Derivation of Jacobian Formula with Dirac Delta Function
aa r X i v : . [ phy s i c s . g e n - ph ] D ec KPOP E -2020-07 Derivation of Jacobian Formula with Dirac Delta Function
Dohyun Kim, ∗ June-Haak Ee, † Chaehyun Yu, ‡ and Jungil Lee § KPOP E Collaboration, Department of Physics, Korea University, Seoul 02841, Korea
We demonstrate how to make the coordinate transformation or change of variables from Cartesiancoordinates to curvilinear coordinates by making use of a convolution of a function with Diracdelta functions whose arguments are determined by the transformation functions between the twocoordinate systems. By integrating out an original coordinate with a Dirac delta function, we replacethe original coordinate with a new coordinate in a systematic way. A recursive use of Dirac deltafunctions allows the coordinate transformation successively. After replacing every original coordinateinto a new curvilinear coordinate, we find that the resultant Jacobian of the corresponding coordinatetransformation is automatically obtained in a completely algebraic way. In order to provide insightson this method, we present a few examples of evaluating the Jacobian explicitly without resort tothe known general formula. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected]; Director of the Korea Pragmatist Organization for Physics Education ( KPOP E ) I. INTRODUCTION
A coordinate transformation or change of variables from a coordinate system to another in multi-dimensionalintegrals has widely been applied to a variety of fields in mathematics and physics. This transformation alwaysinvolves a factor called the
Jacobian , which is the determinant of the Jacobian matrix. The matrix elements of theJacobian matrix are the first-order partial derivatives of the new coordinates with respect to the original coordinates.The formula for the change of variables from n -dimensional variables x , x , · · · , x n to q , q , · · · , q n is expressed interms of the Jacobian J : Z dx dx · · · dx n f ( x , · · · , x n ) = Z dq dq · · · dq n J F ( q , · · · , q n ) , (1)where the integrand f ( x , · · · , x n ) is a function of the independent variables x , · · · , x n and F ( q , · · · , q n ) = f [ x ( q , · · · , q n ) , · · · , x n ( q , · · · , q n )]. In general, the variables x , · · · , x n can be treated as the Cartesian coordi-nates of an n -dimensional Euclidean space and the variables q , q , · · · , q n form a set of curvilinear coordinatesrepresenting the same Euclidean space. We assume that each curvilinear coordinate q i is uniquely defined by theCartesian coordinates: q i = q i ( x , · · · , x n ). We also assume that the inverse transformation is uniquely defined as x i = x i ( q , · · · , q n ) . Then the Jacobian J can be expressed as J = (cid:12)(cid:12)(cid:12)(cid:12) ∂ ( x , x , · · · , x n ) ∂ ( q , q , · · · , q n ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D e t ∂x ∂q ∂x ∂q · · · ∂x n ∂q ∂x ∂q ∂x ∂q · · · ∂x n ∂q ... ... . . . ... ∂x ∂q n ∂x ∂q n · · · ∂x n ∂q n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (2)where D e t stands for the determinant.In physics, the Jacobian appears frequently when one makes the change of variables between Cartesian and curvilin-ear coordinates in various physical quantities involving surface or volume integrals. However, in most physics textbookincluding classical mechanics and electromagnetism usually abstract descriptions are provided. In many textbooks ofcalculus or mathematical physics [1], the Jacobian formula is derived in the following way: First, one transforms themultivariable differential volume by applying the change of variables. Next, one imposes a geometrical argument thatthe infinitesimal volume is invariant under the transformation [2]. The invariance of the volume can also be confirmedby applying Green’s theorem to show that R S dx dx = R T J dq dq , where S is a rectangular region in the x x planeand T is the corresponding region [3]. Although experienced teachers or researchers may follow the abstract logic inthis kind of Jacobian derivation without difficulties, the concept is rather unclear or less intuitive to undergraduatephysics-major students who are not familiar with advanced mathematical concepts of multi-variable calculus.The Dirac delta function δ ( x ) is not a well-defined function but a distribution defined only through integration: R ∞−∞ dx δ ( x ) = 1. For any smooth function f ( x ), R ∞−∞ dx δ ( x − y ) f ( x ) = f ( y ). This property can be applied tochanging the integration variable. Recently, some of us introduced an alternative proof of Cramer’s rule by makinguse of Dirac delta functions [4]. It turns out that the method with a convolution of a coordinate vector with Diracdelta functions provides a systematic way to change integration variables from original coordinates to new coordinates.This change of variables enables us to reproduce Cramer’s rule.The Dirac delta function technique exploited in the derivation of Cramer’s rule in Ref. [4] can be immediatelyapplicable to the evaluation of the Jacobian for the coordinate transformation or change of variables after replacingthe coordinate vector with an arbitrary function. However, the direct application of the approach in Ref. [4] is limitedto a linear transformation between two coordinate systems because Cramer’s rule applies only to the system of linearequations. Thus the method is not applicable to the transformation involving a set of curvilinear coordinates whichare frequently used in physics.The main goal of this work is to present a more intuitive derivation of the Jacobian involving any coordinatetransformation and to demonstrate how it works with heuristic examples. We develop an alternative derivation ofthe Jacobian formula as well as the coordinate transformation by convolving a function with Dirac delta functions.Our derivation relies only on the direct integration of Dirac delta functions whose arguments involve the coordinatetransformation rules. Hence, we expect that students who are familiar with the Dirac delta function can compute theJacobian formula in any coordinate transformations by themselves without referring to a reference. The approachthat we present in this paper is quite straightforward and requires mostly algebraic computation skills.In this paper, we derive the Jacobian for a coordinate transformation or change of variables in the case of a non-lineartransformation by convolving an arbitrary function with Dirac delta functions. While the basic strategy to perform thecoordinate transformation is similar to that employed in Ref. [4], the integration of the original coordinates is ratherinvolved because of the non-linear property of the transformation functions between the two coordinate systems. Theintegration of the original coordinates can be carried out by making use of Dirac delta functions. An extra factorcontaining partial derivatives of corresponding coordinate variables appears in front of the original integrand afterthat multiple integration. It turns out that the extra factor can be evaluated by making use of the chain rule of partialderivatives. A recursive