Derivation of Jacobian Formula with Dirac Delta Function

We demonstrate how to make the coordinate transformation or change of variables from Cartesian coordinates to curvilinear coordinates by making use of a convolution of a function with Dirac delta functions whose arguments are determined by the transformation functions between the two coordinate systems. By integrating out an original coordinate with a Dirac delta function, we replace the original coordinate with a new coordinate in a systematic way. A recursive use of Dirac delta functions allows the coordinate transformation successively. After replacing every original coordinate into a new curvilinear coordinate, we find that the resultant Jacobian of the corresponding coordinate transformation is automatically obtained in a completely algebraic way. In order to provide insights on this method, we present a few examples of evaluating the Jacobian explicitly without resort to the known general formula.


I. INTRODUCTION
A coordinate transformation or change of variables from a coordinate system to another in multi-dimensional integrals has widely been applied to a variety of fields in mathematics and physics. This transformation always involves a factor called the Jacobian, which is the determinant of the Jacobian matrix. The matrix elements of the Jacobian 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 1 , x 2 , · · · , x n to q 1 , q 2 , · · · , q n is expressed in terms of the Jacobian J: dx 1 dx 2 · · · dx n f (x 1 , · · · , x n ) = dq 1 dq 2 · · · dq n JF (q 1 , · · · , q n ), (1) where the integrand f (x 1 , · · · , x n ) is a function of the independent variables x 1 , · · · , x n and F (q 1 , · · · , q n ) = f [x 1 (q 1 , · · · , q n ), · · · , x n (q 1 , · · · , q n )]. In general, the variables x 1 , · · · , x n can be treated as the Cartesian coordinates of an n-dimensional Euclidean space and the variables q 1 , q 2 , · · · , q n form a set of curvilinear coordinates representing the same Euclidean space. We assume that each curvilinear coordinate q i is uniquely defined by the Cartesian coordinates: q i = q i (x 1 , · · · , x n ). We also assume that the inverse transformation is uniquely defined as x i = x i (q 1 , · · · , q n ). Then the Jacobian J can be expressed as where Det stands for the determinant.
In physics, the Jacobian appears frequently when one makes the change of variables between Cartesian and curvilinear coordinates in various physical quantities involving surface or volume integrals. However, in most physics textbook including classical mechanics and electromagnetism usually abstract descriptions are provided. In many textbooks of calculus or mathematical physics [1], the Jacobian formula is derived in the following way: First, one transforms the multivariable differential volume by applying the change of variables. Next, one imposes a geometrical argument that the infinitesimal volume is invariant under the transformation [2]. The invariance of the volume can also be confirmed by applying Green's theorem to show that S dx 1 dx 2 = T Jdq 1 dq 2 , where S is a rectangular region in the x 1 x 2 plane and T is the corresponding region [3]. Although experienced teachers or researchers may follow the abstract logic in this kind of Jacobian derivation without difficulties, the concept is rather unclear or less intuitive to undergraduate physics-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: ∞ −∞ dx δ(x) = 1. For any smooth function f (x), ∞ −∞ dx δ(x − y)f (x) = f (y). This property can be applied to changing the integration variable. Recently, some of us introduced an alternative proof of Cramer's rule by making use of Dirac delta functions [4]. It turns out that the method with a convolution of a coordinate vector with Dirac delta 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 immediately applicable to the evaluation of the Jacobian for the coordinate transformation or change of variables after replacing the coordinate vector with an arbitrary function. However, the direct application of the approach in Ref. [4] is limited to a linear transformation between two coordinate systems because Cramer's rule applies only to the system of linear equations. Thus the method is not applicable to the transformation involving a set of curvilinear coordinates which are frequently used in physics.
