculus and analysis. Example 1 concerns line integrals, in particular the arc length of a curve in R n. Example 2 is an application to surface integration in R 3 using cylindrical coordinates. Examples 3 4 concern integration on an axially symmetric surface (or surface of revolution) in R 3. In these integrals, the volume of J φ contains the necessary information on the surface symmetry. Example 5 shows integration on an (n 1) dimensional surface in R n. The area of the unit sphere in R n, a classical exercise, is computed in Example 6. Example 7 uses (2) to compute the Radon transform of a function f : R n R. Example 8 computes an integral over R n as an integral on R n 1 followed by an integral on R. Example 9 applies this to the computation of the Fourier transforms of a function f : R n R. The last two examples concern simplex faces in R n. Example 10 is the generalized Pythagoras theorem. Example 11 computes the largest face of the n dimensional regular simplex. These examples show that the full rank assumption for J φ is quite natural, and presents no real restriction in applications. The solutions given here should be compared with the classical solutions, as taught in calculus. We see that (2) offers a unified method for a variety of curve and surface integrals, and coordinate systems, without having to construct (and understand) the differential geometry in each application. The computational tractability of (2) is illustrated in Appendix A. g x g y

3 THE CHANGE OF ARIABLES FORMLA SING MATRIX OLME 3 A blanket assumption: Throughout this paper, all functions are continuously differentiable as needed, all surfaces are smooth, and all curves are rectifiable. If the mapping φ : is given by we denote its Jacobi matrix J φ by 2. Examples and applications y i = φ i (x 1, x 2,, x n ), i 1, m (y 1, y 2,, y m ) (x 1, x 2,, x n ) = the customary notation for Jacobi matrices (in the square case). ( ) φi, i 1, m, j 1, n (10) x j Example 1. Let C be an arc on a curve in R n, represented in parametric form as C := φ([0, 1]) = (x 1, x 2,, x n ) : x i := φ i (t), 0 t 1} (11) The Jacobi matrix J φ (t) = (x 1, x 2,, x n ) is the column matrix (φ t i(t)), and its volume is vol J φ = n (φ i (t))2. The line integral (assuming it exists) of a function f along C, f, is given in terms of the volume C of J φ as follows 1 f = f(φ 1 (t),, φ n (t)) n (φ i (t))2 dt. (12) In particular, f 1 gives C 0 arc length C = 1 0 n (φ i (t))2 dt. (13) If one of the variables, say a x 1 b, is used as parameter, (13) gives the familiar result b ( ) 2 arc length C = dxi 1 + dx 1. dx 1 Example 2. Let S be a surface in R 3 represented by where r, θ, z} are cylindrical coordinates The Jacobi matrix of the mapping (15a),(15b) and (14) is (x, y, z) = cos θ r sin θ sin θ r cos θ (r, θ) a i=2 z = z(r, θ) (14) x = r cos θ (15a) y = r sin θ (15b) z = z (15c) z r see also (A.4), Appendix A. The volume of (16) is ( ) 2 ( ) 2 ( ) 2 z z z vol J φ = r 2 + r 2 + = r r θ r r 2 z θ (16) ( ) 2 z, (17) θ

