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.