Question

The divergence of the vector field \( \mathbf{P} = x^2 y \hat{i} + xyj \) is:

• A. \( 2xy + x \)
• B. \( 2xy + z \)
• C. \( x^2 + y \)
• D. \( x^2 y + xz \)

Hint:

Divergence measures the rate of expansion at a point in a vector field.

Answer 1

The Correct Option is A

The divergence of a vector field \( \mathbf{P} = P_x \hat{i} + P_y \hat{j} + P_z \hat{k} \) is given by:
\[ \nabla \cdot \mathbf{P} = \frac{\partial P_x}{\partial x} + \frac{\partial P_y}{\partial y} + \frac{\partial P_z}{\partial z} \] Given \( P_x = x^2 y \) and \( P_y = xy \):
\[ \frac{\partial (x^2 y)}{\partial x} = 2xy, \quad \frac{\partial (xy)}{\partial y} = x \] \[ \nabla \cdot \mathbf{P} = 2xy + x \]