4 4 ADI BEN-ISRAEL see also (A.7), Appendix A. An integral over a domain S is therefore ( ) 2 z f(x, y, z) d = f(r cos θ, r sin θ, z(r, θ)) r r r 2 ( ) 2 z dr dθ. (18) θ Example 3. Let S be a surface in R 3, symmetric about the z axis. This axial symmetry is expressed in cylindrical coordinates by z = z(r), or z = 0 θ in (16) (18). The volume (17) thus becomes vol J φ = r 1 + z (r) 2 (19) with the axial symmetry built in. An integral over a domain in a z symmetric surface S is therefore f(x, y, z) d = f(r cos θ, r sin θ, z(r)) r 1 + z (r) 2 dr dθ. Example 4. Again let S be a z symmetric surface in R 3. We use spherical coordinates x = ρ sin φ cos θ (20a) y = ρ sin φ sin θ (20b) z = ρ cos φ (20c) The axial symmetry is expressed by ρ := ρ(φ) showing that S is given in terms of the two variables φ and θ. The volume of the Jacobi matrix is easily computed (x, y, z) vol = ρ ρ (φ, θ) 2 + (ρ (φ)) 2 sin φ and the change of variables formula gives f(x, y, z) d = f(ρ(φ) sin φ cos θ, ρ(φ) sin φ sin θ, ρ(φ) cos φ) ρ(φ) ρ(φ) 2 + (ρ (φ)) 2 sin φ dφ dθ (21) Example 5. Let a surface S in R n be given by x n := g(x 1, x 2,..., x n 1 ), (22) let be a subset on S, and let be the projection of on R n 1, the space of variables (x 1,..., x n 1 ). The surface S is the graph of the mapping φ :, given by its components φ := (φ 1, φ 2,..., φ n ), The Jacobi matrix of φ is φ i (x 1,..., x n 1 ) := x i, i = 1,..., n 1 φ n (x 1,..., x n 1 ) := g(x 1,..., x n 1 ) J φ = g g g g x 1 x 2 x n 2 x n 1

5 and its volume is THE CHANGE OF ARIABLES FORMLA SING MATRIX OLME 5 vol J φ = For any function f integrable on we therefore have f(x 1, x 2,, x n 1, x n ) d = ( ) 2 g 1 + (23) x i ( ) 2 f(x 1, x 2,, x n 1, g(x 1, x 2,, x n 1 )) g 1 + dx 1 dx 2 dx n 1 x i In particular, f 1 gives the area of ( ) 2 1 d = g 1 + dx 1 dx 2 dx n 1 (25) x i Example 6. Let B n be the unit ball in R n, S n the unit sphere, and a n the area of S n. Integrals on S n, in particular the area a n, can be computed using spherical coordinates, e.g. [12, II.2], or the surface element of S n, e.g. [11]. An alternative, simpler, approach is to use the results of Example 5, representing the upper hemisphere as φ(b n 1 ), where φ = (φ 1, φ 2,, φ n ) is φ i (x 1, x 2,, x n 1 ) = x i, i 1, n 1, φ n (x 1, x 2,, x n 1 ) = 1 x 2 i. (24) The Jacobi matrix is J φ = and its volume is easily computed vol J φ = x 1 x 2 x n 1 x n x n x n ( xi x n ) 2 = 1 x n = 1 1 n 1 x2 i. (26) The area a n is twice the area of the upper hemisphere. Therefore, by (26), dx 1 dx 2 dx n 1 a n = 2 1, (27) n 1 x2 i which is easily integrated to give B n 1 a n = 2 π n 2 Γ ( ), (28) n 2

6 6 ADI BEN-ISRAEL using well-known properties of the beta function B(p, q) := and the gamma function Γ(p) := (1 x) p 1 x q 1 dx, x p 1 e x dx. Example 7 (Radon transform). Let H ξ,p be a hyperplane in R n represented by } H ξ,p := x R n : ξ i x i = p = x : <ξ, x>= p} (29) where ξ n 0 in the normal vector ξ = (ξ 1,, ξ n ) of H ξ,p (such hyperplanes are called non vertical). Then H ξ,p is given by x n := p ξ n which is of the form (22). The volume (23) is here vol J φ = 1 + ( ξi ξ n ξ i ξ n x i (30) ) 2 = ξ ξ n The Radon transform (Rf)(ξ, p) of a function f : R n R is its integral over the hyperplane H ξ,p, see [4], (Rf)(ξ, p) := f(x) dx. (32) x: <ξ,x> = p} sing (30) (31), the Radon transform can be computed as an integral in R n 1 (Rf)(ξ, p) = ξ ( ) f x 1,, x n 1, p ξ i x i dx 1 dx 2 dx n 1 (33) ξ n ξ n ξ n R n 1 See (A.15) and (A.17), Appendix A, for Radon transform in R 2, and (A.20) for R 3. In tomography applications the Radon transforms (Rf)(ξ, p) are computed by the scanning equipment, so (33) is not relevant. The issue is the inverse problem, of reconstructing f from its Radon transforms (Rf)(ξ, p) for all ξ, p. The inverse Radon transform is also an integral, see e.g. [4],[12], and can be expressed analogously to (33), using the method of the next example. Example 8. Consider an integral over R n, f(x) dx = R n (31) R n f(x 1, x 2,, x n ) dx 1 dx 2 dx n (34) Since R n is a union of (parallel) hyperplanes, R n = x : <ξ, x>= p}, where ξ 0, (35) p= we can compute (34) iteratively: an integral over R n 1 (Radon transform), followed by an integral on R, dp f(x) dx = (Rf)(ξ, p) (36) ξ R n

