Question

The vector \(\vec{v}\) is defined as \[ \vec{v} = zx\,\hat{i} + 2xy\,\hat{j} + 3yz\,\hat{k}. \] Which one of the following is the CORRECT value of divergence of \(\vec{v}\), evaluated at the point \((x,y,z) = (3,2,1)\)?

• A. 0
• B. 3
• C. 14
• D. 13

Hint:

Divergence measures the net flux density of a vector field at a point.
Always apply \(\nabla \cdot \vec{v} = \partial v_x/\partial x + \partial v_y/\partial y + \partial v_z/\partial z\).
Carefully substitute the given point values after simplification.

Answer 1

The Correct Option is D

Step 1: Recall that the divergence of a vector field \(\vec{v} = v_x \hat{i} + v_y \hat{j} + v_z \hat{k}\) is \[ \nabla \cdot \vec{v} = \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z}. \] Step 2: Identify the components: \[ v_x = zx,\quad v_y = 2xy,\quad v_z = 3yz. \] Step 3: Compute partial derivatives. \[ \frac{\partial v_x}{\partial x} = \frac{\partial (zx)}{\partial x} = z, \] \[ \frac{\partial v_y}{\partial y} = \frac{\partial (2xy)}{\partial y} = 2x, \] \[ \frac{\partial v_z}{\partial z} = \frac{\partial (3yz)}{\partial z} = 3y. \] Step 4: Add them: \[ \nabla \cdot \vec{v} = z + 2x + 3y. \] Step 5: Substitute \((x,y,z) = (3,2,1)\): \[ \nabla \cdot \vec{v} = 1 + 2(3) + 3(2) = 1 + 6 + 6 = 13. \] Step 6: Correct divergence value is \(\boxed{13}\). Hence the correct option is (D).