To find the curl of the vector field $\vec{F}(x, y, z) = i + (x + yz) j + \left( xy - 3\sqrt{z} \right) k$, we use the formula:

$\text{curl } \vec{F} = \nabla \times \vec{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \right) \hat{i} - \left( \frac{\partial F_z}{\partial x} - \frac{\partial F_x}{\partial z} \right) \hat{j} + \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) \hat{k}$

Step 1: Identify Components of $\vec{F}$

The components of the vector field $\vec{F}$ are:

• $F_x = 1$
• $F_y = x + yz$
• $F_z = xy - 3\sqrt{z}$

Step 2: Compute Partial Derivatives

1. Partial derivatives for the $\hat{i}$ component:

   • $\frac{\partial F_z}{\partial y} = \frac{\partial}{\partial y} (xy - 3\sqrt{z}) = x$
   • $\frac{\partial F_y}{\partial z} = \frac{\partial}{\partial z} (x + yz) = y$

   Thus, the $\hat{i}$ component of the curl is: $\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} = x - y$

2. Partial derivatives for the $\hat{j}$ component:

   • $\frac{\partial F_z}{\partial x} = \frac{\partial}{\partial x} (xy - 3\sqrt{z}) = y$
   • $\frac{\partial F_x}{\partial z} = \frac{\partial}{\partial z} (1) = 0$

   Thus, the $\hat{j}$ component of the curl is: $- \left( \frac{\partial F_z}{\partial x} - \frac{\partial F_x}{\partial z} \right) = -(y - 0) = -y$

3. Partial derivatives for the $\hat{k}$ component:

   • $\frac{\partial F_y}{\partial x} = \frac{\partial}{\partial x} (x + yz) = 1$
   • $\frac{\partial F_x}{\partial y} = \frac{\partial}{\partial y} (1) = 0$

   Thus, the $\hat{k}$ component of the curl is: $\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} = 1 - 0 = 1$

Step 3: Combine the Components

The curl of the vector field is: $\nabla \times \vec{F} = (x - y) \hat{i} - y \hat{j} + 1 \hat{k}$

or, in vector form: $\nabla \times \vec{F} = (x - y) \, \vec{i} - y \, \vec{j} + \vec{k}$

Would you like further details on any specific steps?

Here are five additional questions you might explore:

1. How does the curl change if we alter the components of the vector field?
2. What are the physical interpretations of the curl in various contexts?
3. How would we find the divergence of this vector field?
4. How can we interpret the result of a zero or non-zero curl?
5. How does this curl computation relate to applications in electromagnetism?

Tip: Understanding the curl's geometric meaning can enhance your grasp of fluid rotation and circulation in vector fields!