7 THE CHANGE OF ARIABLES FORMLA SING MATRIX OLME 7 where dp/ ξ is the differential of the distance along ξ (i.e. dp times the distance between the parallel hyperplanes H ξ,p and H ξ,p+1 ). Combining (33) and (36) we get the integral of f on R n, 1 ( ) f(x) dx = x 1,, x n 1, p ξ i x i dx 1 dx 2 dx n 1 dp (37) ξ n ξ n ξ R n n R n 1 f see (A.21), Appendix A, for integration over R 3. It is possible to derive (37) directly from the classical change-of-variables formula (1), by changing variables from x 1,, x n 1, x n } to x 1,, x n 1, p := n ξ i x i }, and using det ( (x1,, x n 1, x n ) (x 1,, x n 1, p) ) = 1. ξ n An advantage of our development is that it can be used recursively, i.e. the inner integral in (37) can again be expressed as an integral in R n 2 followed by an integral in R, etc. Example 9 (Fourier transform). In particular, the Fourier transform (Ff)(ξ) of f is the integral (Ff)(ξ) := (2 π) n/2 e i <ξ,x> f(x) dx = R n } (38) (2 π) n/2 f (x 1,, x n ) exp i ξ k x k dx 1 dx 2 dx n. For ξ n 0 we can compute (38) analogously to (37) as ( (Ff)(ξ) = (2π) n/2 e ip f x 1,, x n 1, p ξ n ξ n R n R n 1 k=1 ξ i ξ n x i dx k dp. (39) ) n 1 The Fourier transform of a function of n variables is thus computed as an integral over R n 1 followed by an integral on R. The inverse Fourier transform of a function g(ξ) is of the same form as (38), (F 1 g)(x) := (2 π) n/2 R n e i <x,ξ> g(ξ) dξ, (40) and can be computed as in (39). Example 10. (The generalized Pythagorean theorem, [10]). Consider an n dimensional simplex } n := (x 1, x 2,, x n ) : a i x i a 0, x i 0, i 1, n, (41) with all a j > 0, j 0, n. We denote the n + 1 faces of n by } F 0 := (x 1, x 2,, x n ) n : a i x i = a 0 k=1 (42a) F j := (x 1, x 2,, x n ) n : x j = 0}, j 1, n (42b) and denote their areas by A 0, A j respectively. The generalized Pythagorean theorem (see [10]) states that A 2 0 = A 2 j (43) j=1

