Vector Differentiation: D.N.

Ghayatadak
gdatta3.14@gmail.com

1. For three vectors show that : ā × (b̄ × c̄) + b̄ × (c̄ × ā) + c̄ × (ā × b̄) = 0.

2. For vectors ~a and ~b given respectively by ~a = 5t2 î + tĵ − t3 k̂ and ~b = sin tî − cos tĵ determine:
d d
(A) (~a · ~b) (B) (~a × ~b).
dt dt
3. Find the gradient of function f (x, y, z) = x2 + 3y 2 + z 3 at the point (1, 1, 1).

4. Find the divergence of the vector field V~ = x2 î + 2y 3 ĵ + z 4 k̂ at (1, 2, 3).

5. Find the divergence of the velocity field V~ = (x2 + y)î + (z − 2xy)ĵ + xy k̂ at (1, 1, 1).

6. Find the divergence of the velocity field V~ = x2 y î − (z 3 − 3x)ĵ + 4y 2 k̂ at (1, 2, 3).

7. The velocity field of an incompressible flow is given by F~ = 2xî + by ĵ − 4z k̂. Find the value of b.

8. Let F~ (x, y, z) = 2xî + 3y ĵ + 4z k̂ and u = x2 + y 2 + z 2 , then find div(uF~ ) at (1, 1, 1).

9. Determine the divergence of the vector field F~ = ρ sin φâρ + ρ2 z aˆφ + z cos φaˆz , where âρ , âφ and
âz are unit vectors in cylindrical co-ordinate system.

10. For what values of the constant a, b and c the vector function F~ = (x + y − az)î + (bx + 2y −
z)ĵ + (−x + cy + 2z)k̂ is irrotational.

11. If the vector function F~ = (3y − az)î + (bx − 2z)ĵ − (cy + z)k̂ is irrotational, then find the values
of the constants a, b and c.

12. If ~r be the position vector of a point, find the value(s) of n for which the vector function rn~r is

(A) irrotational, (B) solenoidal.

13. Find the equation of tangent and outward unit normal to the curve Γ, given by the equation
√

1
x2 + 4y 2 = 4 at a point P = 3,
2
14. Find the equation of the tangent plane at point (1, 1, 1) to the conicoild 3x2 − y 2 = 2z.

15. Find the angle between the planes 2x − y + z = 6 and x + y + 2z = 3 in R3 .

16. Find the angle between the surfaces x2 + y 2 + z 2 − 9 = 0 and z = x2 + y 2 − 3 at (2, −1, 2).

17. Find the directional derivative of the field u(x, y, z) = x2 − 3xz in the direction of the vector
(î + ĵ − 2k̂) at point (2, −1, 4).

18. Find the directional derivative of f (x, y) = x2 y 2 + xy at the point (2, 1) in the direction of a unit
vector which makes an angle of π/3 with the x-axis.

19. Find the directional derivative of the function xy 2 + yz 2 + zx2 along the tangent to the curve
x = t, y = t2 , z = t3 at the point (1, 1, 1).

1
20. Let φ(x, y, z) = 3y 2 + 3yz for (x, y, z) ∈ R3 . Find the direction derivative of φ in the direction of
x−1 y−2 z
the line = = at point (1, −2, 1).
2 −1 −2
21. Find the directional derivative of x2 yz + 4xz 2 at (1, −2, 1) in the direction of 2î − ĵ − 2k̂.
xy
22. Find the directional derivatives of f (x, y) = √ (x + y) at (1, 1) in the direction of the unit vector
2
at an angle of π/4 with y-axis.

23. Let u(x, y, z) = x2 − 2y + 4z 2 for (x, y, z) ∈ R3 . Then find the directional derivative of u in the
3 4
direction î − k̂ at the point (5, 1, 0).
5 5
R3 → Rbe defined by f (x, y, z) = sin x + 2ey/2 + z 2 . Find the maximum rate of change
24. Let f : 
π
of f at , 0, 1 .
4
p
25. Show that div(grad rn ) = n(n + 1)rn−2 where r = x2 + y 2 + z 2 .
 
2 p
2
26. Show that, ∇ f (r) = f 0 (r) + f 00 (r), where r = x2 + y 2 + z 2 .
r
~r
27. Find ϕ(r) such that ∇ϕ = and ϕ(1) = 0.
r5
28. Calculate ∇2 (rn ) and find its expression in terms of r and n, r being the distance of any point
(x, y, z) from the origin, n being a constant and ∇2 being the Laplace operator.

29. Show that if ϕ(x, y, z) is any solution of Laplace’s quation, then ∇ϕ is a vector that is both
solenoidal and irrotational.
∂2 ∂2 ∂2
30. Show that ∇ · ∇ϕ = ∇2 ϕ where ∇2 = + + denotes the Laplacian Operator.
∂x2 ∂y 2 ∂z 2
31. Let ϕ and ψ be two smooth scalar valued functions. Compute div(∇ϕ × ∇ψ)
p √
32. Let v=(v1 , v2 , v3 ) be a vector field on R3 where v1 = 1 + x2 + y 2 , v2 = 1 + z 2 and v3 =
p
1 + x2 y 2 z 2 . Evaluate div(curl v)

∂2 ∂2 ∂2
33. F~ being a vector, prove that curl (curl F~ ) = grad (div F~ ) − ∇2 F~ where ∇2 = + + .
∂x2 ∂y 2 ∂z 2
p
34. Show that div(grad rn ) = n(n + 1)rn−2 where r = x2 + y 2 + z 2 .
√ √ √
35. Find the lenght of the curve y = 4 − x2 form x = − 2 to x = 2.

36. Consdier a curve in three-dimensional space given in parametric form by x(t) = cos t, y(t) =
2t π
sin t, z(t) = ; 0 ≤ t ≤ . Find the length of the curve.
π 2
37. Find the arc length of the curve in the plane, whose
 equation in polar coordinates is given by
−π π
r = a cos θ, where θ varies over the interval , .
2 2

If you found any mistake(s) please report me at dng.maths@coep.ac.in n