Understanding the Curl of a Vector Field

The question asks us to find the curl of the given vector field $ \mathbf{v} = yz \hat{i} + 3zx \hat{j} + z \hat{k} $. The curl is a vector operation that describes the infinitesimal rotation of the vector field. It is often denoted as $ \nabla \times \mathbf{v} $.

Calculating the Curl

The curl of a vector field $ \mathbf{v} = v_x \hat{i} + v_y \hat{j} + v_z \hat{k} $ is calculated using the determinant of a special matrix:

$$ \nabla \times \mathbf{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ v_x & v_y & v_z \end{vmatrix} $$

For the given vector field $ \mathbf{v} = yz \hat{i} + 3zx \hat{j} + z \hat{k} $, we have:

• $ v_x = yz $
• $ v_y = 3zx $
• $ v_z = z $

Step-by-Step Curl Calculation

Let's substitute the components into the determinant formula:

Now, we expand the determinant:

• $ \hat{i} $ component: $ \left( \frac{\partial}{\partial y}(z) - \frac{\partial}{\partial z}(3zx) \right) $
• $ \hat{j} $ component: $ - \left( \frac{\partial}{\partial x}(z) - \frac{\partial}{\partial z}(yz) \right) $
• $ \hat{k} $ component: $ \left( \frac{\partial}{\partial x}(3zx) - \frac{\partial}{\partial y}(yz) \right) $

Let's calculate the partial derivatives:

• $ \frac{\partial}{\partial y}(z) = 0 $
• $ \frac{\partial}{\partial z}(3zx) = 3x $
• $ \frac{\partial}{\partial x}(z) = 0 $
• $ \frac{\partial}{\partial z}(yz) = y $
• $ \frac{\partial}{\partial x}(3zx) = 3z $
• $ \frac{\partial}{\partial y}(yz) = z $

Substitute these back into the expansion:

• $ \hat{i} $ component: $ (0 - 3x) = -3x $
• $ \hat{j} $ component: $ - (0 - y) = - (-y) = y $
• $ \hat{k} $ component: $ (3z - z) = 2z $

Combining these components, the curl of the vector field is:

$$ \nabla \times \mathbf{v} = -3x \hat{i} + y \hat{j} + 2z \hat{k} $$

Matching with Options

Comparing our result $ -3x \hat{i} + y \hat{j} + 2z \hat{k} $ with the given options:

• Option 1: $ -3x \hat{i} + y \hat{j} + 2z \hat{k} $ (Matches)
• Option 2: $ 3x \hat{i} - y \hat{j} + 2z \hat{k} $
• Option 3: $ -3x \hat{i} - y \hat{j} - 2x \hat{k} $
• Option 4: $ 3x \hat{i} + y \hat{j} - 2z \hat{k} $

The calculated curl matches the first option.