Dr B R Ambedkar National Institute of Technology, Jalandhar
B.Tech. (MAFC0102) Mathematics-II
Tutorial Sheet: Gradient, Divergence and Curl
1. If F = xy 2 i + 2x2 yz j − 3yz 2 k, find ∇ · F and ∇ × F at the point (1, −1, 1).
2. If F = (x2 − y 2 + 2xz) i + (xz − xy + yz) j + (z 2 + x2 ) k, then find
∇ · F,
∇(∇ · F),
∇ × F,
∇ · (∇ × F),
∇ × (∇ × F)
at the point (1, 1, 1).
3. Find ∇ · (∇ϕ) and ∇ × (∇ϕ) at (1, 1, 1) for ϕ = x2 y 3 z 4 .
4. Find |∇ϕ| if ϕ = 2xz 4 − x2 y at (2, −2, −1).
5. If ϕ = log x2 + y 2 + z 2 , then find ∇ϕ.
6. Find ∇(r), ∇ 1r , ∇(log r) where r = |r| and r = xi + yj + zk.
7. Prove that ∇(rn ) = nr n−2 r.
8. Find the unit normal to the surface x2 + xy + y 2 + xyz = 0 at the point (1, −2, 1).
9. Find the normal derivative to the surface x2 y + xz 2 = 0 at the point (−1, 1, 1).
10. Show that the vector F = 3y 4 z 2 i + 4x3 z 2 j − 3x2 y 2 k is solenoidal.
11. If F = (x + 3y) i + (y − 2z) j + (x + λz) k is solenoidal, find the value of λ.
12. Show that the vector F = (6xy + z 3 ) i + (3x2 − z) j + (3xz 2 − y) k is irrotational.
13. Let φ = x3 + y 3 + z 3 − 3xyz. Compute ∇φ, ∇ · ∇φ, ∇ × ∇φ at the point (1, 2, 3).
14. Show that
2
r
= ,
∇·
r
r
r = (x, y, z), r = |r| =
p
x2 + y 2 + z 2 , r ̸= 0.
15. Let F = xz 3 i − 2x2 yz j + 2yz 4 k. Find ∇ · F and ∇ × F at the point (1, −1, 1).
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