The main goal of this work is to present a more intuitive derivation of the Jacobian involving any coordinate transformation and to demonstrate how it works with heuristic examples. We develop an alternative derivation of the 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 coordinate transformation rules. Hence, we expect that students who are familiar with the Dirac delta function can compute the Jacobian formula in any coordinate transformations by themselves without referring to a reference. The approach that 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-linear transformation by convolving an arbitrary function with Dirac delta functions. While the basic strategy to perform the coordinate transformation is similar to that employed in Ref. [4], the integration of the original coordinates is rather involved because of the non-linear property of the transformation functions between the two coordinate systems. The integration of the original coordinates can be carried out by making use of Dirac delta functions. An extra factor containing partial derivatives of corresponding coordinate variables appears in front of the original integrand after that multiple integration. It turns out that the extra factor can be evaluated by making use of the chain rule of partial derivatives. 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 with the 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 remarkable that our derivation is free of borrowing abstract and advanced mathematical concepts unlike the other derivations available. Instead, we exploit a simple concept of integration of the one-dimensional Dirac delta function repeatedly in combination with a purely algebraic manipulation in reorganizing the extra factor by applying chain rules. This intuitive and systematic approach is expected to be pedagogically useful in upper-level mathematics or physics courses in 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 the paper, make a rough sketch of our strategy and present a formal derivation of the Jacobian factor for the coordinate transformation from the n-dimensional Cartesian coordinates to a set of curvilinear coordinates. Section III is devoted to the explicit evaluation of the Jacobians for a few examples. Conclusions are given in section IV and a rigorous derivation of the chain rule for partial derivatives is given in Appendices.

A. Strategy and Notation
In this subsection, we present our strategy to derive the Jacobian for a coordinate transformation or change of variables from the Cartesian coordinates x i to the curvilinear coordinates q i with transformation functions for i = 1 through n, where n is a positive integer. We assume that the two sets of coordinates describe a single point uniquely and, therefore, the two sets of coordinates have a one-to-one correspondence although the transformation is in general non-linear. Thus the transformation in Eq. (3) is invertible: the inverse transformation from the curvilinear coordinates to the Cartesian coordinates exists. If the curvilinear coordinates are a linear combination of the Cartesian coordinates, then the linear transformation is invertible if the transformation matrix is non-singular: the determinant of the matrix is not vanishing. Then, the inverse transformation can be written as The basic strategy to derive the Jacobian with Dirac delta functions is the same as that for the derivation of Cramer's rule for a partial set of a coordinate transformation or change of variables given in Ref. [4]. One could immediately 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] in order to consider arbitrary curvilinear coordinates.
We define an n-dimensional integral I n , where d n x ≡ dx 1 · · · dx n is the n-dimensional differential volume element and the integrand f (x 1 , · · · , x n ) is an arbitrary function of the Cartesian coordinates. We assume that every Cartesian coordinate is integrated over the region (−∞, ∞). We define the unity for i = 1 through n. Multiplying the unity ½ i to the integral I n , one can integrate out the integration variable x i by making use of the Dirac delta function keeping the q i integral unevaluated. By applying this process to I n recursively from i = 1 through n, we complete the change of the integration variables from the Cartesian coordinates to the curvilinear coordinates. While we have suppressed the bounds of the integration for the curvilinear coordinate q i 's in Eq. (6), the curvilinear coordinates are assumed to be integrated over the entire region to cover the whole Euclidean space represented by the Cartesian coordinates by a single time.
We first compute ½ 1 × I n : By definition, we integrate over x 1 by making use of the delta function δ[q 1 − q 1 (x 1 , · · · , x n )] keeping the q 1 integral unevaluated: where x 1 is expressed in terms of q 1 and x j 's for j = 2 through n satisfying the condition that the argument of the delta function vanishes, q 1 − q 1 (x 1 , · · · , x n ) = 0. Because the explicit forms of f (x i ) and x i vary depending on the integration step, we adopt the notation f (q 1 , x 2 , · · · , x n ) after the x 1 integration. We will present more detailed explanations for this notation in the later part of this subsection. The extra factor G 1 is the remnant of the integration of the delta function and its explicit form will be given later in this paper.
Next, we multiply ½ 2 to I n in Eq. (8) to find that where every x 1 in the argument of the delta function as well as the integrand function is replaced with the expression in terms of q 1 and x j 's for j = 2 through n. Performing the integration over x 2 by making use of the delta function, we find After the x 2 integration every x 2 in the integrand is expressed in terms of q 1 , q 2 and x j 's for j = 3 through n. Again, G 2 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 find that I n reduces into the n-dimensional multiple integral over the curvilinear coordinates q i for i = 1 through n only.
where G n is the remnant of the integration of the delta functions. At this stage, the integrand acquires an additional factor 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 1 , · · · , x j in the integrand must be replaced with the expressions in terms of q 1 , · · · , q j and x j+1 , · · · , x n as for k = 1 through j. Each x k in Eq. (12) is determined by the condition that the argument of the corresponding Dirac delta function vanishes: One must keep in mind that the explicit form of each x k varies depending on the integration step as is displayed in Eq. (12). Thus, one must distinguish, for example, where the former is the expression after the integration over x j and the latter is that after the integration over x j−1 . 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 coordinates and the polar coordinates with the transformation functions r = x 2 + y 2 and θ = arctan y x and the inverse transformation functions x = r cos θ and y = r sin θ.
