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.

我们演示了如何通过使用函数与狄拉克 delta 函数的卷积来进行坐标转换或变量从笛卡尔坐标到曲线坐标的变化，狄拉克 delta 函数的参数由两个坐标系之间的转换函数确定。通过将原始坐标与 Dirac delta 函数积分，我们以系统的方式将原始坐标替换为新坐标。Dirac delta 函数的递归使用允许连续进行坐标变换。在将每个原始坐标替换为新的曲线坐标后，我们发现相应坐标变换的合成雅可比行列式是以完全代数的方式自动获得的。为了提供有关此方法的见解，