use of Dirac delta functions enables us to achieve the coordinate transformation successively.After replacing every coordinate with new coordinates, we identify the resultant extra factor in the integrand withthe Jacobian for the coordinate transformation or change of variables.The derivation of the Jacobian formula presented in this paper is new to our best knowledge. It is remarkablethat our derivation is free of borrowing abstract and advanced mathematical concepts unlike the other derivationsavailable. Instead, we exploit a simple concept of integration of the one-dimensional Dirac delta function repeatedlyin combination with a purely algebraic manipulation in reorganizing the extra factor by applying chain rules. Thisintuitive and systematic approach is expected to be pedagogically useful in upper-level mathematics or physics coursesin practice of the recursive use of both the Dirac delta function and the chain rule of partial derivatives.This paper is organized as follows: In section II, we introduce notations that are frequently used in the rest of thepaper, make a rough sketch of our strategy and present a formal derivation of the Jacobian factor for the coordinatetransformation from the n -dimensional Cartesian coordinates to a set of curvilinear coordinates. Section III is devotedto the explicit evaluation of the Jacobians for a few examples. Conclusions are given in section IV and a rigorousderivation of the chain rule for partial derivatives is given in Appendices. II. DERIVATION OF THE JACOBIANA. Strategy and Notation
In this subsection, we present our strategy to derive the Jacobian for a coordinate transformation or change ofvariables from the Cartesian coordinates x i to the curvilinear coordinates q i with transformation functions q i = q i ( x , · · · , x n ) (3)for i = 1 through n , where n is a positive integer. We assume that the two sets of coordinates describe a single pointuniquely and, therefore, the two sets of coordinates have a one-to-one correspondence although the transformation isin general non-linear. Thus the transformation in Eq. (3) is invertible: the inverse transformation from the curvilinearcoordinates to the Cartesian coordinates exists. If the curvilinear coordinates are a linear combination of the Cartesiancoordinates, then the linear transformation is invertible if the transformation matrix is non-singular: the determinantof the matrix is not vanishing. Then, the inverse transformation can be written as x i = x i ( q , · · · , q n ) . (4)The basic strategy to derive the Jacobian with Dirac delta functions is the same as that for the derivation ofCramer’s rule for a partial set of a coordinate transformation or change of variables given in Ref. [4]. One couldimmediately apply the approach in Ref. [4] to find the Jacobian as long as the transformation (3) is linear. In general,the transformation functions (3) are non-linear. Here we develop a generalized version of the approach in Ref. [4] inorder to consider arbitrary curvilinear coordinates.We define an n -dimensional integral I n , I n = Z ∞−∞ d n x f ( x , · · · , x n ) , (5)where d n x ≡ dx · · · dx n is the n -dimensional differential volume element and the integrand f ( x , · · · , x n ) is anarbitrary function of the Cartesian coordinates. We assume that every Cartesian coordinate is integrated over theregion ( −∞ , ∞ ). We define the unity i ≡ Z dq i δ [ q i − q i ( x , · · · , x n )] = 1 (6)for i = 1 through n . Multiplying the unity i to the integral I n , one can integrate out the integration variable x i bymaking use of the Dirac delta function keeping the q i integral unevaluated. By applying this process to I n recursivelyfrom i = 1 through n , we complete the change of the integration variables from the Cartesian coordinates to thecurvilinear coordinates. While we have suppressed the bounds of the integration for the curvilinear coordinate q i ’s inEq. (6), the curvilinear coordinates are assumed to be integrated over the entire region to cover the whole Euclideanspace represented by the Cartesian coordinates by a single time.We first compute × I n : I n = × I n = Z dq Z ∞−∞ d n x f ( x , · · · , x n ) δ [ q − q ( x , · · · , x n )] . (7)By definition, we integrate over x by making use of the delta function δ [ q − q ( x , · · · , x n )] keeping the q integralunevaluated: I n = Z dq Z ∞−∞ dx · · · dx n G f ( q , x , · · · , x n ) , (8)where x is expressed in terms of q and x j ’s for j = 2 through n satisfying the condition that the argument ofthe delta function vanishes, q − q ( x , · · · , x n ) = 0. Because the explicit forms of f ( x i ) and x i vary depending onthe integration step, we adopt the notation f ( q , x , · · · , x n ) after the x integration. We will present more detailedexplanations for this notation in the later part of this subsection. The extra factor G is the remnant of the integrationof the delta function and its explicit form will be given later in this paper.Next, we multiply to I n in Eq. (8) to find that I n = × I n = Z dq dq Z ∞−∞ dx · · · dx n G f ( q , · · · , x n ) δ [ q − q ( q , x , · · · , x n )] , (9)where every x in the argument of the delta function as well as the integrand function is replaced with the expressionin terms of q and x j ’s for j = 2 through n . Performing the integration over x by making use of the delta function,we find I n = Z dq dq Z ∞−∞ dx · · · dx n G f ( q , q , x , · · · , x n ) . (10)After the x integration every x in the integrand is expressed in terms of q , q and x j ’s for j = 3 through n . Again, G is the remnant of the integration of the delta functions.In this way, we integrate over x k for k = 1 through n successively. Finally, after the integration over x n we findthat I n reduces into the n -dimensional multiple integral over the curvilinear coordinates q i for i = 1 through n only. I n = Z dq · · · dq n G n f ( q , · · · , q n ) , (11)where G n is the remnant of the integration of the delta functions. At this stage, the integrand acquires an additionalfactor G n in front of the original integrand f in Eq. (5). This extra factor is identified with the Jacobian.In an intermediate step, for example, where the integration over x j (1 ≤ j ≤ n ) is carried out, x , · · · , x j in theintegrand must be replaced with the expressions in terms of q , · · · , q j and x j +1 , · · · , x n as x k = x k ( q , · · · , q j , x j +1 , · · · , x n ) (12)for k = 1 through j . Each x k in Eq. (12) is determined by the condition that the argument of the correspondingDirac delta function vanishes: q k − q k [ x ( q , · · · , q j , x j +1 , · · · , x n ) , · · · , x j ( q , · · · , q j , x j +1 , · · · , x n ) , x j +1 , · · · , x n ] = 0 . (13)One must keep in mind that the explicit form of each x k varies depending on the integration step as is displayedin Eq. (12). Thus, one must distinguish, for example, x k ( q , · · · , q j , x j +1 , · · · , x n ) from x k ( q , · · · , q j − , x j , · · · , x n ),where the former is the expression after the integration over x j and the latter is that after the integration over x j − .This notation is also applied to the original integrand function