8 8 ADI BEN-ISRAEL We prove it here using the change of variables formula (2). For any j 1, n we can represent the (largest) face F 0 as F 0 = φ j} (F j ) where φ j} = (φ j} 1,, φ j} n ) is φ j} i (x 1, x 2,, x n ) = x i, i j, φ j} j (x 1, x 2,, x n ) = a 0 a j i j a i a j x i. The Jacobi matrix of φ j} is an n (n 1) matrix with the i th unit vector in row i j, and ( a 1, a 2,, a j 1, a j+1,, a n 1, a ) n a j a j a j a j a j a j in row j. The volume of the Jacobi matrix of φ j} is computed as ( ) 2 n ai vol J φ j} = = 1 + i j a j a2 i a 2 j = a a j where a is the vector (a 1,, a n ). Therefore, the area of F 0 is ( ) a ( ) a A 0 = dx i = A j, j 1, n (44) F j a j a j i j n n j=1 A2 j j=1 = a j 2 a 2 A 2 0 and the generalized Pythagorean theorem (43) reduces to the ordinary Pythagorean theorem a 2 = a j 2. j=1 Example 11. The simplex (41) with a j = 1 for all j 0, n is the n dimensional regular simplex } n := (x 1, x 2,, x n ) : 1, x i 0, i 1, n. (45) The area A 0 of its largest face in given by (44) as F 0 = (x 1,, x n ) : A 0 x i x i = n n 1 = 1, x i 0, i 1, n where n 1, the volume of the (n 1) dimensional regular simplex, is Therefore n 1 = 1 (n 1)! A 0 =, see e.g. [9, 47]. }. (46) n (n 1)!. (47) Note that A 0 0 as n, although the side faces of the unit cube (of which the face F 0 is a diagonal section ) have areas 1.

9 THE CHANGE OF ARIABLES FORMLA SING MATRIX OLME 9 References [1] A. Ben-Israel, A volume associated with m n matrices, Lin. Algeb. Appl. 167(1992), [2] M. Berger and B. Gostiaux, Differential Geometry: Manifolds, Curves and Surfaces (translated by S. Levy), Graduate Texts in Mathematics No. 115, Springer-erlag, New York 1988 [3] R. Courant, Differential and Integral Calculus, (translated by E.J. McShane), Interscience Publishers, New York, 1936 [4] S.R. Deans, The Radon Transform and Some of its Applications, Wiley, [5] Derive for Windows, Soft Warehouse, Inc Waialae Avenue, Suite 304, Honolulu, Hawaii , SA. [6] H. Federer, Geometric Measure Theory, Springer-erlag, New York, [7] W. Fleming, Functions of Several ariables (2nd edition), Springer-erlag, New York, 1977 [8] C.G.J. Jacobi, De Determinantibus Functionalibus, Crelle Journal für die reine und angewandte Mathematik 22(1841), Reprinted in C.G.J. Jacobi s Gesammelte Werke (K. Weierstrass, editor), ol. 3, , Berlin [9] M.G. Kendall, A Course in the Geometry of n Dimensions, Griffin s Statistical Monographs No. 8, Hafner, [10] S.-Y.T. Lin and Y.-F. Lin, The n-dimensional Pythagorean theorem, Lin. and Multilin. Algeb. 26(1990), [11] C. Müller, Spherical Harmonics, Lecture Notes in Mathematics No. 17, Springer-erlag, [12] F. Natterer, The Mathematics of Computerized Tomography, Wiley, [13] M. Schreiber, Differential Forms: A Heuristic Introduction, Springer-erlag, Appendix A: Illustrations with Derive. The integrations of this paper can be done symbolically. We illustrate this for the symbolic package Derive, [5], omitting details such as the commands (e.g. Simplify, Approximate) and settings (e.g. Trigonometry:=Expand) that are used to obtain these results. The Jacobi matrix is computed by the function JACOBIAN(u, α) := ECTOR(GRAD(u m, α), m, DIMENSION(u)) found in the Derive utility file ECTOR.mth. Example: Define the surface as a function of r and θ, as in Example 2. Then Z(r, θ) := JACOBIAN([r COS(θ), r SIN(θ), Z(r, θ)], [r, θ]) gives (16), COS(θ) r SIN(θ) SIN(θ) r COS(θ) Z(r, θ) Z(r, θ) r θ The volume of an (n + 1) n matrix of full rank is computed by For example, gives (17), OL(a) := SQRT(SM(DET(DELETE ELEMENT(a, k )) 2, k, 1, DIMENSION(a)))) OL(JACOBIAN([r COS(θ), r SIN(θ), Z(r, θ)], [r, θ])) (A.1) (A.2) (A.3) (A.4) (A.5) (A.6) ( ) 2 ( ) 2 Z(r, θ) Z(r, θ) r r r θ 2 (A.7) In Example 3 the surface is symmetric about the z-axis, and we use Z(r) instead of Z(r, θ) in (A.6) to get (19), OL(JACOBIAN([r COS(θ), r SIN(θ), Z(r)], [r, θ])) r Z (r) (A.8) (A.9)

