Let's recompute the curl of the vector field \( F(x, y, z) \) step by step for clarity.

### Given Vector Field
The vector field \( F(x, y, z) \) is:
\[
F(x, y, z) = (F_1, F_2, F_3) = (x^2 - y^2, 2xy, 2z)
\]

### Curl of \( F \)
The curl of a vector field \( F = (F_1, F_2, F_3) \) is given by:
\[
\nabla \times F = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right)
\]

Let's compute each component:

1. **First component**:
   \[
   \frac{\partial F_3}{\partial y} = \frac{\partial (2z)}{\partial y} = 0
   \]
   \[
   \frac{\partial F_2}{\partial z} = \frac{\partial (2xy)}{\partial z} = 0
   \]
   Therefore, the first component is:
   \[
   \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z} = 0 - 0 = 0
   \]

2. **Second component**:
   \[
   \frac{\partial F_1}{\partial z} = \frac{\partial (x^2 - y^2)}{\partial z} = 0
   \]
   \[
   \frac{\partial F_3}{\partial x} = \frac{\partial (2z)}{\partial x} = 0
   \]
   Therefore, the second component is:
   \[
   \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x} = 0 - 0 = 0
   \]

3. **Third component**:
   \[
   \frac{\partial F_2}{\partial x} = \frac{\partial (2xy)}{\partial x} = 2y
   \]
   \[
   \frac{\partial F_1}{\partial y} = \frac{\partial (x^2 - y^2)}{\partial y} = -2y
   \]
   Therefore, the third component is:
   \[
   \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} = 2y - (-2y) = 2y + 2y = 4y
   \]

### Final Curl Expression

Putting it all together, the curl of the vector field \( F(x, y, z) \) is:
\[
\nabla \times F = (0, 0, 4y)
\]

This vector field has only a non-zero component in the \( z \)-direction, and it is proportional to \( y \).

Would you like to proceed with the integration over the surface \( A \) using this curl, or would you like more details on this part?

Learn how to compute the curl of the vector field F(x, y, z) = (x^2 - y^2, 2xy, 2z) step-by-step with detailed explanations. This problem covers vector calculus topics including the curl formula and partial derivatives. Suitable for advanced undergraduate students.

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