Question

Let $\vec{F}(x, y, z) = x^2 y \hat{i} + z \hat{j} + x y z \hat{k}$, then $\text{div} \vec{F}$ is given by

• A. $x y$
• B. 0
• C. $2 x y + x y z$
• D. $3 x y$

Hint:

Divergence returns a scalar field ($\nabla \cdot \vec{F}$), whereas Curl returns a vector field ($\nabla \times \vec{F}$). Pay close attention to the dot vs cross product!

Answer 1

The Correct Option is D

Step 1: Concept
The divergence of a vector field $\vec{F} = F_x \hat{i} + F_y \hat{j} + F_z \hat{k}$ is a scalar field representing the net flux per unit volume exiting an infinitesimal region. It is calculated via the dot product $\nabla \cdot \vec{F}$.

Step 2: Key Formulas and Approach
\[ \text{div} \vec{F} = \nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \] Here $F_x = x^2 y$, $F_y = z$, and $F_z = x y z$.

Step 3: Step-by-step Explanation

• Differentiate $F_x$ with respect to $x$: \[ \frac{\partial}{\partial x}(x^2 y) = 2 x y \]
• Differentiate $F_y$ with respect to $y$: \[ \frac{\partial}{\partial y}(z) = 0 \]
• Differentiate $F_z$ with respect to $z$: \[ \frac{\partial}{\partial z}(x y z) = x y \]
• Sum the partial derivatives: \[ \text{div} \vec{F} = 2 x y + 0 + x y = 3 x y \]

Step 4: Final Answer
The divergence of $\vec{F}$ is $3 x y$. Thus, Option (D) is correct.