If 𝑨=𝑥2𝑦𝑖̂−2𝑥𝑧𝑗̂+2𝑦𝑧𝑘,̂ then identify the value of 𝑐𝑢𝑟𝑙 (𝑐𝑢𝑟𝑙 𝑨).

Given the vector field \(\mathbf{A} = x^2y \hat{i} - 2xz \hat{j} + 2yz \hat{k}\), we are tasked with finding the curl of \(\mathbf{A}\), denoted as \(\nabla \times \mathbf{A}\).

The curl of a vector field \(\mathbf{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}\) is calculated as:

\[
\nabla \times \mathbf{A} = \left( \frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z} \right) \hat{i} - \left( \frac{\partial A_z}{\partial x} - \frac{\partial A_x}{\partial z} \right) \hat{j} + \left( \frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y} \right) \hat{k}
\]

### Components of \(\mathbf{A}\):
- \(A_x = x^2y\)
- \(A_y = -2xz\)
- \(A_z = 2yz\)

Now, let's calculate the curl component by component.

1. **For the \( \hat{i} \)-component**:
\[
\frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z} = \frac{\partial (2yz)}{\partial y} - \frac{\partial (-2xz)}{\partial z} = 2z - (-2x) = 2z + 2x
\]
Thus, the \( \hat{i} \)-component is \( (2z + 2x) \).

2. **For the \( \hat{j} \)-component**:
\[
\frac{\partial A_z}{\partial x} - \frac{\partial A_x}{\partial z} = \frac{\partial (2yz)}{\partial x} - \frac{\partial (x^2y)}{\partial z} = 0 - 0 = 0
\]
Thus, the \( \hat{j} \)-component is \( 0 \).

3. **For the \( \hat{k} \)-component**:
\[
\frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y} = \frac{\partial (-2xz)}{\partial x} - \frac{\partial (x^2y)}{\partial y} = -2z - x^2 = -(2z + x^2)
\]
Thus, the \( \hat{k} \)-component is \( -(2z + x^2) \).

### Final Result:
\[
\nabla \times \mathbf{A} = (2z + 2x) \hat{i} + 0 \hat{j} - (2z + x^2) \hat{k}
\]

Therefore, the curl of \(\mathbf{A}\) is:
\[
\nabla \times \mathbf{A} = (2z + 2x) \hat{i} - (2z + x^2) \hat{k}
\]

Do you want further details or have any questions?

Here are some related questions to deepen your understanding:
1. How is the curl of a vector field related to its divergence?
2. What is the physical interpretation of curl in vector calculus?
3. Can you find the divergence of the given vector field \(\mathbf{A}\)?
4. What conditions must hold for a vector field to have zero curl?
5. How does the curl operator behave in different coordinate systems, such as cylindrical or spherical?

**Tip:** When dealing with vector calculus, remember that the curl is often used to describe rotational motion in a field, especially in physics.

If 𝑨 = 𝑥²𝑦𝑖̂ − 2𝑥𝑧𝑗̂ + 2𝑦𝑧𝑘̂, then identify the value of 𝑐𝑢𝑟𝑙 (𝑐𝑢𝑟𝑙 𝑨).

Explore how to calculate the curl of the vector field A = x²y î − 2xz ĵ + 2yz k̂ using vector calculus. This problem involves applying the curl formula and understanding the rotational properties of vector fields. Ideal for students in advanced calculus or physics courses.

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