Let's analyze the given mathematical problems from the image:

Problem (a):

Show that
$\delta \mu = \frac{1}{2} (\Delta + \nabla)$

Problem (b):

Show that
$\Delta - \nabla^2 = \Delta \nabla'$

To solve these problems, we need to know the definitions of the symbols used:

• $\delta \mu$: Could be a variation of some quantity $\mu$.
• $\Delta$: Often denotes the Laplace operator.
• $\nabla$: Represents the gradient operator.
• $\nabla^2$: Represents the Laplacian operator.
• $\Delta \nabla'$: Could denote a combination of operators acting on a function.

(a) Solution:

Given expression: $\delta \mu = \frac{1}{2} (\Delta + \nabla)$

To show this, we'll need to work with the definitions of $\Delta$ and $\nabla$ and any rules that might apply. This expression might relate to a decomposition of an operator into its symmetric and antisymmetric parts.

(b) Solution:

Given expression: $\Delta - \nabla^2 = \Delta \nabla'$

Here, we need to prove the equivalence of these expressions. This might involve applying definitions of $\Delta$, $\nabla^2$, and potentially the divergence or gradient operators.

Could you please provide additional context or definitions related to these symbols? If these expressions come from a specific field (like vector calculus, differential geometry, or physics), knowing that will help tailor the explanation accordingly.

Questions for Further Exploration:

1. What are the specific definitions of $\delta \mu$, $\Delta$, and $\nabla$ in this context?
2. Are there any specific conditions or constraints given for these problems?
3. Are these problems part of a particular topic like quantum mechanics, differential equations, or tensor calculus?
4. Is there a specific function or vector field that these operators are acting upon?
5. Could you provide any additional information or context from the source material?

Tip:

When dealing with abstract symbols in mathematical problems, it is crucial to have a clear understanding of the definitions and the field of application. This helps in applying the correct theorems and properties to solve the problem accurately.