f and the extra factor G i .As an explicit example, we consider the coordinate transformation between the 2-dimensional Cartesian coordinatesand the polar coordinates with the transformation functions r = p x + y and θ = arctan yx (14)and the inverse transformation functions x = r cos θ and y = r sin θ. (15)In an intermediate step, y can be expressed in terms of θ and x as y ( θ, x ) = x tan θ , which must be distinguished from y ( θ, r ) = r sin θ .Since the dependence of a coordinate variable varies according to the integration step, one must take special carein dealing with their partial derivatives. In order to avoid such an ambiguity, we introduce a notation for the partialderivative with subscripts of variables that are held constant. In the above 2-dimensional transformation, the partialderivative of θ with respect to y holding x fixed is denoted by (cid:18) ∂θ∂y (cid:19) x = xx + y , (16)while the partial derivative of θ with respect to y holding r fixed is represented by (cid:18) ∂θ∂y (cid:19) r = 1 p r − y . (17)It is apparent from Eqs. (16) and (17) that (cid:18) ∂θ∂y (cid:19) x = (cid:18) ∂θ∂y (cid:19) r . (18)In a general case, for a variable q j ( q , · · · , q i , x i +1 , · · · , x n ), we denote the partial derivative of q j with respect to x a holding q , · · · , q i , x i +1 , · · · , x n fixed with the subscript ( i ) as (cid:18) ∂q j ∂x a (cid:19) ( i ) = ∂q j ( q , · · · , q i , x i +1 , · · · , x n ) ∂x a , (19)where i + 1 ≤ j ≤ n and i + 1 ≤ a ≤ n . Similarly, for a variable x k ( q , · · · , q i , x i +1 , · · · , x n ), the partial derivative of x k with respect to q b holding q , · · · , q i , x i +1 , · · · , x n fixed is denoted by (cid:18) ∂x k ∂q b (cid:19) ( i ) = ∂x k ( q , · · · , q i , x i +1 , · · · , x n ) ∂q b , (20)where 1 ≤ k ≤ i and 1 ≤ b ≤ i . We denote the partial derivative of x ℓ with respect to x c holding q , · · · , q i , x i +1 , · · · , x n fixed by (cid:18) ∂x ℓ ∂x c (cid:19) ( i ) = ∂x ℓ ( q , · · · , q i , x i +1 , · · · , x n ) ∂x c , (21)where 1 ≤ ℓ ≤ i and i + 1 ≤ c ≤ n . Finally, we express the partial derivative of q ℓ with respect to x c holding x , · · · , x n without subscript: ∂q ℓ ∂x c = ∂q ℓ ( x , · · · , x n ) ∂x c , (22)where 1 ≤ ℓ ≤ n and 1 ≤ c ≤ n .In the coordinate transformation from ( x , · · · , x n ) to ( q , · · · , q n ), we define the function G k , which is relevant fora partial set of integral variables corresponding to a transformation from ( x , · · · , x k ) to ( q , · · · , q k ), as G k = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D e t ∂q ∂x ∂q ∂x · · · ∂q k ∂x ∂q ∂x ∂q ∂x · · · ∂q k ∂x ... ... . . . ... ∂q ∂x k ∂q ∂x k · · · ∂q k ∂x k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (23)for k = 1 through n . Note that G n is the inverse of the Jacobian J n which is defined by J n = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D e t (cid:18) ∂x ∂q (cid:19) ( n ) (cid:18) ∂x ∂q (cid:19) ( n ) · · · (cid:18) ∂x k ∂q (cid:19) ( n ) (cid:18) ∂x ∂q (cid:19) ( n ) (cid:18) ∂x ∂q (cid:19) ( n ) · · · (cid:18) ∂x k ∂q (cid:19) ( n ) ... ... . . . ... (cid:18) ∂x ∂q k (cid:19) ( n ) (cid:18) ∂x ∂q k (cid:19) ( n ) · · · (cid:18) ∂x k ∂q k (cid:19) ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (24) B. 2-dimensional case
In this subsection, we consider a 2-dimensional integral I for the integration of an arbitrary function f ( x , x ): I = Z ∞−∞ dx dx f ( x , x ) . (25)The Cartesian coordinates x and x are transformed into curvilinear coordinates q and q with the transformationrelations q = q ( x , x ) , q = q ( x , x ) , (26)which are assumed to be invertible and non-singular. Thus, x and x can be expressed in terms of q and q : x = x ( q , q ) , x = x ( q , q ) . (27)We multiply the unities i = Z dq i δ [ q i − q i ( x , x )] = 1 (28)to I for i = 1, 2, sequentially. First, we multiply in Eq. (28) to I . Then, I can be expressed as I = Z dq Z ∞−∞ dx dx δ [ q − q ( x , x )] f ( x , x ) . (29)The integration over x can be carried out by making use of the delta function for q as Z ∞−∞ dx δ [ q − q ( x , x )] = 1 (cid:12)(cid:12)(cid:12)(cid:12) ∂q ∂x (cid:12)(cid:12)(cid:12)(cid:12) x = x ( q ,x ) = 1 G ( q , x ) , (30)which leads to I = Z dq Z ∞−∞ dx G ( q , x ) f ( q , x ) . (31)After the integration over x , every x on the right-hand side of Eq. (30) and δ [ q − q ( x , x )] f ( x , x ) in Eq. (28) mustbe replaced with x ( q , x ) satisfying the condition that the argument of the delta function vanishes, q − q ( x , x ) = 0.Then the delta function for q and f ( x , x ) can be expressed as δ [ q − q ( q , x )] and f ( q , x ), respectively.After multiplying in Eq. (28) to I in Eq. (31), the integration over x can be performed by making use of theremaining delta function for q as Z ∞−∞ dx δ [ q − q ( q , x )] = 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ∂q ∂x (cid:19) (1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = x ( q ,q ) = G ( q , q ) G ( q , q ) . (32)After the integration over x , every x on the right-hand sides of Eqs. (30) and (32) and in f ( q , x ) is replaced with x ( q , q ), which can be obtained from the condition that the argument of the delta function vanishes, q − q ( q , x ) =0. Then, we can express x in terms of q and q by replacing x with x ( q , q ) in x ( q , x ). We have obtained thelast equality of Eq. (32) by making use of the chain rule for the partial derivatives. A rigorous proof of this formulais given in Eq. (A5) of Appendix A.After both x and x are integrated out, the integral I reduces into I = Z dq dq G × G G f [ x ( q , q ) , x ( q , q )] = Z dq dq J f [ x ( q , q ) , x ( q , q )] , (33)where the Jacobian for the change of variables is identified as J = J = 1 / G . This completes the proof of theJacobian for a 2-dimensional coordinate transformation or change of variables. C. 3-dimensional case
In this subsection, we extend the results in the previous subsection to the 3-dimensional case. This is a special caseof the n -dimensional coordinate transformation or change of variables, which we will prove in the next subsection.However, it is worthwhile to prove the 3-dimensional case in detail for a pedagogical purpose.We consider a 3-dimensional integral I for an arbitrary function f ( x , x , x ): I = Z ∞−∞ dx dx dx f ( x , x , x ) . (34)The Cartesian coordinates x i for i = 1 through 3 are transformed into the curvilinear coordinates q i ’s as q = q ( x , x , x ) , q = q ( x , x , x ) , q = q ( x , x , x ) . (35)The inverse transformation can be expressed as x = x ( q , q , q ) , x = x ( q , q , q ) , x = x ( q , q , q ) . (36)We carry out the change of variables by multiplying the unites i = Z dq i δ [ q i − q i ( x , x , x )] = 1 (37)for i = 1 through 3 to I , sequentially. First, after multiplying in Eq. (37) to I , we find that I can be expressedas I = Z dq Z ∞−∞ dx dx dx δ [ q − q ( x , x , x )] f ( x , x , x ) . (38)Similarly to the 2-dimensional case, we perform the integration over x by making use of the Dirac delta functionfor q as Z ∞−∞ dx δ [ q − q ( x , x , x )] = 1 (cid:12)(cid:12)(cid:12)(cid:12) ∂q ∂x (cid:12)(cid:12)(cid:12)(cid:12) x = x ( q ,x ,x ) = 1 G ( q , x , x ) , (39)which leads to I = Z dq Z ∞−∞ dx dx G ( q , x , x ) f ( q , x , x ) . (40)After the integration over x , every x in the integrand and remaining delta functions is replaced with x ( q , x , x )satisfying the condition that the argument of the Dirac delta function vanishes, q − q ( x , x , x ) = 0. Then the deltafunction for q in Eq. (37) can be expressed as δ [ q − q ( q , x , x )].After multiplying in Eq. (37) to I in Eq. (40), the integration over x can be carried out by making use of theDirac delta function for q as Z ∞−∞ dx δ [ q − q ( q , x , x )] = 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ∂q ∂x (cid:19) (1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = x ( q ,q ,x ) = G ( q , q , x ) G ( q , q , x ) , (41)where the last equality comes from Eq. (B5). Then, Eq. (41) is expressed as I = Z dq dq Z ∞−∞ dx G ( q , q , x ) × G ( q , q , x ) G ( q , q , x ) f ( q , q , x ) . (42) x ( q , q , x ) is determined from the condition that the argument of the Dirac delta function vanishes, q − q ( q , x , x ) = 0. Substituting x ( q , q , x ) into x ( q , x , x ), we obtain x = x ( q , q , x ) and the argumentof the Dirac delta function for q in Eq.(37) is expressed as δ [ q − q ( q , q , x )].Finally, after multiplying in Eq. (37) to I in Eq. (42), we integrate over x by taking into account the deltafunction for q as Z ∞−∞ dx δ [ q − q ( q , q , x )] = 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ∂q ∂x (cid:19) (2) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = x ( q ,q ,q ) = G ( q , q , q ) G ( q , q , q ) , (43)where the last equality comes from Eq. (B11). After the integration over x , every x in the integrand and theright-hand sides of Eqs. (39), (41) and (43) is replaced with x ( q , q , q ) which is determined from the condition thatthe argument of the delta function vanishes, q − q ( q , q , x ) = 0. Substituting x ( q , q , q ) into x ( q , q , x ) and x ( q , q , x ), we can obtain the expressions for x = x ( q , q , q ) and x = x ( q , q , q ), respectively. Then, we canexpress the integrand f ( x , x , x ) in terms of q , q and q and the integral I is expressed as I = Z dq dq dq G × G G × G G f [ x ( q , q , q ) , x ( q , q , q ) , x ( q , q , q )]= Z dq dq dq J f [ x ( q , q , q ) , x ( q , q , q ) , x ( q , q , q )] , (44)where the Jacobian for the change of variables is identified as J = J = 1 / G . This completes the proof of theJacobian for a 3-dimensional coordinate transformation or change of variables. D. n -dimensional case In this subsection, we consider the n -dimensional integral I n for a function f ( x , · · · , x n ) defined in Eq. (5) bymultiplying the unities i in Eq. (6) to I n in Eq. (5) sequentially. Then, we integrate I n , which is multiplied by theunity, over x i for i = 1 through n successively by making use of the Dirac delta function Z ∞−∞ dx i δ [ q i − q i ( x , · · · , x n )] . (45)After the integration of all x i variables, the corresponding Jacobian formula is obtained by employing mathematicalinduction.First, the integration over x can be carried out from Eq. (7) and the result for the integration over x can easilybe generalized from the 2-dimensional version in Eq. (30) as Z ∞−∞ dx δ [ q − q ( x , · · · , x n )] = 1 (cid:12)(cid:12)(cid:12)(cid:12) ∂q ∂x (cid:12)(cid:12)(cid:12)(cid:12) x = x ( q ,x , ··· ,x n ) = 1 G ( q , x , · · · , x n ) , (46)which leads to I n = Z dq Z ∞−∞ dx · · · dx n G ( q , x , · · · , x n ) f ( q , x , · · · , x n ) . (47)After the x integration, every x in the integrand of Eq. (7) and G in Eq. (46) is replaced with x = x ( q , x , · · · , x n ) . (48)The constraint equation coming from the convolution with the Dirac delta function in Eq. (46) is q − q [ x ( q , x , · · · , x n ) , x , · · · , x n ] = 0 . (49)Then, we carry out the integration over x i for i = 1 through n − i to I n in Eq. (47) sequentially.We assume that, after the integration over x i for i = 1 through n −
1, the result of the integration of Dirac deltafunctions is Z ∞−∞ dx · · · dx i i Y j =1 δ [ q j − q j ( x , · · · , x n )] = 1 G × G G × · · · × G i − G i = 1 G i ( q , · · · , q i , x i +1 , · · · , x n ) , (50)which leads to I n = Z dq · · · dq i Z ∞−∞ dx i +1 · · · dx n G i ( q , · · · , q i , x i +1 , · · · , x n ) f ( q , · · · , q i , x i +1 , · · · , x n ) . (51)For any j ≤ i every x j in the integrand of Eq. (5) and G j ’s in Eq. (50) is replaced with x j = x j ( q , · · · , q i , x i +1 , · · · , x n ) (52)after the integrations over x through x i . There are i constraint equations coming from the convolution with Diracdelta functions in Eq. (50): q j − q j [ x ( q , · · · , q i , x i +1 , · · · , x n ) , · · · , x i ( q , · · · , q i , x i +1 , · · · , x n ) , x i +1 , · · · , x n ] = 0 , (53)where j runs from 1 through i .After multiplying i +1 to I n in Eq. (51), we integrate out one more Cartesian coordinate x i +1 to find that Z ∞−∞ dx i +1 δ [ q i +1 − q i +1 ( q , · · · , q i , x i +1 , · · · , x n )] x j = x j ( q , ··· ,q i ,x i +1 , ··· ,x n ) = 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ∂q i +1 ∂x i +1 (cid:19) ( i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x k = x k ( q , ··· ,q i +1 ,x i +2 , ··· ,x n ) = G i ( q , · · · , q i +1 , x i +2 , · · · , x n ) G i +1 ( q , · · · , q i +1 , x i +2 , · · · , x n ) , (54)where 1 ≤ j ≤ i and 1 ≤ k ≤ i + 1. The proof of the last equality can be found in Eq. (C8) in Appendix C. Incombination with Eqs. (50) and (54), we find that Z ∞−∞ dx · · · dx i dx i +1 i +1 Y j =1 δ [ q j − q j ( x , · · · , x n )] = 1 G × G G × · · · × G i − G i × G i G i +1 = 1 G i +1 ( q , · · · , q i +1 , x i +2 , · · · , x n ) . (55)For any j ≤ i + 1 every x j in the integrand of Eq. (7) and G j ’s in Eq. (55) is replaced with x j = x j ( q , · · · , q i +1 , x i +2 , · · · , x n ) (56)after the integrations over x through x i +1 . There are i + 1 constraint equations coming from the convolution withDirac delta functions in Eq. (55): q j − q j [ x ( q , · · · , q i +1 , x i +2 , · · · , x n ) , · · · , x i ( q , · · · , q i +1 , x i +2 , · · · , x n ) , x i +1 , · · · , x n ] = 0 , (57)where j runs from 1 through i + 1. According to mathematical induction, this proves that the assumption in Eq. (50)with the constraints (52) and (53) is true for all i = 1 through n .Finally, the integral I n can be expressed in terms of q , · · · , q n as I n = Z d n q G n f ( x , · · · , x n ) | x k = x k ( q , ··· ,q n ) = Z d n q J n f ( x , · · · , x n ) | x k = x k ( q , ··· ,q n ) , (58)where 1 ≤ k ≤ n . This completes the proof of the Jacobian J = J n = 1 / G n for an n -dimensional coordinatetransformation or change of variables.0 III. APPLICATION
Since the proof of the Jacobian formula in the previous section is rather abstract, readers who are not familiarwith the notation might be confused. In this section, we present a few explicit examples of deriving the Jacobianwithout resorting to the general formula derived in the previous section. We expect that the explicit examples willhelp readers to understand the method presented in the previous section more intuitively and to apply it to a specificchange of variables.
