This question involves vector calculus and asks us to evaluate $\nabla \times (\vec{r} \times \vec{F})$, where:

• $\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}$, the position vector.
• $\vec{F}$ is a general three-dimensional vector field.

Formula Used:

We use the vector triple product expansion for $\nabla \times (\vec{r} \times \vec{F})$. The formula is:

$\nabla \times (\vec{A} \times \vec{B}) = (\vec{B} \cdot \nabla) \vec{A} - (\vec{A} \cdot \nabla) \vec{B} + \vec{A} (\nabla \cdot \vec{B}) - \vec{B} (\nabla \cdot \vec{A})$

Here:

• $\vec{A} = \vec{r}$,
• $\vec{B} = \vec{F}$.

Applying the Formula:

Breaking it into terms:

1. First term: $(\vec{F} \cdot \nabla) \vec{r}$:

   • Since $\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}$,
   • $(\vec{F} \cdot \nabla) \vec{r} = \vec{F}$ (as differentiation gives back components of $\vec{F}$).

2. Second term: $(\vec{r} \cdot \nabla) \vec{F}$:

   • This term represents the directional derivative of $\vec{F}$ along $\vec{r}$.

3. Third term: $\vec{r} (\nabla \cdot \vec{F})$:

   • $\nabla \cdot \vec{F}$ is the divergence of $\vec{F}$, so $\vec{r} (\nabla \cdot \vec{F})$ scales $\vec{r}$ by that value.

4. Fourth term: $- \vec{F} (\nabla \cdot \vec{r})$:

   • $\nabla \cdot \vec{r} = 3$ (since $\partial x/\partial x + \partial y/\partial y + \partial z/\partial z = 1 + 1 + 1 = 3$).
   • So, this term becomes $-3 \vec{F}$.

Combining the Results:

$\nabla \times (\vec{r} \times \vec{F}) = (\nabla \cdot \vec{F}) \vec{r} - (\vec{r} \cdot \nabla) \vec{F} - 3 \vec{F}$

This matches option (d): $(\nabla \cdot \vec{F}) \vec{r} - (\vec{r} \cdot \nabla) \vec{F} - 3 \vec{F}.$

Final Answer:

(d)

Let me know if you'd like a more detailed breakdown or examples! Here are five related questions to expand your understanding:

1. What is the physical significance of the divergence of a vector field?
2. How does the vector triple product formula generalize for various vector fields?
3. Can you derive $\nabla \times (\nabla \times \vec{F})$ using similar principles?
4. What is the importance of the position vector $\vec{r}$ in vector calculus?
5. How can the curl of a vector field be interpreted geometrically?

Tip: Always remember the identity $\nabla \cdot \vec{r} = 3$ in three dimensions—it’s a useful shortcut!