2004

1) Evaluate ∫e→F⋅→dr for the field →F= grad (xy2z3) where e is the ellipse in which the plane z=2x+3y cuts the cylinder x2+y2=12 counterclockwise as viewed from the positive end of the z-axis looking towards the origin.

[10M]

2) Show that div(→A×→B)=→B⋅curl→A−→A⋅curl→B.

[10M]

3) Evaluate curl [(2→i−→j+3→k)×→rr3], where →r=x→i+y→j+z→k and r3=x3+y3+z3.

[10M]

4) Evaluate ∬S(x→i+y→j+z→k)⋅→nds, where S is the surface x+y+z=1 lying in the first octant.

[10M]

5) Evaluate ∇2u in spherical polar coordinates.

[10M]

2003

1) Find expressions for curvature and torsion at a point on the curve x=acosθ, y=asinθ, z=aθcotβ

[10M]

2) If ˉr is the position vector of the point (x,y,2) with respect to the origin, prove that ∇2f(r)=f′′(r)+2rf′(r)
Find f(r) such that Δ2f(r)=0.

[10M]

3) If →F is solenoidal, prove that Curl Curl Curl Curl →F=∇4→F.

[10M]

4) Verify Stoke’s Theorem when
¯F=(2xy−x2)ˉi−(x2−y2)ˉd
and C is the boundary of the region enclosed by the parabolas y2=x and x2=y

[10M]

TBC

5) Express ∇×→F and ∇2Φ in cylindrical co-ordinates,

[10M]

2002

1) Find the eurvature and torsion of the curve x=(2t+1)t−1, y=t2t−1, z=t+2. Interpret your answer.

[10M]

2) State Stoke’s theorem and then verify it for ˉA=(x2+1)ˉi+xyˉj integrated round the square in the plane

z=0 whose sides are along the lines

[10M]

1. Prove that: (i) →∇×(→A×→B)=→A(→∇⋅→B)−→B(→∇⋅→A)+(→B⋅→∇)→A−(→A⋅→∇)→B

[10M]

(ii) curl→a×→rr3= −→ar3+3→rr3(→a⋅→r), →a = constant vector.

4) Show that if →A≠→0 and both of the conditions →A⋅→B=→A→C and →A×→B=→A×→C hold simultancously then →B=→C but if only one of these conditions holds then ˉB≠ˉC necessarily.

[10M]

5) Prove the following
(i) If u1,u2,u3 are general coordinates, then
∂→r∂u1×∂→r∂u2×∂vecr∂u3 and →∇u1,→∇u2,→∇u3 are reciprocal system of vectors.

[5M]

(ii) (∂→ru1⋅∂→ru2×∂→ru3)(→∇u1⋅→∇u2×→∇u3)=1.

[5M]

2001

1) Find an equation for the plane passing through the points P1(3,1,−2), P2(−1,2,4), P3(2,−1,1) by using vector method.

[10M]

2) Prove that ∇×(∇×ˉA)=−∇2ˉA+∇(∇⋅ˉA).

[10M]

3) If ∇⋅ˉE,∇⋅ˉH,∇×ˉE=−∂ˉH∂t, ∇×ˉH=∂ˉE∂t Show that ˉE & ˉH satisfy ∇2u=−∂2ˉu∂t2

[10M]

4) Given the space curve x=t, y=t2, z=23t3. Find
(i) the curvature ρ (ii) the torsion τ.

[10M]

5) If F=(y2+z2−x2)i+(z2+x2−y2)j+(x2+y2−z2)k evaluate ∬ScurlˉF⋅ˆnds, taken over the portion of the surface x2+y2+z2−2ax+az=0 above the plane z=0 and verify Stokes’ theorem.

[10M]

2000

1) Solve for x the vector equation px+x×a=b,p≠0

[10M]

2) Prove the identities:
(i) Curl grad ϕ=0,(ii) div curl f=0.

If →OA=ai,→OB=aj,→OC=ak form three coterminous edges of a cube and s denotes the surface of the cube, evaluate ∫′1(x3−yz)i−2x2yj+2k} nds by expressing it as volume integral, Where n is the unit outward normal to ds.

[20M]