论文标题
带有Dirac Delta功能的Jacobian公式的推导
Derivation of Jacobian Formula with Dirac Delta Function
论文作者
论文摘要
我们通过利用具有DIRAC DELTA函数的函数的卷积，其参数由两个坐标系统之间的转换函数确定，我们演示了如何使变量从笛卡尔坐标到曲线坐标的变量变量的变量。通过将原始坐标与Dirac Delta功能集成，我们以系统的方式替换了原始坐标。递归的狄拉克三角洲函数的使用允许连续的坐标转换。将每个原始坐标替换为新的曲线坐标后，我们发现相应坐标转换的所得jacobian将以完全代数的方式自动获得。为了提供有关此方法的见解，我们提供了一些明确评估Jacobian的示例，而无需诉诸已知的通用公式。
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.
