用Dirac δ函数推导雅可比公式
Derivation of Jacobian Formula with Dirac Delta Function
摘要：
我们演示了如何利用一个函数与Dirac δ函数的卷积来实现从笛卡尔坐标到曲线坐标的坐标变换或变量变换，其中Dirac δ函数的自变量由两个坐标系之间的变换函数确定。通过用狄拉克δ函数积分出原始坐标，我们以系统的方式用新坐标替换原始坐标。递归使用狄拉克δ函数允许坐标变换连续。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矩阵以完全代数的方式自动获得。为了提供这种方法的见解，我们提出了一些例子，明确评估雅可比矩阵，而不诉诸已知的一般公式。
英文摘要：
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.
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