Question

In a three-dimensional \( xyz \)-space, if \( \vec{v} = 3z \hat{i} + 2z \hat{j} + z \hat{k} \), and \( \text{curl}(\vec{v}) = a \hat{i} + b \hat{j} + c \hat{k} \), then the value of \( (a + b + c) \) is ________________ (in integer).

Hint:

The curl of a vector field measures the rotation of the field at each point. The components are derived from the partial derivatives of the field components.

Answer 1

Correct Answer: 1

The curl of a vector field \( \vec{v} = P \hat{i} + Q \hat{j} + R \hat{k} \) is given by: \[ \text{curl}(\vec{v}) = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \hat{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \hat{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \hat{k} \] Here, \( P = 3z \), \( Q = 2z \), and \( R = z \). Now, calculating the partial derivatives: For \( \hat{i} \)-component: \[ \frac{\partial R}{\partial y} = 0, \quad \frac{\partial Q}{\partial z} = 2 \quad \Rightarrow \quad \text{Component of curl in } \hat{i} = 0 - 2 = -2 \] For \( \hat{j} \)-component: \[ \frac{\partial P}{\partial z} = 3, \quad \frac{\partial R}{\partial x} = 0 \quad \Rightarrow \quad \text{Component of curl in } \hat{j} = 3 - 0 = 3 \] For \( \hat{k} \)-component: \[ \frac{\partial Q}{\partial x} = 0, \quad \frac{\partial P}{\partial y} = 0 \quad \Rightarrow \quad \text{Component of curl in } \hat{k} = 0 - 0 = 0 \] Thus, \[ \text{curl}(\vec{v}) = -2 \hat{i} + 3 \hat{j} + 0 \hat{k} \] Therefore, \( a = -2 \), \( b = 3 \), and \( c = 0 \), and \[ a + b + c = -2 + 3 + 0 = 1 \] Final Answer: 1