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 care in dealing with their partial derivatives. In order to avoid such an ambiguity, we introduce a notation for the partial derivative with subscripts of variables that are held constant. In the above 2-dimensional transformation, the partial derivative of θ with respect to y holding x fixed is denoted by while the partial derivative of θ with respect to y holding r fixed is represented by It is apparent from Eqs. (16) and (17) that In a general case, for a variable q j (q 1 , · · · , q i , x i+1 , · · · , x n ), we denote the partial derivative of q j with respect to x a holding q 1 , · · · , q i , x i+1 , · · · , x n fixed with the subscript (i) as where 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 1 , · · · , x n without subscript: where 1 ≤ ℓ ≤ n and 1 ≤ c ≤ n.
In the coordinate transformation from (x 1 , · · · , x n ) to (q 1 , · · · , q n ), we define the function G k , which is relevant for a partial set of integral variables corresponding to a transformation from (x 1 , · · · , x k ) to (q 1 , · · · , q k ), as for k = 1 through n. Note that G n is the inverse of the Jacobian J n which is defined by In this subsection, we consider a 2-dimensional integral I 2 for the integration of an arbitrary function f (x 1 , x 2 ): The Cartesian coordinates x 1 and x 2 are transformed into curvilinear coordinates q 1 and q 2 with the transformation relations which are assumed to be invertible and non-singular. Thus, x 1 and x 2 can be expressed in terms of q 1 and q 2 : We multiply the unities to I 2 for i = 1, 2, sequentially. First, we multiply ½ 1 in Eq. (28) to I 2 . Then, I 2 can be expressed as The integration over x 1 can be carried out by making use of the delta function for q 1 as which leads to After the integration over x 1 , every x 1 on the right-hand side of Eq. (30) and δ[q 2 −q 2 (x 1 , x 2 )]f (x 1 , x 2 ) in Eq. (28) must be replaced with x 1 (q 1 , x 2 ) satisfying the condition that the argument of the delta function vanishes, q 1 −q 1 (x 1 , x 2 ) = 0. Then the delta function for q 2 and f (x 1 , x 2 ) can be expressed as δ[q 2 − q 2 (q 1 , x 2 )] and f (q 1 , x 2 ), respectively.
After multiplying ½ 2 in Eq. (28) to I 2 in Eq. (31), the integration over x 2 can be performed by making use of the remaining delta function for q 2 as After the integration over x 2 , every x 2 on the right-hand sides of Eqs. (30) and (32) and in f (q 1 , x 2 ) is replaced with x 2 (q 1 , q 2 ), which can be obtained from the condition that the argument of the delta function vanishes, q 2 −q 2 (q 1 , x 2 ) = 0. Then, we can express x 1 in terms of q 1 and q 2 by replacing x 2 with x 2 (q 1 , q 2 ) in x 1 (q 1 , x 2 ). We have obtained the last equality of Eq. (32) by making use of the chain rule for the partial derivatives. A rigorous proof of this formula is given in Eq. (A5) of Appendix A.
