For a position vector $\mathbf{r} = x\hat{i} + y\hat{j} + z\hat{k}$ with magnitude $r = |\mathbf{r}| = \sqrt{x^2 + y^2 + z^2}$, which of the following vector identities is/are CORRECT?
Options:
(A) $\nabla r = \frac{\mathbf{r}}{r}$
(B) $\nabla \cdot \mathbf{r} = 3$
(C) $\nabla \times \mathbf{r} = \mathbf{0}$
(D) $\nabla(r^n) = n r^{n-2} \mathbf{r}$
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
1. **Gradient of Magnitude (Option A)**: $\frac{\partial r}{\partial x} = \frac{x}{r}$, so $\nabla r = \frac{x\hat{i} + y\hat{j} + z\hat{k}}{r} = \frac{\mathbf{r}}{r}$ (True).
2. **Divergence of Position Vector (Option B)**: $\nabla \cdot \mathbf{r} = \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial z}{\partial z} = 1 + 1 + 1 = 3$ (True).
3. **Curl of Position Vector (Option C)**: $\nabla \times \mathbf{r} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \partial/\partial x & \partial/\partial y & \partial/\partial z \\ x & y & z \end{vmatrix} = \mathbf{0}$ (True).
4. **Power Law Gradient (Option D)**: $\nabla(r^n) = n r^{n-1} \nabla r = n r^{n-1} \left( \frac{\mathbf{r}}{r} \right) = n r^{n-2} \mathbf{r}$ (True).