2000

1) In what direction from the point (-1,1,1) is the directional derivative of f=x2yz3 a maximum ? Compute its magnitude.

[10M]

2) Show that

(i) (A+B)⋅(B+C)×(C+A)=2A⋅B×C

(ii)∇×(A×B)=(B.∇)A−B(∇.A)−(A.∇)B+A(∇.B)

[10M]

3) Evaluate ∬S→F⋅ˆndS, where F=2xyi+yz2j+xzk and S is the surface of the parallelcpiped bounded by x=0, y=0,z=0, x=2, y=1 and z=3.

[10M]

1999

1) If →a,→b,→c are the position vectors of A,B,C prove that →a×→b+→b×→c+→c×→a is a vector perpendicuar to the plane ABC.

[10M]

2) If →F=∇(x3+y3+z3−3xyz), find ∇×→F.

[10M]

3) Evaluate ∫C(e−xsinydx+e−xcosydy) (Green’s theorem), where C is the rectangle whose vertices are (0,0),(π,0),(π,π2) and (0,π2).

[10M]

4) If X,Y,Z are the components of contravariant vector in rectangular Cartesian coordinates x,y,z in a three dimensional space, show that the components of the vectors in cylinderal coordinates r,θ,z are Xcosθ+Ysinθ,−Xrsinθ+Yrcosθ,Z

[10M]

5) Show that ∇2ϕ=gij(∂2ϕ∂xi∂xj−∂ϕ∂xl{1ij}) where ϕ is a scalar function of coordinates xi.

[10M]

1998

1) If r1 and r2 are the vectors joining the fixed points A(x1,y1,z1) and B(x2,y2,z2) respectively to a variable point P(x,y,z), then find the value of grad(r1⋅r2) and curl(r1×r2).

[10M]

2) Show that (a×b)×c=a×(b×c) if either b=0 (or any other vector is 0) or c is collinear with a or b is orthogonal to a and c(both).

[10M]

1997

1) Prove that if →A,→B and ˙C are three given non-coplanar vectors, then any vector →F can be put in the form F=α→B×→C+β→C×→A+γ→A×→B. For given determine α, β, γ.

[10M]

2) Verify Gauss theorem for →F=4xi−2y2ˆj+z2ˆk taken over the region bounded by x2+y2=4 z=0 and z=3.

[10M]

1996

1) If x→i+yˆj+zˆk and r=|→r|, show that:
(i) →r× grad f (r)=0
(ii) div(rn→r)=(n+3)rn

[15M]

2) Verify Gauss’ divergence theorem for →F=xyˆi+z2ˆj+2yzˆk on the tetrahedron x=y=z=0,x+y+z=1

[15M]

1995

1) Let the region V be bounded by the smooth surface S and let n denote outward drawn unit normal vector at a point on S. If ϕ is harmonic in V, show that ∫S∂ϕ∂ndS=0

[20M]

2) In the vector field v(x), let there exist a surface on which v=0$.$Showthat,atanarbitrarypointofthissurface,curlv=0$$ is tangential to the surface or vanishes.

[20M]