Question

Consider a cube of unit edge length and sides parallel to co-ordinate axes, with its centroid at the point (1, 2, 3). The surface integral \[ \int_A \vec{F} \cdot d\vec{A} \] of the vector field \[ \vec{F} = 3x\,\hat{i} + 5y\,\hat{j} + 6z\,\hat{k} \] over the entire surface A of the cube is ________________.

• A. 14
• B. 27
• C. 28
• D. 31

Hint:

Surface integrals over closed surfaces are easiest using the Divergence Theorem: convert to a volume integral of divergence.

Answer 1

The Correct Option is A

We apply the Divergence Theorem:
\[ \int_A \vec{F}\cdot d\vec{A} = \iiint_V (\nabla\cdot \vec{F})\, dV. \] For the field \(\vec{F}=3x\hat{i}+5y\hat{j}+6z\hat{k}\), the divergence is \[ \nabla\cdot\vec{F} = 3 + 5 + 6 = 14. \] The cube has unit edge length, so its volume is \[ V = 1. \] Hence the flux equals \[ 14 \times 1 = 14. \] Final Answer: 14