https://file.aiquickdraw.com/mathbot/file/1725511006402byfoa13y.jpg

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.

Show that (a) δμ = 1/2(Δ + ∇) and (b) Δ - ∇^2 = Δ∇'

Explore the detailed solution to prove operator equations involving δμ, Δ, and ∇ in vector calculus. Understand the decomposition of operators, properties of gradient, and Laplacian operators, suitable for undergraduate students.

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