A. Spherical coordinates
In this subsection, we consider the change of variables from the 3-dimensional Cartesian coordinates ( x, y, z ) to thespherical coordinates ( r, θ, φ ). The Cartesian coordinates can be expressed in terms of the radius r , the polar angle θ and the azimuthal angle φ as x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ. (59)We use cos θ instead of θ as the integration variable and reorganize the order of multiple integrations in order tosimplify the computation. That is, ( x , x , x ) in section II C corresponds to ( z, y, x ) while ( q , q , q ) correspondsto (cos θ, φ, r ), respectively. However, it turns out that the integral is invariant under this reordering. We also notethat one can use, for instance, sin θ instead of cos θ as an integration variable and it will not alter the result of theintegration. Then the inverse transformation of Eq. (59) is expressed ascos θ = z p x + y + z , (60a) φ = δ (cid:16) φ − arctan yx (cid:17) Θ( x ) + δ (cid:16) φ − arctan yx − π (cid:17) Θ( − x ) , (60b) r = p x + y + z , (60c)where the Heaviside step function Θ( x ) is defined byΘ( x ) = , for x > , , for x = 0 , , for x < . (61)Conventionally, the arctangent function is defined in the region [ − π , π ]. The period of the tangent function is π ,while the azimuthal angle ranges from 0 to 2 π . Thus the angle φ for x > yx ∈ [ − π , π ] while thatfor x < π + arctan yx ∈ [ π , π ] in order to make the transformation function invertible in the entirerange: φ = arctan yx ∈ ( − π , π ) , for x > ,π + arctan yx ∈ ( π , π ) , for x < , π , for x = 0 and y > , − π , for x = 0 and y < , , for x = y = 0 . (62)We consider a 3-dimensional integral J with an arbitrary integrand f ( x, y, z ) J = Z ∞−∞ dx dy dz f ( x, y, z ) . (63)We multiply the unities = Z − d cos θ δ cos θ − z p x + y + z ! = 1 , (64a) = Z π/ − π/ dφ h δ (cid:16) φ − arctan yx (cid:17) Θ( x ) + δ (cid:16) φ − arctan yx − π (cid:17) Θ( − x ) i = 1 , (64b) = Z ∞ dr δ (cid:16) r − p x + y + z (cid:17) = 1 (64c)1to J without changing the value of the integral sequentially.First, we integrate out the x = z coordinate by multiplying in Eq. (64a) to J in Eq. (63). The integrationover z can be performed as Z ∞−∞ dz δ cos θ − z q x + y + z ! = p x + y sin θ . (65)After the integration over z , every z in the integrand of J is replaced with the expression in terms of cos θ , y and x : z = p x + y / tan θ , where we omit the simple conversion between trigonometric functions here and after. Theintegrand of the remaining double integral over x and y is a function of θ , x and y : J = Z − d cos θ Z ∞−∞ dx dy f ( x, y, p x + y / tan θ ) p x + y sin θ . (66)This can also be obtained by substituting cos θ , y and x into Eq. (39).Next, we integrate out the x = y coordinate by multiplying in Eq. (64b) into J in Eq. (66). The integrationover y can be performed as Z ∞−∞ dy h δ (cid:16) φ − arctan yx (cid:17) Θ( x ) + δ (cid:16) φ − arctan yx − π (cid:17) Θ( − x ) i = 1 | ddy arctan yx | [Θ( x ) + Θ( − x )] (cid:12)(cid:12)(cid:12) y = x tan φ = | x | cos φ , (67)where we have used the identity Θ( x ) + Θ( − x ) = 1 . (68)Equation (67) can also be obtained by substituting cos θ , φ and x into Eq. (41). After the integration over y , every y in the integrand of J is replaced with the expression in terms of cos θ , φ and x . The integrand of the remainingintegral over x is a function of cos θ , φ and x : J = Z − d cos θ Z π dφ Z ∞−∞ dx f ( x, y, p x + y / tan θ ) p x + y sin θ | x | cos φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y = x tan φ = Z − d cos θ Z π dφ Z ∞−∞ dx f [ x, x tan φ, | x | / ( | cos φ | tan θ )] x sin θ | cos φ | , (69)where y = x tan φ . After the integrations over both z and y , y = x tan φ and z = x/ (cos φ tan θ ).Finally, after multiplying in Eq. (64c) into J in Eq. (69), the integration over x = x can be performed as Z ∞−∞ dx δ ( r − p x + y + z ) (cid:12)(cid:12)(cid:12) y = x tan φ,z = x cos φ tan θ = Z ∞−∞ dx δ (cid:20) r − (cid:12)(cid:12)(cid:12)(cid:12) x cos φ sin θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) = | cos φ | sin θ. (70)This can also be obtained by substituting cos θ , φ and r into Eq. (43). Here, the radius r is non-negative because theDirac delta function requires that r = p x + y + z and x + y + z is non-negative. After the integration over allof the Cartesian coordinates, x , y and z are expressed as in Eq. (59).Substituting Eq. (59) into (65), (67), (70), and f ( x, y, z ), we find that J can be expressed in terms of the sphericalcoordinates as J = Z ∞ dr Z − d cos θ Z π dφ r sin θ sin θ | r sin θ cos φ | cos φ | cos φ | sin θ f ( r sin θ cos φ, r sin θ sin φ, r cos θ )= Z ∞ dr Z − d cos θ Z π dφ r f ( r sin θ cos φ, r sin θ sin φ, r cos θ )= Z ∞ dr Z π dθ Z π dφ r sin θ f ( r sin θ cos φ, r sin θ sin φ, r cos θ ) . (71)The overall factor r sin θ is identified to be the Jacobian J = r sin θ. (72)This exactly reproduces the result which can be obtained by applying the formula (44) [1].2 B. n -dimensional polar coordinates In this subsection, we extend the 3-dimensional case to the n -dimensional coordinate transformation from the n -dimensional Cartesian coordinates ( x , · · · , x n ) to the n -dimensional polar coordinates ( r, θ , θ , · · · , θ n − , φ ). Here, r is the radius and there are n − θ i ’s and a single azimuthal angle φ . The