GATE 2020 MA – Question 60
Let $\mathbf F=(2x-2y\cos x)\hat i+(2y-y^2\sin x)\hat j+4z\hat k$. S is the closed surface of the tetrahedron bounded by x = 0,y = 0,z = 0,x+y+z = 1, with outward normal. Find $\iint_S\mathbf F\cdot\hat n\,dS$, to two decimal places.
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Correct answer: 1.30 to 1.40
Explanation
The divergence is $(2+2y\sin x)+(2-2y\sin x)+4=8$. By the divergence theorem the outward flux equals 8 times the tetrahedron’s volume. Its intercepts are all 1, so volume is 1/6. Therefore flux is $8/6=4/3=1.3333$, rounding to 1.33.