论文标题
用Dirac Delta功能的Cramer规则的证明
Proof of Cramer's rule with Dirac Delta Function
论文作者
论文摘要
我们通过将线性方程式系统解释为$ n $维的笛卡尔坐标矢量的转换，向Cramer规则提供了新的证明。为了找到解决方案，我们通过将原始坐标矢量与Dirac Delta函数进行卷积，并将集成变量从原始坐标从原始坐标变为新坐标来进行反向转换。作为副产品，我们得出了Cramer规则的广义版本，该版本适用于部分变量，这是我们最佳知识的新变量。我们为多变量函数找到转换规则的表述对于更改机械系统的部分广义坐标特别有用。
We present a new proof of Cramer's rule by interpreting a system of linear equations as a transformation of $n$-dimensional Cartesian-coordinate vectors. To find the solution, we carry out the inverse transformation by convolving the original coordinate vector with Dirac delta functions and changing integration variables from the original coordinates to new coordinates. As a byproduct, we derive a generalized version of Cramer's rule that applies to a partial set of variables, which is new to our best knowledge. Our formulation of finding a transformation rule for multi-variable functions shall be particularly useful in changing a partial set of generalized coordinates of a mechanical system.
