Find the value of n for which the vector r ^ n * r solenoidal, where r =x hat i +y hat j +z hat k .

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.

Explore how to find the value of n for which the vector field r^n * r is solenoidal in Cartesian coordinates. Understand the application of vector calculus and the divergence theorem. Suitable for advanced undergraduate students and anyone interested in vector field analysis.

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