After both x 1 and x 2 are integrated out, the integral I 2 reduces into where the Jacobian for the change of variables is identified as J = J 2 = 1/ G 2 . This completes the proof of the Jacobian 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 case of 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 3 for an arbitrary function f (x 1 , x 2 , x 3 ): The Cartesian coordinates x i for i = 1 through 3 are transformed into the curvilinear coordinates q i 's as The inverse transformation can be expressed as We carry out the change of variables by multiplying the unites for i = 1 through 3 to I 3 , sequentially. First, after multiplying ½ 1 in Eq. (37) to I 3 , we find that I 3 can be expressed as Similarly to the 2-dimensional case, we perform the integration over x 1 by making use of the Dirac delta function for q 1 as which leads to After the integration over x 1 , every x 1 in the integrand and remaining delta functions is replaced with x 1 (q 1 , x 2 , x 3 ) satisfying the condition that the argument of the Dirac delta function vanishes, q 1 − q 1 (x 1 , x 2 , x 3 ) = 0. Then the delta function for q 2 in Eq. (37) can be expressed as After multiplying ½ 2 in Eq. (37) to I 3 in Eq. (40), the integration over x 2 can be carried out by making use of the Dirac delta function for q 2 as where the last equality comes from Eq. (B5). Then, Eq. (41) is expressed as x 2 (q 1 , q 2 , x 3 ) is determined from the condition that the argument of the Dirac delta function vanishes, q 2 − q 2 (q 1 , x 2 , x 3 ) = 0. Substituting x 2 (q 1 , q 2 , x 3 ) into x 1 (q 1 , x 2 , x 3 ), we obtain x 1 = x 1 (q 1 , q 2 , x 3 ) and the argument of the Dirac delta function for q 3 in Eq.(37) is expressed as δ[q 3 − q 3 (q 1 , q 2 , x 3 )].
Finally, after multiplying ½ 3 in Eq. (37) to I 3 in Eq. (42), we integrate over x 3 by taking into account the delta function for q 3 as where the last equality comes from Eq. (B11). After the integration over x 3 , every x 3 in the integrand and the right-hand sides of Eqs. (39), (41) and (43) is replaced with x 3 (q 1 , q 2 , q 3 ) which is determined from the condition that the argument of the delta function vanishes, q 3 − q 3 (q 1 , q 2 , x 3 ) = 0. Substituting x 3 (q 1 , q 2 , q 3 ) into x 1 (q 1 , q 2 , x 3 ) and x 2 (q 1 , q 2 , x 3 ), we can obtain the expressions for x 1 = x 1 (q 1 , q 2 , q 3 ) and x 2 = x 2 (q 1 , q 2 , q 3 ), respectively. Then, we can express the integrand f (x 1 , x 2 , x 3 ) in terms of q 1 , q 2 and q 3 and the integral I 3 is expressed as where the Jacobian for the change of variables is identified as J = J 3 = 1/ G 3 . This completes the proof of the Jacobian 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 1 , · · · , x n ) defined in Eq. (5) by multiplying the unities ½ i in Eq. (6) to I n in Eq. (5) sequentially. Then, we integrate I n , which is multiplied by the unity, over x i for i = 1 through n successively by making use of the Dirac delta function After the integration of all x i variables, the corresponding Jacobian formula is obtained by employing mathematical induction. First, the integration over x 1 can be carried out from Eq. (7) and the result for the integration over x 1 can easily be generalized from the 2-dimensional version in Eq. (30) as which leads to After the x 1 integration, every x 1 in the integrand of Eq. (7) and G 1 in Eq. (46) is replaced with The constraint equation coming from the convolution with the Dirac delta function in Eq. (46) is Then, we carry out the integration over x i for i = 1 through n − 1 by multiplying ½ 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 delta functions is which leads to For any j ≤ i every x j in the integrand of Eq. (5) and G j 's in Eq. (50) is replaced with after the integrations over x 1 through x i . There are i constraint equations coming from the convolution with Dirac delta functions in Eq. (50): where j runs from 1 through i.
After multiplying ½ i+1 to I n in Eq. (51), we integrate out one more Cartesian coordinate x i+1 to find that where 1 ≤ j ≤ i and 1 ≤ k ≤ i + 1. The proof of the last equality can be found in Eq. (C8) in Appendix C. In combination with Eqs. (50) and (54), we find that For any j ≤ i + 1 every x j in the integrand of Eq. (7) and G j 's in Eq. (55) is replaced with after the integrations over x 1 through x i+1 . There are i + 1 constraint equations coming from the convolution with Dirac delta functions in Eq. (55): 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 1 , · · · , q n as where 1 ≤ k ≤ n. This completes the proof of the Jacobian J = J n = 1/ G n for an n-dimensional coordinate transformation or change of variables.

