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The image you provided contains three mathematical or physics-related questions. Here is what I can extract and explain from the content:

1. **Q1** appears to involve a problem related to **Poisson's equation and Laplace's equation**, likely asking to **differentiate between the two** and explain their physical significance.
   
2. **Q2** seems to be about vector calculus and vector algebra:
   - It starts with a vector **A = 4xî + 2yĵ + z**.
   - You're asked to **express this vector in cylindrical coordinates** and prove the identity involving cross products:
     \[
     A \cdot (B \times C) = B \cdot (C \times A) = C \cdot (A \times B)
     \]
   
3. **Q3** involves gradient and curl calculations:
   - You are required to **calculate the divergence** (\( \nabla \cdot V \)), **gradient** (\( \nabla V \)), and **curl** (\( \nabla \times V \)) for a given vector field.

Let me know if you'd like to delve into the solutions for any specific question. 

Do you want detailed steps on any part of this problem set? Here are some related questions you might want to explore:

1. How are Poisson's and Laplace's equations applied in electrostatics?
2. What are the cylindrical coordinates for a given Cartesian vector?
3. What is the geometric interpretation of a cross product?
4. How do you prove the identity \( A \cdot (B \times C) \) algebraically?
5. How can we use divergence and curl to analyze vector fields?

**Tip:** Cylindrical coordinates are expressed as \( (r, \theta, z) \), where \( r = \sqrt{x^2 + y^2} \) and \( \theta = \tan^{-1}(y/x) \).

Q2: Given a vector A = 4xî + 2yĵ + z, express it in the cylindrical coordinate system and prove the following identity: A ⋅ (B × C) = B ⋅ (C × A) = C ⋅ (A × B).

Explore the transformation of a vector A = 4xî + 2yĵ + z into the cylindrical coordinate system and the proof of the vector triple product identity: A ⋅ (B × C) = B ⋅ (C × A) = C ⋅ (A × B). This problem involves key concepts in vector calculus and algebra, suitable for undergraduate-level students.

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