Let's rewrite the formulas using the new notation $\nabla^*$ in place of both $\text{div}$ and $\text{curl}$. Based on your instructions, the $\nabla^*$ will now represent these two different operations depending on the context. Here's the rewritten set of equations:

1. (i) $\nabla^* (\nabla^* \vec{G}) = 0$

2. (ii) $\nabla^* (\nabla f) = \vec{0}$

3. (iii) $\nabla^* (f \vec{G}) = f (\nabla^* \vec{G}) + (\nabla f) \cdot \vec{G}$

4. (iv) $\nabla^* (\nabla f) = \nabla^2 f$

5. (v) $\nabla^* (\vec{F} + \vec{G}) = \nabla^* \vec{F} + \nabla^* \vec{G}$

6. (vi) $\nabla^* (f \vec{G}) = f (\nabla^* \vec{G}) + (\nabla f) \cdot \vec{G}$

Notice that equations (iii) and (vi) end up looking identical because they both involve the product of a scalar function $f$ and a vector field $\vec{G}$. In both cases, $\nabla f$ is the gradient of $f$, and $\cdot$ indicates a dot product between $\nabla f$ and the vector $\vec{G}$.

Would you like further details on any of these steps, or have any questions?

Here are 5 related questions to explore further:

1. How does the divergence of a vector field relate to the curl of the same field?
2. Can you provide a physical interpretation of $\nabla^*$ in the context of fluid dynamics?
3. What is the difference between the Laplacian $\nabla^2 f$ and the gradient $\nabla f$?
4. How does the product rule apply in vector calculus, as seen in equation (vii)?
5. What are some common applications of these vector calculus identities in physics?

Tip: When dealing with vector calculus, always keep in mind that $\nabla$ behaves differently when applied to scalars vs. vectors.