III. APPLICATION
Since the proof of the Jacobian formula in the previous section is rather abstract, readers who are not familiar with the notation might be confused. In this section, we present a few explicit examples of deriving the Jacobian without resorting to the general formula derived in the previous section. We expect that the explicit examples will help readers to understand the method presented in the previous section more intuitively and to apply it to a specific change of variables.

A. Spherical coordinates
In this subsection, we consider the change of variables from the 3-dimensional Cartesian coordinates (x, y, z) to the spherical 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 to simplify the computation. That is, (x 1 , x 2 , x 3 ) in section II C corresponds to (z, y, x) while (q 1 , q 2 , q 3 ) corresponds to (cos θ, φ, r), respectively. However, it turns out that the integral is invariant under this reordering. We also note that one can use, for instance, sin θ instead of cos θ as an integration variable and it will not alter the result of the integration. Then the inverse transformation of Eq. (59) is expressed as cos θ = z where the Heaviside step function Θ(x) is defined by Conventionally, the arctangent function is defined in the region [− π 2 , π 2 ]. The period of the tangent function is π, while the azimuthal angle ranges from 0 to 2π. Thus the angle φ for x > 0 is set to be arctan y x ∈ [− π 2 , π 2 ] while that for x < 0 is set to be π + arctan y x ∈ [ π 2 , 3π 2 ] in order to make the transformation function invertible in the entire range: , for x > 0, π + arctan y x ∈ ( π 2 , 3π 2 ), for x < 0, π 2 , for x = 0 and y > 0, − π 2 , for x = 0 and y < 0, 0, for x = y = 0.
We consider a 3-dimensional integral J 3 with an arbitrary integrand f (x, y, z) We multiply the unities to J 3 without changing the value of the integral sequentially.
First, we integrate out the x 1 = z coordinate by multiplying ½ 1 in Eq. (64a) to J 3 in Eq. (63). The integration over z can be performed as After the integration over z, every z in the integrand of J 3 is replaced with the expression in terms of cos θ, y and x: z = x 2 + y 2 / tan θ, where we omit the simple conversion between trigonometric functions here and after. The integrand of the remaining double integral over x and y is a function of θ, x and y: This can also be obtained by substituting cos θ, y and x into Eq. (39).
Next, we integrate out the x 2 = y coordinate by multiplying ½ 2 in Eq. (64b) into J 3 in Eq. (66). The integration over y can be performed as where we have used the identity 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 3 is replaced with the expression in terms of cos θ, φ and x. The integrand of the remaining integral over x is a function of cos θ, φ and x: 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 ½ 3 in Eq. (64c) into J 3 in Eq. (69), the integration over x 3 = x can be performed as This can also be obtained by substituting cos θ, φ and r into Eq. (43). Here, the radius r is non-negative because the Dirac delta function requires that r = x 2 + y 2 + z 2 and x 2 + y 2 + z 2 is non-negative. After the integration over all of 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 3 can be expressed in terms of the spherical coordinates as The overall factor r 2 sin θ is identified to be the Jacobian B. n-dimensional polar coordinates In this subsection, we extend the 3-dimensional case to the n-dimensional coordinate transformation from the ndimensional Cartesian coordinates (x 1 , · · · , x n ) to the n-dimensional polar coordinates (r, θ 1 , θ 2 , · · · , θ n−2 , φ). Here, r is the radius and there are n − 2 polar angles θ i 's and a single azimuthal angle φ. The corresponding transformation functions are expressed as x 1 = r sin θ 1 sin θ 2 · · · sin θ n−3 sin θ n−2 cos φ, x 2 = r sin θ 1 sin θ 2 · · · sin θ n−3 sin θ n−2 sin φ, x 3 = r sin θ 1 sin θ 2 · · · sin θ n−3 cos θ n−2 , . . .
Our strategy to integrate out x i 's is as follows: According to the integrand of the right-hand side of Eq. (75a), the integration over x n−i+1 in the integral J n provides the constraint to the polar angle θ i . Thus we choose to integrate over from x n to x 3 to express them in terms of the polar angles from θ 1 through θ n−2 . Then we integrate out x 2 to express x n through x 2 in terms of the n − 2 polar angles and the azimuthal angle φ by making use of Eq. (75b). As the last step, we integrate out x 1 to determine all of the Cartesian coordinates in terms of the spherical polar coordinates by making use of (75c).