corresponding transformationfunctions are expressed as x = r sin θ sin θ · · · sin θ n − sin θ n − cos φ,x = r sin θ sin θ · · · sin θ n − sin θ n − sin φ,x = r sin θ sin θ · · · sin θ n − cos θ n − , ... x n − = r sin θ cos θ ,x n = r cos θ . (73)We reorganize the order of integrations of the Cartesian coordinates as ( x n , x n − , · · · , x ) for simplicity and the cor-responding curvilinear coordinates are reorganized as (cos θ , cos θ , · · · , cos θ n − , φ, r ). We consider an n -dimensionalintegral J n with an arbitrary integrand f ( x , x , · · · , x n ): J n = Z ∞−∞ d n x f ( x , x , · · · , x n ) , (74)where d n x = dx dx · · · dx n . We multiply the unities i = Z − d cos θ i δ cos θ i − x n − i +1 q x + · · · + x n − i +1 = 1 , i = 1 , · · · , n − , (75a) n − = Z π − π dφ (cid:20) δ (cid:18) φ − arctan x x (cid:19) Θ( x ) + δ (cid:18) φ − arctan x x − π (cid:19) Θ( − x ) (cid:21) = 1 , (75b) n = Z ∞ dr δ (cid:18) r − q x + · · · + x n (cid:19) = 1 (75c)for i = 1 through n to J n , sequentially, keeping the integral invariant. Θ( x ) in Eq. (75b) is the Heaviside step functiondefined in Eq. (60b). There are numerous ways to perform the multiple integrations over the n Cartesian coordinates.Our strategy to integrate out x i ’s is as follows: According to the integrand of the right-hand side of Eq. (75a), theintegration over x n − i +1 in the integral J n provides the constraint to the polar angle θ i . Thus we choose to integrateover from x n to x to express them in terms of the polar angles from θ through θ n − . Then we integrate out x to express x n through x in terms of the n − φ by making use of Eq. (75b).As the last step, we integrate out x to determine all of the Cartesian coordinates in terms of the spherical polarcoordinates by making use of (75c).First, after multiplying i in Eq. (75a) into J n , the integration over x n − i +1 for i = 1 through n − Z ∞−∞ dx n − i +1 δ cos θ i − x n − i +1 q x + · · · + x n − i +1 = q x + · · · + x n − i sin θ i , (76)where one can obtain the same results from Eqs. (46) and (54) taking care of the order of integration. After theintegration over x n − i +1 , we make the replacement x n − i +1 = q x + · · · + x n − i tan θ i . (77)Then, J n is expressed as J n = Z − d cos θ · · · Z − d cos θ n − Z ∞−∞ dx dx f ( x , x , cos θ , · · · , cos θ n − ) n − Y i =1 q x + · · · + x n − i sin θ i , (78)3where every x i in the last factor for i = 3 through n is replaced by that in Eq. (77).After multiplying n − in Eq. (75b) into J n in Eq. (78), the integration over x can be carried out in a similarmanner as is done in Eq. (67). The result is Z ∞−∞ dx (cid:20) δ (cid:18) φ − arctan x x (cid:19) Θ( x ) + δ (cid:18) φ − arctan x x − π (cid:19) Θ( − x ) (cid:21) = | x | cos φ , (79)where x = x tan φ after the integration. This can also be obtained from Eq. (54) while keeping the results inEqs. (76) and (77). Then, x n − i +1 for i = 1 through n − x n − i +1 = | x || cos φ | sin θ n − · · · sin θ i +1 tan θ i . (80)Then, we find that J n = Z − d cos θ · · · Z − d cos θ n − Z π dφ Z ∞−∞ dx f ( x , φ, cos θ , · · · , cos θ n − ) | x | cos φ n − Y i =1 q x + · · · + x n − i sin θ i , (81)where every x i in the last factor for i = 2 through n is replaced by that in Eq. (80).Finally, after multiplying n in Eq. (75c) to J n in Eq. (81), the integration over x can be performed like Eq. (70)and we obtain Z ∞−∞ dx δ (cid:18) r − q x + · · · + x n (cid:19) = Z ∞−∞ dx δ (cid:20) r − (cid:12)(cid:12)(cid:12)(cid:12) x cos φ sin θ n − · · · sin θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) = | cos φ | sin θ n − · · · sin θ , (82)where we have omitted the replacements of x = x tan φ and x through x n that can be obtained from Eq. (80) onthe left-hand side. This results can also be obtained from Eq. (54) while keeping the results in Eqs. (76), (77), (79)and (80). After integrating out all of the Cartesian coordinates, we reproduce the expression for every x i that is givenin Eq. (73).Combining all of the results listed above, we find that the n -dimensional coordinate transformation or change ofvariables is carried out as J n = Z ∞ dr Z − d cos θ · · · Z − d cos θ n − Z π dφ × r sin θ sin θ · · · r sin θ · · · sin θ n − sin θ n − r sin θ · · · sin θ n − | cos φ | cos φ | cos φ | sin θ · · · sin θ n − × f ( r sin θ · · · sin θ n − cos φ, r sin θ · · · sin θ n − sin φ, r sin θ · · · sin θ n − cos θ n − , · · · , r sin θ cos θ , r cos θ )= Z ∞ dr Z π dθ · · · Z π dθ n − Z π dφ r n − sin n − θ · · · sin θ n − sin θ n − × f ( r sin θ · · · sin θ n − cos φ, r sin θ · · · sin θ n − sin φ, r sin θ · · · sin θ n − cos θ n − , · · · , r sin θ cos θ , r cos θ ) . (83)The extra factor r n − sin n − θ · · · sin θ n − sin θ n − in front of the original integrand f is identified with the Jacobian J = r n − sin n − θ · · · sin θ n − sin θ n − . (84)This exactly reproduces the result in Refs. [5, 6], which can be obtained by applying the general formula (58). IV. CONCLUSIONS
