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The Electric field due to a volume charge distribution can be written as \mathbf{E}(\mathbf{r})=\frac{1}{4 \pi \epsilon_{0}} \int \mathrm{d}^{3} r^{\prime} \rho\left(\mathbf{r}^{\prime}\right) \frac{\mathbf{r}-\mathbf{r}^{\prime}}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|^{3}} and using them show that \nabla \frac{1}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}=-\frac{\mathbf{r}-\mathbf{r}^{\prime}}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|^{3}} \text { and } \nabla^{2} \frac{1}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}=-4 \pi \delta\left(\mathbf{r}-\mathbf{r}^{\prime}\right) and using them show that \nabla \cdot \mathbf{E}=\rho / \epsilon_{0} \text { and } \nabla \times \mathbf{E}=\mathbf{0} \text {. }

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