While solving some problem i encounter the following step. Given $$ T^{\mu\nu}_{,\nu}=\sum_i m_i\int d\tau_i U^{\mu}_i U^{\nu}_i \frac{\partial}{\partial{x^{\nu}}}\delta^4[x-x_i(\tau_i)]$$
The delta function depends only on $x -x_i$ so we can replace $ \frac{\partial}{\partial{x^{\nu}}}$ by $ -\frac{\partial}{\partial{x^{\nu}_i}}$
Can anyone explain why there is a $\textbf{negative sign}$ while changing from $ \frac{\partial}{\partial{x^{\nu}}}$ to $ \frac{\partial}{\partial{x^{\nu}_i}}$ ?
We have $$ \varphi(y) = \int \delta(x-y) \, \varphi(x) \, dx = \{ x = y+z \} = \int \delta(z) \, \varphi(y+z) \, dz . $$
Therefore, we have $$ \varphi'(y) = \int \partial_y(\delta(x-y)) \, \varphi(x) \, dx \\ $$ but also $$ \varphi'(y) = \int \delta(z) \, \partial_y(\varphi(y+z)) \, dz = \int \delta(z) \, \varphi'(y+z) \, dz = \{ x = y+z \} = \int \delta(x-y) \, \varphi'(x) \, dx \\ = - \int \partial_x(\delta(x-y)) \, \varphi(x) \, dx . $$
Therefore, $\partial_y(\delta(x-y)) = - \partial_x(\delta(x-y)).$