Question

Match the LIST-I with LIST-II - Let $\vec{F}(x,y,z) = x\hat{i} + y\hat{j} + z\hat{k}$ and $\vec{G}(x,y,z) = xz\hat{i} + xy\hat{j} + yz\hat{k}$

Choose the correct answer from the options given below:

• A. A-IV, B-III, C-II, D-I
• B. A-IV, B-III, C-I, D-II
• C. A-III, B-IV, C-I, D-II
• D. A-II, B-IV, C-III, D-I

Hint:

For position vector $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$: $\text{div } \vec{r} = 3$ and $\text{curl } \vec{r} = \vec{0}$.

Answer 1

The Correct Option is A

Step 1 : Concept:
This question involves computing divergence and curl for vector fields $\vec{F}$ and $\vec{G}$, and using vector operator expansion rules.

Step 2 : Key Formulas and Approach:
1. $\text{div } \vec{V} = \frac{\partial V_x}{\partial x} + \frac{\partial V_y}{\partial y} + \frac{\partial V_z}{\partial z}$ 2. $\text{curl } \vec{V} = \nabla \times \vec{V}$ 3. $\text{div}(\vec{F} \times \vec{G}) = \vec{G} \cdot \text{curl } \vec{F} - \vec{F} \cdot \text{curl } \vec{G}$

Step 3 : Step-by-step Explanation:

• Item A: $\vec{F} = x\hat{i} + y\hat{j} + z\hat{k}$. \[ \text{div } \vec{F} = \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial z}{\partial z} = 1 + 1 + 1 = 3 \] Matches with IV.

• Item B: $\text{curl } \vec{F} = 0$. $\text{curl } \vec{G} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k}
\partial_x & \partial_y & \partial_z
xz & xy & yz \end{vmatrix} = \hat{i}(z - 0) - \hat{j}(0 - x) + \hat{k}(y - 0) = z\hat{i} + x\hat{j} + y\hat{k}$. So $\text{curl}(\vec{F} + \vec{G}) = \text{curl } \vec{F} + \text{curl } \vec{G} = z\hat{i} + x\hat{j} + y\hat{k}$. Matches with III.

• Item C: $\vec{G} = xz\hat{i} + xy\hat{j} + yz\hat{k}$. \[ \text{div } \vec{G} = \frac{\partial (xz)}{\partial x} + \frac{\partial (xy)}{\partial y} + \frac{\partial (yz)}{\partial z} = z + x + y \] Matches with II.

• Item D: Using vector identity: \[ \text{div}(\vec{F} \times \vec{G}) = \vec{G} \cdot \text{curl } \vec{F} - \vec{F} \cdot \text{curl } \vec{G} \] Since $\text{curl } \vec{F} = \vec{0}$: \[ \text{div}(\vec{F} \times \vec{G}) = -\vec{F} \cdot \text{curl } \vec{G} = -(x\hat{i} + y\hat{j} + z\hat{k}) \cdot (z\hat{i} + x\hat{j} + y\hat{k}) = -(xz + yx + zy) = -(xy + yz + zx) \] Matches with I.

Step 4 : Final Answer:
The correct matching is A-IV, B-III, C-II, D-I, which corresponds to option (A).