Question

If $\phi = 2x^{3}y^{2}z^{4}$ then $\nabla^{2}\phi = $

• A. $12xy^{2}z^{4} + 4x^{2}z^{4} + 20x^{3}y^{2}z^{3}$
• B. $12xy^{2}z^{4} + 4x^{3}z^{4} + 24x^{3}y^{2}z^{2}$
• C. $2x^{2}y^{2}z + 4x^{3}z^{4} + 24x^{3}y^{2}z^{2}$
• D. $4xy^{2}z + 4x^{2}z^{4} + 24x^{3}y^{2}z^{2}$

Hint:

$\nabla^2$ is simply the sum of the second partial derivatives of the scalar field.

Answer 1

The Correct Option is B

Step 1: Concept
The Laplacian operator is $\nabla^{2}\phi = \frac{\partial^{2}\phi}{\partial x^{2}} + \frac{\partial^{2}\phi}{\partial y^{2}} + \frac{\partial^{2}\phi}{\partial z^{2}}$.

Step 2: Meaning
We perform second-order partial differentiation for each variable:
$\frac{\partial \phi}{\partial x} = 6x^{2}y^{2}z^{4} \rightarrow \frac{\partial^{2}\phi}{\partial x^{2}} = 12xy^{2}z^{4}$.
$\frac{\partial \phi}{\partial y} = 4x^{3}yz^{4} \rightarrow \frac{\partial^{2}\phi}{\partial y^{2}} = 4x^{3}z^{4}$.
$\frac{\partial \phi}{\partial z} = 8x^{3}y^{2}z^{3} \rightarrow \frac{\partial^{2}\phi}{\partial z^{2}} = 24x^{3}y^{2}z^{2}$.

Step 3: Analysis
Combine the three terms: $12xy^{2}z^{4} + 4x^{3}z^{4} + 24x^{3}y^{2}z^{2}$.

Step 4: Conclusion
Final Answer: (B)