10 10 ADI BEN-ISRAEL For example, the volume associated with the surface z = 1/r OL(JACOBIAN([r COS(θ), r SIN(θ), 1/r], [r, θ])) gives r4 + 1 r (A.10) The surface area of z = 1/r is 2π times the integral of (A.10). In particular, the surface area from r = 1 to r = 100 [ ] SQRT(r 4 + 1) 2 π INT, r, 1, 100 gives r Consider now the Radon Transform in R 2. A line can be solved for y a x + b y = p y = p b a b x if b 0. Then OL(JACOBIAN ([x, p b a ] ) b x a2 + b, [x] gives 2 b in agreement with (31). The Radon transform (33) is then computed by RADON AX(f, x, y, a, b, p) := For example, the Radon transform of e x2 y 2, INT(LIM(f, y, p/b a/b x), x,, ) SQRT(a 2 + b 2 )/ b RADON AX(EXP( x 2 y 2 ), x, y, a, b, p) is ] π EXP [ p2 a 2 + b 2 (A.11) (A.12) (A.13) (A.14) The function (A.13) requires b 0. We modify it for lines with b = 0 (i.e. vertical lines) as follows RADON(f, x, y, a, b, p) := The Radon transform of e x2 y 2 IF(b = 0, INT(LIM(f, x, p/a), y,, ), RADON AX(f, x, y, a, b, p)) w.r.t. a vertical line RADON(EXP( x 2 y 2 ), x, y, a, 0, p) gives ] π EXP [ p2 a 2 (A.15) which is (A.14) with b = 0. An alternative way is to normalize the coefficients of the lines (A.11), say a = cos α and b = sin α. The Radon transform is then RADON NORMAL AX(f, x, y, α, p) := (A.16) INT(LIM(f, y, p/sin(α) COT(α) x), x,, )/ABS(SIN(α)) provided α 0. The following function treats α = 0 separately RADON NORMAL(f, x, y, α, p) := IF(α = 0, INT(LIM(f, x, p), y,, ), RADON NORMAL AX(f, x, y, α, p)) For example, the Radon transform of is computed by f(x, y) := 1 if 0 x, y 1 0 otherwise RADON NORMAL(CHI(0, x, 1) CHI(0, y, 1), x, y, α, p) (A.17) (A.18)

11 THE CHANGE OF ARIABLES FORMLA SING MATRIX OLME 11 giving [ p SIGN(COS(α) p) SIN(2α) 0.5 ] SIN(α) The Radon transform in R 3 is computed w.r.t. a plane as follows (ignoring the case c = 0), RADON 3(f, x, y, z, a, b, c, p) := a x + b y + c z = p + p SIN(2α) INT(INT(LIM(f, z, p/c a/c x b/c y), x,, ), y,, ) SQRT(a 2 + b 2 + c 2 )/ c Finally, an integral over R 3 is computed, using (37) as follows: INT 3(f, x, y, z, a, b, c) := INT(RADON 3(f, x, y, z, a, b, c, p), p,, )/SQRT(a 2 + b 2 + c 2 ) For example, the integral over R 3 of f(x, y, z) = e x+y+z x2 y 2 z 2, INT 3(EXP(x + y + z x 2 y 2 z 2 ), x, y, z, a, b, c) gives π 3/2 e 3/4 (A.19) (A.20) (A.21) RTCOR Rutgers Center for Operations Research, Rutgers niversity, 640 Bartholomew Rd, Piscataway, NJ , SA address: bisrael@rutcor.rutgers.edu