This document contains a mathematics tutorial sheet with 42 problems involving vector calculus concepts like derivatives, gradients, divergence, curl, and directional derivatives. The problems cover calculating derivatives of vector functions, finding velocities and accelerations of moving particles, determining irrotational and solenoidal vector fields, and applying vector calculus operations like gradient, divergence, and curl to scalar and vector functions.

1. Pandit Deendayal Petroleum University
School of Petroleum Technology
Tutorial Sheet- II
Mathematics –II
  
1. If r  i cos 2 t  3 j sin 2 t , find
 
dr d 2r
(i) , when t=0, (ii) , when t=1.
dt dt 2
 
dr 1 d 2r 2
(iii) , when t  and (iv) 2
, when t  .
dt 6 dt 3
 d   

         
2. If a  ti  t 2 j  t 3k , b  i cos t  j sin t , c  3t 2i  4tk , find a  b  c , where
dt
r  .
3. A particles moves along the curve x  3t 2 , y  t 2  2t , z  t 2 , where t is the time.
(a) Determine its velocity and acceleration at any time.
(b) Find the magnitudes of velocity and acceleration at time t  1 .
4. Find a unit tangent vector to any point on the space curve x  a cos t , y  a sin t , z  bt ,
where a and b are constants and t is the time.

3 d 1  dr 
5. Prove that r     r . , where r is a vector function of the scalar variable t.
dt  r  dt
  
r r a   
6. Differentiate     2 with respect to r , where r is a function of t and a is a
a.r r
constant vector.
 
  da da
7. If a is a unit vector, prove that a   .
dt dt
 
d 2   dr  d 2 r 
8. Evaluate 2  r    2  .
dt  dt  dt 
9. A particle moves along the curve x  4cos t , y  4sin t , z  6t , where t denotes the time.
Find (a) the velocity and the acceleration at any time t and (b) the magnitudes of the
velocity and the acceleration at time t   .

10. A particle moves so that its position vector r is given by  cos t,sin t,0 , where  is a
constant and t is the time. Show that
 
(a) The velocity v of the particle is perpendicular to r .

(b) The acceleration a is directed towards the origin and has magnitude proportional to
the distance from the origin.
 
(c) r  v  a constant vector.

2. 11. Find a unit tangent vector at the point t  1 of the space curve x  t , y  t 2 , z  t 3 where t
denotes the time.
    3 
12. If   x, y, z   xy z and u  xzi  xyj  yz k , find 2  u  at the point (2,-1,1).
2 2
x z
13. If u  x  y  z, v  x 2  y 2  z 2 , w  yz  zx  xy then prove that u.v w  0.
   
14. Show that grad  r .a   a, where a is a constant vector and r is position vector of the
point  x, y, z  .
 
15. Show that  a.  a. .
   
16. Show that  a. r  a where r   x, y, z  .
17. Prove that grad f u   f ' u  grad u.
    
18. If r  r , where r  xi  yj  zk , prove that f  r   f '  r  r.

   r
19. If r  r and r   x, y, z  , prove that  log r  2 .
r
  
20. If r   x, y, z  and r  r , Prove that f  r   r  0.
    
21. If f   y 2  z 2  x2  i   z 2  x 2  y 2  j   x 2  y 2  z 2  k , find div f . Also find its value
at the point (2,-1,-1).
     
  
22. Prove that div  r  a   b  2b .a, where r   x, y, z  and a , b are constant vectors.

23. Prove that r n r is solenoidal, when n  3 .
    
24. If u  x 2 zi  2 y 3 z 2 j  xy 2 zk , find curl u at the point (1,-1,1)
      
  
25. Prove that   r  a   b  a  b , where r   x, y, z  and a , b are constant vectors.
26. Find the constants a, b, c so that the vector
   
F   x  2 y  az  i  bx  3 y  z  j   4x  cy  2z  k is irrotational.
 
27. Find div f , where f  grad  x3  y3  z 3  3xyz  .
28. If    x3  y3  z 3  3xyz  , find curl  grad  .
  
29. If a and b are constant vectors and r   x, y, z  then prove that
 
(i)
  1 
. a      0
    1   3  a.r  b .r
(ii) a. b .    

a.b 
 3 .
  r    r  r5 r
30. Find the unit normal vector to the level surface x 2  y  z  4 at the point (2,0,0).
31. Find the directional derivative of f  x, y, z   x2 yz  4xz 2 at the point (1, -2, -1) in the
  
direction of the vector 2i  j  2k .

3. 32. Find the greatest value of the directional derivative of the function 2x 2  y  z 4 at the
point (2, -1, 1).
        2  
33. If A  x yzi  2 xz j  xz k and B  2 zi  yj  x k , Find the value of
2 3 2 2
xy

A  B at 
(1,0,-2).
 
   2  2 f 2 f
34. If f   2 x y  x  i   e  y sin x  j  x cos yk , Verify that
2 4 xy
 .
xy yx
   
1 r  
35. Prove that      3 , where r  r and r  xi  yj  zk .
r r
     
36. Prove that r n  nr n2 r , where r  r and r  xi  yj  zk .
     
37. Prove that .r   a  r   2r .a where r   x, y, z  and a is a constant vector.

38. Prove that the vector r 3r is solenoidal.
    
39. Prove that the vector curl  r n r   0, where r  r and r   x, y, z 
40. Prove that div grad r n  n  n  1 r n2 . or Prove that 2r n  n  n  1 r n2 .
41. What is the greatest rate of increase of u  xyz 2 at the point (1, 0, 3).
42. Find the maximum value of the directional derivatives of   x 2 yz at the point (1,4,1).