GATE 2026 EE Q.32: solution and answer

Official answer: 0Source: Official answer key, GATE 2026 EE (IIT Guwahati)

1. The surface $S$, the unit sphere, is closed, so the Gauss divergence theorem applies: $\oint {\oint }_S\vec{F}\cdot \overrightarrow{ds}={\iiint }_V(\nabla \cdot \vec{F})dV$, where $V$ is the unit ball $x^2+y^2+z^2\le 1$. $\vec{F}$ is smooth everywhere, so the theorem holds without exceptions.

2. Take the divergence one term at a time. $\frac{\partial }{\partial x}\sin (y)=0$, because $F_x=\sin (y)$ does not depend on $x$.

3. $\frac{\partial }{\partial y}\cos (x)=0$, because $F_y=\cos (x)$ does not depend on $y$.

4. $\frac{\partial }{\partial z}(5)=0$, because $F_z=5$ is a constant.

5. So $\nabla \cdot \vec{F}=0+0+0=0$ at every point, and $$\oint {\oint }_S\vec{F}\cdot \overrightarrow{ds}={\iiint }_{V}0dV=0$$

6. Check it directly. On the unit sphere the outward unit normal is $\hat{n}=x\hat{x}+y\hat{y}+z\hat{z}$, so $\vec{F}\cdot \hat{n}=x\sin (y)+y\cos (x)+5z$. The sphere is symmetric under $x\to -x$, $y\to -y$ and $z\to -z$. The term $x\sin (y)$ is odd in $x$, $y\cos (x)$ is odd in $y$ and $5z$ is odd in $z$, so each term integrates to zero over the sphere. Physically, the constant field $5\hat{z}$ flows in through the lower hemisphere and out through the upper hemisphere in equal amounts. The flux is $0.0$.