To determine the value of $n$ for which the vector $\mathbf{r}^n \mathbf{r}$ is solenoidal, we follow these steps:

Definitions:

• A vector field is solenoidal if its divergence is zero, i.e., $\nabla \cdot \mathbf{V} = 0$.
• Here, $\mathbf{V} = \mathbf{r}^n \mathbf{r}$, where $\mathbf{r} = x\hat{i} + y\hat{j} + z\hat{k}$ and $\mathbf{r} = \sqrt{x^2 + y^2 + z^2}$ is the magnitude.

Step 1: Write $\mathbf{V}$:

$\mathbf{V} = \mathbf{r}^n \mathbf{r} = \mathbf{r}^{n+1} \hat{r},$ where $\hat{r} = \frac{\mathbf{r}}{|\mathbf{r}|}$ is the unit vector in the direction of $\mathbf{r}$.

Step 2: Compute $\nabla \cdot \mathbf{V}$:

Using the formula for the divergence of $\mathbf{r}^m$, we write: $\nabla \cdot (\mathbf{r}^m \mathbf{r}) = (m + 3)\mathbf{r}^{m-1} = 0,$ where $m = n + 1$.

For the divergence to be zero: $m + 3 = 0.$

Step 3: Solve for $n$:

Since $m = n + 1$, we substitute $m$ into the equation: $n + 1 + 3 = 0,$ $n = -4.$

Final Answer:

The vector $\mathbf{r}^n \mathbf{r}$ is solenoidal when $n = -4$.

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Would you like me to expand or explain specific steps in more detail?

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Here are 5 follow-up questions to deepen your understanding:

1. What does it mean for a vector field to be solenoidal in physical contexts like fluid dynamics?
2. How is the divergence operator applied to vector fields in cylindrical or spherical coordinates?
3. Can you derive the solenoidal condition for other types of vector fields, such as $\mathbf{r}^n \mathbf{A}$, where $\mathbf{A}$ is another vector field?
4. How does the value of $n$ affect the physical interpretation of the field $\mathbf{r}^n \mathbf{r}$?
5. Can you verify this result by explicitly computing the divergence in Cartesian coordinates?

Tip: Solenoidal fields often correspond to incompress