The expression on the whiteboard reads:

$\nabla \cdot \left( \vec{r} \, \nabla \left( \frac{1}{r^3} \right) \right) = 3 r^{-4}$

It appears to ask for a verification of this result, as indicated by the Turkish phrase "olduğunu gösteriniz," which translates to "show that this is true."

Solution Outline

To solve this problem, we need to:

1. Use vector calculus to expand and simplify the expression on the left-hand side.
2. Apply the definition of $\vec{r} = r \hat{r}$, where $r = |\vec{r}|$, and perform the gradient operation on $\frac{1}{r^3}$.
3. Confirm that the result matches $3 r^{-4}$.

Would you like a detailed step-by-step solution for this problem?

Here are some related questions that might deepen your understanding:

1. How do you compute the gradient of $\frac{1}{r^n}$ for a general $n$?
2. What is the significance of the Laplacian operator in spherical coordinates?
3. Can we apply this approach to find electric fields from potentials in physics?
4. How does this computation change if we consider the vector field in cylindrical coordinates?
5. What other physical scenarios involve operations like $\nabla \cdot (\vec{r} f(r))$?

Tip

Understanding vector calculus operations like divergence and gradient in spherical coordinates can simplify many physics problems, especially in fields like electromagnetism.