Question:
If $r$ is the position vector of any point on a closed surface S that encloses the volume V then $\iint_{S}(\overline{r} \cdot ds)$ is equal to}
If $r$ is the position vector of any point on a closed surface S that encloses the volume V then $\iint_{S}(\overline{r} \cdot ds)$ is equal to}
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$\text{div}(r) = 3$ is a fundamental constant in 3D vector calculus.
Updated On: Jul 3, 2026
- $\frac{1}{2}V$
- V
- 2V
- 3V
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The Correct Option is D
Solution and Explanation
Step 1: Concept
This is a direct application of Gauss Divergence Theorem: $\iint_{S} \overline{r} \cdot dS = \iiint_{V} (\nabla \cdot \overline{r}) dV$.
Step 2: Meaning
The position vector is $\overline{r} = xi + yj + zk$.
Step 3: Analysis
Calculate $\nabla \cdot \overline{r} = \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial z}{\partial z} = 1 + 1 + 1 = 3$.
Step 4: Conclusion
The integral becomes $\iiint_{V} 3 dV = 3 \iiint_{V} dV = 3V$. Final Answer: (D)
This is a direct application of Gauss Divergence Theorem: $\iint_{S} \overline{r} \cdot dS = \iiint_{V} (\nabla \cdot \overline{r}) dV$.
Step 2: Meaning
The position vector is $\overline{r} = xi + yj + zk$.
Step 3: Analysis
Calculate $\nabla \cdot \overline{r} = \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial z}{\partial z} = 1 + 1 + 1 = 3$.
Step 4: Conclusion
The integral becomes $\iiint_{V} 3 dV = 3 \iiint_{V} dV = 3V$. Final Answer: (D)
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