To find the value of $n$ for which the vector $\mathbf{r}^n \cdot \mathbf{r}$ is solenoidal, where $\mathbf{r} = x \hat{i} + y \hat{j} + z \hat{k}$, we need to ensure that the divergence of the vector field is zero. A vector field $\mathbf{F}$ is solenoidal if $\nabla \cdot \mathbf{F} = 0$.

Given: $\mathbf{r} = x \hat{i} + y \hat{j} + z \hat{k}$

We need to evaluate the divergence of: $\mathbf{F} = r^n \mathbf{r}$

Where $r = \sqrt{x^2 + y^2 + z^2}$.

So, $\mathbf{F} = r^n (x \hat{i} + y \hat{j} + z \hat{k}) = x r^n \hat{i} + y r^n \hat{j} + z r^n \hat{k}$

To find the divergence, we calculate: $\nabla \cdot \mathbf{F} = \frac{\partial}{\partial x} (x r^n) + \frac{\partial}{\partial y} (y r^n) + \frac{\partial}{\partial z} (z r^n)$

Each term involves the product rule for differentiation: $\frac{\partial}{\partial x} (x r^n) = r^n + x \frac{\partial}{\partial x} (r^n)$

Using the chain rule: $\frac{\partial}{\partial x} (r^n) = n r^{n-1} \frac{\partial r}{\partial x} = n r^{n-1} \frac{x}{r} = n x r^{n-2}$

Thus, $\frac{\partial}{\partial x} (x r^n) = r^n + x \cdot n x r^{n-2} = r^n + n x^2 r^{n-2}$

Similarly: $\frac{\partial}{\partial y} (y r^n) = r^n + n y^2 r^{n-2}$ $\frac{\partial}{\partial z} (z r^n) = r^n + n z^2 r^{n-2}$

Summing these partial derivatives: $\nabla \cdot \mathbf{F} = (r^n + n x^2 r^{n-2}) + (r^n + n y^2 r^{n-2}) + (r^n + n z^2 r^{n-2})$ $\nabla \cdot \mathbf{F} = 3r^n + n (x^2 + y^2 + z^2) r^{n-2}$

Since $r^2 = x^2 + y^2 + z^2$, we can substitute: $\nabla \cdot \mathbf{F} = 3r^n + n r^2 r^{n-2}$ $\nabla \cdot \mathbf{F} = 3r^n + n r^n$ $\nabla \cdot \mathbf{F} = (3 + n) r^n$

For the vector field to be solenoidal, the divergence must be zero: $(3 + n) r^n = 0$

Since $r^n \neq 0$ for any non-zero $\mathbf{r}$, we have: $3 + n = 0$ $n = -3$

Therefore, the value of $n$ for which the vector field $\mathbf{r}^n \cdot \mathbf{r}$ is solenoidal is $n = -3$.

Do you want more details or have any questions?

Here are some related questions to explore:

1. What is the definition of a solenoidal vector field?
2. How do you compute the divergence of a vector field in Cartesian coordinates?
3. What is the physical significance of a solenoidal vector field?
4. How would the solution change if we considered a different coordinate system, like spherical coordinates?
5. Can you derive the expression for the gradient of a scalar field?
6. What is the curl of a vector field and how is it related to solenoidal fields?
7. How can we apply the concept of solenoidal fields in fluid dynamics?
8. What are some real-world examples of solenoidal vector fields?

Tip: When dealing with vector calculus, always remember to use the appropriate coordinate system for simplifying your calculations.