First, after multiplying ½ i in Eq. (75a) into J n , the integration over x n−i+1 for i = 1 through n− 2 can be performed where one can obtain the same results from Eqs. (46) and (54) taking care of the order of integration. After the integration over x n−i+1 , we make the replacement Then, J n is expressed as where every x i in the last factor for i = 3 through n is replaced by that in Eq. (77).
After multiplying ½ n−1 in Eq. (75b) into J n in Eq. (78), the integration over x 2 can be carried out in a similar manner as is done in Eq. (67). The result is where x 2 = x 1 tan φ after the integration. This can also be obtained from Eq. (54) while keeping the results in Eqs. (76) and (77). Then, x n−i+1 for i = 1 through n − 1 can be expressed as Then, we find that where every x i in the last factor for i = 2 through n is replaced by that in Eq. (80).
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 Cartesian coordinates to arbitrary curvilinear coordinates by making use of Dirac delta functions, whose arguments correspond to 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 Cartesian coordinates systematically. By making use of the chain rule for the partial derivatives, we can carry out the integration over the Cartesian coordinates successively and end up with the integral expressed in terms of the curvilinear coordinates. Then, the Jacobian can be read off by comparing the integrands of the resultant integral with the original one.
It turns out that the formula derived in this paper exactly reproduces the Jacobian for the coordinate transformation or change of variables.
We have presented a few examples, where we have integrated out the Cartesian coordinates by making use of Dirac delta functions explicitly without resorting to the general formula for the Jacobian derived in this paper. We find that the formulas obtained in these explicit examples are exactly the same as those in the general formula (58). Since the derivation of the Jacobian in the general case that makes use of the chain rule of the partial derivatives is rather abstract, 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 the coordinate transformation or change of variables from the n-dimensional Cartesian coordinates to the curvilinear coordinates is new. Although there are several ways to derive the Jacobian available in textbooks [1][2][3], our derivation could be pedagogically useful in upper-level mathematics or physics courses in practice using Dirac delta functions successively. Compared with the methods popular in the textbook level, our method is more intuitive because we have employed only the explicit calculation of elementary single-dimensional integrals without relying on abstract geometrical interpretations or more abstract Green's theorem with which undergraduate physics-major students are not usually familiar. Furthermore, a detailed derivation of the chain rule for the partial derivatives, which is employed to prove the Jacobian formula, should be a nice working example with which one can understand a rigorous usage of the partial derivatives with multi-dimensional variables without ambiguity.
(∂q 2 /∂x 2 ) (1) can be expressed as which leads to In this section, we consider a 3-dimensional coordinate transformation from the Cartesian coordinates (x 1 , x 2 , x 3 ) to the curvilinear coordinates (q 1 , q 2 , q 3 ). First, we consider the total differentials of q 1 and x 1 which can be expressed as where with the right-hand side of Eq. (B1b), we obtain Comparing the coefficients of the differentials on both sides, we find that where Eq. (B3a) is trivial. By making use of Eq. (B3b), we find that the factor in the denominator of Eq. (41) (∂q 2 /∂x 2 ) (1) can be expressed as which leads to In order to prove Eq. (B11), we take into account the total differentials of q 1 , x 1 , q 2 and x 2 . The total derivatives can be expressed as where , and Comparing both sides of Eq. (B7), we find two relevant non-trivial equations: By making use of Cramer's rule, we find that (B9) Then, by making use of Eq. (B9), we find that the factor (∂q 3 /∂x 3 ) (2) in the denominator of Eq. (43) can be expressed as which leads to Appendix C: n-dimensional case The total differentials of q 1 , · · · , q i , x 1 , · · · , x i can be expressed as The partial derivative of q i+1 with respect to x i+1 holding q 1 , · · · , q i , x i+2 , · · · , x n fixed is Substituting dx 1 , · · · , dx i into dq j , we obtain where 1 ≤ j ≤ i. Because the coefficient of dx i+1 in (C3) should be 0, we obtain the following equation: By making use of Cramer's rule, we find that and (Â −1 [i×i] except that the jth column is replaced with ∂Õ i . Substituting (C5) into (C2), we obtain Here, the ij minor M ij ( ) of an n × n square matrix is the determinant of a matrix whose ith row and jth column are removed from . Hence, Eq. (C7) leads to