List of top Real Analysis Questions on Vector Calculus (Divergence Theorem) asked in GATE MA

Let \( \alpha = \iint_S \vec{F} \cdot \hat{n} \, dS \), where \( \vec{F} = (2x + 3z)\hat{i} + (xz - y)\hat{j} + (y^2 + 2z)\hat{k} \) and \( S \) is the sphere with centre at \( (3, -1, 2) \) and radius \( 9 \). Here \( \hat{n} \) is the unit normal drawn outward and \( \hat{i}, \hat{j}, \hat{k} \) are unit vectors.

Then the value of \( \dfrac{1}{36\pi}\alpha \) is equal to ______. (Answer in integer)