We have derived the general formula for the Jacobian of the transformation from the n -dimensional Cartesiancoordinates to arbitrary curvilinear coordinates by making use of Dirac delta functions, whose arguments correspondto the transformation functions between the two coordinate systems. The multiplication of the trivial identities (6)to the original integral enables us to integrate out the original integration variables corresponding to the Cartesiancoordinates systematically. By making use of the chain rule for the partial derivatives, we can carry out the integrationover the Cartesian coordinates successively and end up with the integral expressed in terms of the curvilinear coordi-nates. Then, the Jacobian can be read off by comparing the integrands of the resultant integral with the original one.4It turns out that the formula derived in this paper exactly reproduces the Jacobian for the coordinate transformationor change of variables.We have presented a few examples, where we have integrated out the Cartesian coordinates by making use of Diracdelta functions explicitly without resorting to the general formula for the Jacobian derived in this paper. We findthat the formulas obtained in these explicit examples are exactly the same as those in the general formula (58). Sincethe derivation of the Jacobian in the general case that makes use of the chain rule of the partial derivatives is ratherabstract, we expect that these examples will give insights on understanding the derivation concretely.To our best knowledge, this derivation of the Jacobian factor by making use of Dirac delta functions for thecoordinate transformation or change of variables from the n -dimensional Cartesian coordinates to the curvilinearcoordinates is new. Although there are several ways to derive the Jacobian available in textbooks [1–3], our derivationcould be pedagogically useful in upper-level mathematics or physics courses in practice using Dirac delta functionssuccessively. Compared with the methods popular in the textbook level, our method is more intuitive because wehave employed only the explicit calculation of elementary single-dimensional integrals without relying on abstractgeometrical interpretations or more abstract Green’s theorem with which undergraduate physics-major students arenot usually familiar. Furthermore, a detailed derivation of the chain rule for the partial derivatives, which is employedto prove the Jacobian formula, should be a nice working example with which one can understand a rigorous usage ofthe partial derivatives with multi-dimensional variables without ambiguity. ACKNOWLEDGMENTS
As members of the Korea Pragmatist Organization for Physics Education (
KPOP E ), the authors thank the remain-ing members of KPOP E for useful discussions. This work is supported in part by the National Research Foundationof Korea (NRF) under the BK21 FOUR program at Korea University, Initiative for science frontiers on upcom-ing challenges, and by grants funded by the Korea government (MSIT), Grants No. NRF-2017R1E1A1A01074699(J.L.) and No. NRF-2020R1A2C3009918 (J.E. and D.K.). The work of C.Y. is supported by Basic Science Re-search Program through the National Research Foundation (NRF) of Korea funded by the Ministry of Education(2020R1I1A1A01073770). Appendix A: 2-dimensional case
In this section, we consider a 2-dimensional coordinate transformation from the Cartesian coordinates ( x , x ) tothe curvilinear coordinates ( q , q ). The total differentials of q and x can be expressed as dq = ∂q ∂x dx + ∂q ∂x dx , (A1a) dx = (cid:18) ∂x ∂q (cid:19) (1) dq + (cid:18) ∂x ∂x (cid:19) (1) dx , (A1b)where q = q ( x , x ) in Eq. (A1a) and x = x ( q , x ) in Eq. (A1b), respectively. Note that the definition of thepartial derivative with a subscript are given in section II A: ( ∂x /∂q ) (1) , ( ∂x /∂x ) (1) , and ( ∂q /∂x ) (2) are definedin Eq. (20), (21) and (22), respectively.Replacing dx in Eq. (A1a) with the right-hand side of Eq. (A1b), we obtain dq = ∂q ∂x (cid:18) ∂x ∂q (cid:19) (1) dq + " ∂q ∂x (cid:18) ∂x ∂x (cid:19) (1) + ∂q ∂x dx . (A2)Comparing the coefficients of the differentials on both sides, we find that ∂q ∂x (cid:18) ∂x ∂q (cid:19) (1) = 1 , (A3a) ∂q ∂x (cid:18) ∂x ∂x (cid:19) (1) + ∂q ∂x = 0 . (A3b)We note that Eq. (A3a) is trivial. By making use of Eq. (A3b), we find that the factor in the denominator of Eq. (32)5( ∂q /∂x ) (1) can be expressed as (cid:18) ∂q ∂x (cid:19) (1) = ∂q ∂x (cid:18) ∂x ∂x (cid:19) (1) + ∂q ∂x = D e t ∂q ∂x ∂q ∂x ∂q ∂x ∂q ∂x D e t (cid:18) ∂q ∂x (cid:19) , (A4)which leads to (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ∂q ∂x (cid:19) (1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = G G . (A5) Appendix B: 3-dimensional case
In this section, we consider a 3-dimensional coordinate transformation from the Cartesian coordinates ( x , x , x )to the curvilinear coordinates ( q , q , q ). First, we consider the total differentials of q and x which can be expressedas dq = ∂q ∂x dx + ∂q ∂x dx + ∂q ∂x dx , (B1a) dx = (cid:18) ∂x ∂q (cid:19) (1) dq + (cid:18) ∂x ∂x (cid:19) (1) dx + (cid:18) ∂x ∂x (cid:19) (1) dx , (B1b)where q = q ( x , x , x ) in Eq. (B1a) and x = x ( q , x , x ) in Eq. (B1b), respectively. Replacing dx in Eq. (B1a)with the right-hand side of Eq. (B1b), we obtain dq = ∂q ∂x (cid:18) ∂x ∂q (cid:19) (1) dq + " ∂q ∂x (cid:18) ∂x ∂x (cid:19) (1) + ∂q ∂x dx + " ∂q ∂x (cid:18) ∂x ∂x (cid:19) (1) + ∂q ∂x dx . (B2)Comparing the coefficients of the differentials on both sides, we find that ∂q ∂x (cid:18) ∂x ∂q (cid:19) (1) = 1 , (B3a) ∂q ∂x (cid:18) ∂x ∂x (cid:19) (1) + ∂q ∂x = 0 , (B3b) ∂q ∂x (cid:18) ∂x ∂x (cid:19) (1) + ∂q ∂x = 0 , (B3c)where Eq. (B3a) is trivial.By making use of Eq. (B3b), we find