1994

1) Show that rn→r is an irrotational vector for any value of n, but is solenoidal only if n=−3.

[10M]

2) If →F=y→i+(x−2xz)→j−xy→k, evaluate ∬S(∇×→F)⋅→ndS,
where S is the surface of the sphere x2+y2+z2=a2 above the xy plane.

[10M]

1993

1) Prove that the angular velocity of rotation at any point is equal to one half of the curl of the velocity vector V.

[10M]

–

2) Evaluate ∬s∇×FhdS where S is the upper half surface of the unit sphere x2+y2+z2=1 and →F=zˆi+xˆj+yˆk.

[10M]

1992

1) If →f(x,y,z)=(y2+z2)→i+(z2+x2)→j+(x2+y2)→k, then calculate ∫c→f→dx, where C consists of (i) the line segment from (0,0,0) to (1,1,1) (ii) the three line segments AB,BC and CD, where A,B,C and D are respectively the points (0,0,0), (1,0,0), (1,1,0) and (1,1,1) (iii) the curve ˉx+→u+¯u2j+¯u2ku, from 0 to 1

[10M]

2) If →a and →b are constant vectors, show that

(i) div{x×(→a×→x)}=−2→x→a
(ii) div{(→a×→x)×(→b×→x)}=2→a⋅(→b×→x)−2b⋅(→a×x)

[10M]

3) Obtain the formula

[10M]

1991

1) If ϕ be a scalar point function and F be a vector point function, show that the components of F normal and tangential to surface ϕ=0 at any point there of are (F.∇ϕ)∇ϕ(∇ϕ)2 and ∇ϕ×(F×∇ϕ)(∇ϕ)2.

2) Find the value of ∫ curl F. dS taken over the portion of the surface x2+y2−2ax+az=0, for which z≥0 when F=(y2+z2−x2)ˆi+(z2+x2−y2)ˆj+(x2+y2−z2)ˆk

1989

1) Define the curl of a vector point function.

2) Prove that ∇×(→rr2)=0 where →r=(x,y,z) and r=|→r|.