Question

If \( \nabla \phi = 2xy^2 \hat{i} + x^2z^2 \hat{j} + 3x^2y^2z^2 \hat{k} \), then \( \phi(x,y,z) \) is:

• A. \( \phi = xyz^2 + c \)
• B. \( \phi = x^3 y^2 z^2 + c \)
• C. \( \phi = x^2 y^2 z^3 + c \)
• D. \( \phi = x^3 y^2 + c \)

Hint:

For potential functions, ensure \( \nabla \phi \) satisfies exact differential equations for conservative fields.

Answer 1

The Correct Option is B

Step 1: Integrating \( \frac{\partial \phi}{\partial x} = 2xy^2 \). \[ \phi = \int 2xy^2 dx = x^2 y^2 + f(y,z). \]
Step 2: Integrating \( \frac{\partial \phi}{\partial y} = x^2z^2 \). \[ \frac{\partial}{\partial y} (x^2 y^2 + f(y,z)) = x^2 z^2. \] Solving, we find: \[ f(y,z) = y^2 z^2 + g(z). \]
Step 3: Integrating \( \frac{\partial \phi}{\partial z} = 3x^2 y^2 z^2 \). \[ \frac{\partial}{\partial z} (x^2 y^2 + y^2 z^2 + g(z)) = 3x^2 y^2 z^2. \] Solving, we find: \[ \phi = x^3 y^2 z^2 + c. \]
Step 4: Selecting the correct option. Since \( \phi = x^3 y^2 z^2 + c \) matches, the correct answer is (B).