that the factor in the denominator of Eq. (41) ( ∂q /∂x ) (1) can be expressedas (cid:18) ∂q ∂x (cid:19) (1) = ∂q ∂x (cid:18) ∂x ∂x (cid:19) (1) + ∂q ∂x = D e t ∂q ∂x ∂q ∂x ∂q ∂x ∂q ∂x D e t (cid:18) ∂q ∂x (cid:19) , (B4)which leads to (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ∂q ∂x (cid:19) (1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = G G . (B5)6In order to prove Eq. (B11), we take into account the total differentials of q , x , q and x . The total derivativescan be expressed as dq = ∂q ∂x dx + ∂q ∂x dx + ∂q ∂x dx , (B6a) dq = ∂q ∂x dx + ∂q ∂x dx + ∂q ∂x dx , (B6b) dx = (cid:18) ∂x ∂q (cid:19) (2) dq + (cid:18) ∂x ∂q (cid:19) (2) dq + (cid:18) ∂x ∂x (cid:19) (2) dx , (B6c) dx = (cid:18) ∂x ∂q (cid:19) (2) dq + (cid:18) ∂x ∂q (cid:19) (2) dq + (cid:18) ∂x ∂x (cid:19) (2) dx , (B6d)where q = q ( x , x , x ) in Eq. (B6a), q = q ( x , x , x ) in Eq. (B6b), x = x ( q , q , x ) in Eq. (B6c), and x = x ( q , q , x ) in Eq. (B6d), respectively. Substituting Eqs. (B6c) and (B6d) into Eqs. (B6a) and (B6b), weobtain dq = " ∂q ∂x (cid:18) ∂x ∂q (cid:19) (2) + ∂q ∂x (cid:18) ∂x ∂q (cid:19) (2) dq + " ∂q ∂x (cid:18) ∂x ∂q (cid:19) (2) + ∂q ∂x (cid:18) ∂x ∂q (cid:19) (2) dq + " ∂q ∂x (cid:18) ∂x ∂x (cid:19) (2) + ∂q ∂x (cid:18) ∂x ∂x (cid:19) (2) + ∂q ∂x dx ,dq = " ∂q ∂x (cid:18) ∂x ∂q (cid:19) (2) + ∂q ∂x (cid:18) ∂x ∂q (cid:19) (2) dq + " ∂q ∂x (cid:18) ∂x ∂q (cid:19) (2) + ∂q ∂x (cid:18) ∂x ∂q (cid:19) (2) dq + " ∂q ∂x (cid:18) ∂x ∂x (cid:19) (2) + ∂q ∂x (cid:18) ∂x ∂x (cid:19) (2) + ∂q ∂x dx . (B7)Comparing both sides of Eq. (B7), we find two relevant non-trivial equations: ∂q ∂x (cid:18) ∂x ∂x (cid:19) (2) + ∂q ∂x (cid:18) ∂x ∂x (cid:19) (2) = − ∂q ∂x ,∂q ∂x (cid:18) ∂x ∂x (cid:19) (2) + ∂q ∂x (cid:18) ∂x ∂x (cid:19) (2) = − ∂q ∂x . (B8)By making use of Cramer’s rule, we find that (cid:18) ∂x ∂x (cid:19) (2) = D e t ∂q ∂x ∂q ∂x ∂q ∂x ∂q ∂x D e t ∂q ∂x ∂q ∂x ∂q ∂x ∂q ∂x , (cid:18) ∂x ∂x (cid:19) (2) = − D e t ∂q ∂x ∂q ∂x ∂q ∂x ∂q ∂x D e t ∂q ∂x ∂q ∂x ∂q ∂x ∂q ∂x . (B9)Then, by making use of Eq. (B9), we find that the factor ( ∂q /∂x ) (2) in the denominator of Eq. (43)7can be expressed as (cid:18) ∂q ∂x (cid:19) (2) = ∂q ∂x (cid:18) ∂x ∂x (cid:19) (2) + ∂q ∂x (cid:18) ∂x ∂x (cid:19) (2) + ∂q ∂x = D e t ∂q ∂x ∂q ∂x ∂q ∂x ∂q ∂x ∂q ∂x ∂q ∂x ∂q ∂x ∂q ∂x ∂q ∂x D e t ∂q ∂x ∂q ∂x ∂q ∂x ∂q ∂x , (B10)which leads to (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ∂q ∂x (cid:19) (2) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = G G . (B11) Appendix C: n -dimensional case The total differentials of q , · · · , q i , x , · · · , x i can be expressed as dq = n X a =1 ∂q ∂x a dx a , ... dq i = n X a =1 ∂q i ∂x a dx a ,dx = i X a =1 "(cid:18) ∂x ∂q a (cid:19) ( i ) dq a + n X a = i +1 "(cid:18) ∂x ∂x a (cid:19) ( i ) dx a , ... dx i = i X a =1 "(cid:18) ∂x i ∂q a (cid:19) ( i ) dq a + n X a = i +1 "(cid:18) ∂x i ∂x a (cid:19) ( i ) dx a . (C1)The partial derivative of q i +1 with respect to x i +1 holding q , · · · , q i , x i +2 , · · · , x n fixed is (cid:18) ∂q i +1 ∂x i +1 (cid:19) ( i ) = i X a =1 " ∂q i +1 ∂x a (cid:18) ∂x a ∂ i +1 (cid:19) ( i ) + ∂q i +1 ∂x i +1 . (C2)Substituting dx , · · · , dx i into dq j , we obtain dq j = i X a =1 ∂q j ∂x a " i X b =1 (cid:18) ∂x a ∂q b (cid:19) ( i ) dq b + n X b = i +1 (cid:18) ∂x a ∂x b (cid:19) ( i ) dx b + n X a = i +1 ∂q j ∂x a dx a , (C3)where 1 ≤ j ≤ i . Because the coefficient of dx i +1 in (C3) should be 0, we obtain the following equation: ∂q ∂x · · · ∂q ∂x i ... . . . ... ∂q i ∂x · · · ∂q i ∂x i (cid:18) ∂x ∂x i +1 (cid:19) ( i ) ... (cid:18) ∂x i ∂x i +1 (cid:19) ( i ) = − ∂q ∂x i +1 ... ∂q i ∂x i +1 . (C4)By making use of Cramer’s rule, we find that (cid:18) ∂x j ∂x i +1 (cid:19) ( i ) = − D e t [( J − i × i ] ) ( j ) ( ∂ q i )] D e t [ J − i × i ] ] , (C5)8where J − i × i ] = ∂q ∂x · · · ∂q ∂x i ... . . . ... ∂q i ∂x · · · ∂q i ∂x i ,∂ q i = ∂q ∂x i +1 ... ∂q i ∂x i +1 , (C6)and ( J − i × i ] ) ( j ) ( ∂ q i ) is identical to J − i × i ] except that the j th column is replaced with ∂ q i . Substituting (C5) into (C2),we obtain (cid:18) ∂q i +1 ∂x i +1 (cid:19) ( i ) = ∂q i +1 ∂x i +1 − i X j =1 D e t [( J − i × i ] ) ( j ) ( ∂ q i )] D e t [( J − i × i ] )] ∂q i +1 ∂x j = 1 D e t [ J − i × i ] ] i +1 X j =1 ( − i +1 − j M i +1 j ( J − i +1) × ( i +1)] )= D e t [ J − i +1) × ( i +1)] ] D e t [ J − i × i ] ] . (C7)Here, the ij minor M ij ( A ) of an n × n square matrix A is the determinant of a matrix whose i th row and j th columnare removed from A . Hence, Eq. (C7) leads to (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ∂q i +1 ∂x i +1 (cid:19) ( i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = G i +1 G i . (C8) [1] G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists , 7th edition, Elsevier (2012), pp. 227–229.[2] J. Stewart,
Calculus , 8th edition, Cenage Learning (2015), pp. 1052–1059.[3] T. M. Apostol,
Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equationsand Probability , 2nd edition, J. Wiley, New York (1969), pp. 401–409.[4] J.-H. Ee, C. Yu and J. Lee, “Proof of Cramer’s rule with Dirac delta function,” Eur. J. Phys. , 065002 (2020) .[5] L. E. Blumenson, “A Derivation of n-Dimensional Spherical Coordinates,” The American Mathematical Monthly , 63(1960).[6] J.-H. Ee, D.- W. Jung, U-R. Kim and J. Lee, “Combinatorics in tensor-integral reduction,” Eur. J. Phys.38