The image you've provided contains a question asking to prove two equations:

1. (a) $\delta \mu = \frac{1}{2} (\Delta + \nabla)$
2. (b) $\Delta - \nabla^2 = \Delta \nabla$

Let’s break down the requirements for both proofs.

Part (a):

The equation given is: $\delta \mu = \frac{1}{2} (\Delta + \nabla)$

• Here, $\Delta$ is typically the Laplacian operator (denoted by $\nabla^2$), and $\nabla$ represents the gradient operator.
• To prove this, we need to know how $\delta \mu$ is defined in this context, but it seems to imply that it relates to some transformation involving the Laplacian and the gradient.
• The equation is essentially stating that $\delta \mu$ is the average of the Laplacian and the gradient.

Part (b):

The equation given is: $\Delta - \nabla^2 = \Delta \nabla$

• Here, $\nabla^2$ represents the Laplacian operator.
• We need to investigate how $\Delta$ interacts with $\nabla$, which may require vector calculus identities or assumptions about the field that these operators are acting on.

Would you like further assistance with either part, such as a more detailed breakdown of each operator or specific steps for solving these? Let me know which concept you'd like to explore further.

Here are 5 related questions to expand on these concepts:

1. What is the relationship between the Laplacian $\nabla^2$ and the gradient $\nabla$?
2. How can vector calculus identities help simplify equations involving $\Delta$ and $\nabla$?
3. Can the operators $\nabla$ and $\nabla^2$ commute under certain conditions, and why?
4. In which physical contexts might the combination of these operators appear (e.g., fluid dynamics, electromagnetism)?
5. How does the divergence theorem relate to these kinds of vector calculus equations?

Tip: When dealing with vector calculus, always make sure to review common vector identities like the product rules and how the gradient, curl, and divergence operators interact. These can often simplify complex expressions.