Question

Consider the vector field \( \mathbf{F} = a_x (4y - c_1 z) + a_y (4x + 2z) + a_z (2y + z) \) in a rectangular coordinate system \( (x, y, z) \) with unit vectors \( a_x, a_y, a_z \). If the field \( \mathbf{F} \) is irrotational (conservative), then the constant \( c_1 \) (in integer) is _________.

Hint:

To determine the constant in an irrotational vector field, compute the curl and set it equal to zero.

Answer 1

Correct Answer: 0

For a vector field to be irrotational (or conservative), its curl must be zero. The curl of \( \mathbf{F} \) is given by: \[ \nabla \times \mathbf{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \right) a_x + \left( \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \right) a_y + \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) a_z \] For the given vector field \( \mathbf{F} \), we compute each component of the curl. After solving the equations, we find: \[ c_1 = 0 \] Thus, the constant \( c_1 \) is \( 0 \).