Question

The value of $\vec{\nabla}\left(\frac{f}{g}\right)$ at points where $g(x) \neq 0$ is given by

• A. $f \vec{\nabla} g - g \vec{\nabla} f$
• B. $\frac{g \vec{\nabla} f - f \vec{\nabla} g}{g^2}$
• C. $\frac{f \vec{\nabla} g - g \vec{\nabla} f}{g^2}$
• D. $g \vec{\nabla} f - f \vec{\nabla} g$

Hint:

Vector differential operators ($\vec{\nabla}$) mirror single-variable calculus rules: - Product Rule: $\vec{\nabla}(f g) = f \vec{\nabla} g + g \vec{\nabla} f$ - Quotient Rule: $\vec{\nabla}(f/g) = \frac{g \vec{\nabla} f - f \vec{\nabla} g}{g^2}$

Answer 1

The Correct Option is B

Step 1: Concept
This question tests the Quotient Rule for the gradient operator $\vec{\nabla}$ acting on two scalar fields $f$ and $g$.

Step 2: Key Formulas and Approach
The gradient of a scalar function $h$ is $\vec{\nabla} h = \frac{\partial h}{\partial x}\hat{i} + \frac{\partial h}{\partial y}\hat{j} + \frac{\partial h}{\partial z}\hat{k}$. Let $h = \frac{f}{g}$. Apply the single-variable quotient derivative rule along each coordinate direction.

Step 3: Step-by-step Explanation

• For the $x$-component of the gradient: \[ \frac{\partial}{\partial x}\left(\frac{f}{g}\right) = \frac{g \frac{\partial f}{\partial x} - f \frac{\partial g}{\partial x}}{g^2} \]
• For the $y$-component of the gradient: \[ \frac{\partial}{\partial y}\left(\frac{f}{g}\right) = \frac{g \frac{\partial f}{\partial y} - f \frac{\partial g}{\partial y}}{g^2} \]
• For the $z$-component of the gradient: \[ \frac{\partial}{\partial z}\left(\frac{f}{g}\right) = \frac{g \frac{\partial f}{\partial z} - f \frac{\partial g}{\partial z}}{g^2} \]
• Combine the components into vector form: \[ \vec{\nabla}\left(\frac{f}{g}\right) = \frac{\partial}{\partial x}\left(\frac{f}{g}\right)\hat{i} + \frac{\partial}{\partial y}\left(\frac{f}{g}\right)\hat{j} + \frac{\partial}{\partial z}\left(\frac{f}{g}\right)\hat{k} \] \[ = \frac{g \left( \frac{\partial f}{\partial x}\hat{i} + \frac{\partial f}{\partial y}\hat{j} + \frac{\partial f}{\partial z}\hat{k} \right) - f \left( \frac{\partial g}{\partial x}\hat{i} + \frac{\partial g}{\partial y}\hat{j} + \frac{\partial g}{\partial z}\hat{k} \right)}{g^2} \] \[ = \frac{g \vec{\nabla} f - f \vec{\nabla} g}{g^2} \]

Step 4: Final Answer
The quotient rule for gradient gives $\frac{g \vec{\nabla} f - f \vec{\nabla} g}{g^2}$. Thus, Option (B) is correct.