1988

1) Define the divergence of a vector point function, prove that div(→u×→v)=→v⋅curl→u−→ucurl→v.

2) Using Gauss divergence theorem, evaluate ∬s(xˆi+yˆi+z2ˆk)⋅ˆnds where S is the closed surface bounded by the cone x2+y2=2 and the plane Z=1 and ˆn is the outward unit normal to S.

1987

1) Show that for a vector field →f, curl (curl →f)=grad(div→f)−∇2→f.

2) If →r is the position vector to a point whose distance from the origin is r, prove that div→f=0 if →f=→rr3.

3) Prove that for a three vectors →a,→b,→c →a×(→b×→c)=→b(→a⋅→c)−→c(→a⋅→b) and explain its geometric meaning.

1986

1) Let →a,→b be given vectors in the three dimensional Euclidean space E3 and let ϕ(→x) be a scalar field of the vectors →x also of E3. If ϕ(→x)=(→x×→a)⋅(→x×→b), show that
gradϕ(i.e,∇ϕ(→x))=→b×(→x×→a)+→a×(→x×→b)

2) If →f,→g are two vector fields in E3 and if ‘div’, ‘curl’ are defined on an open set S⊂E3 show that div(→f×→g)=→g.curl→f−→f⋅curl→g.

1985

1) If P, Q, R are points (3,−2,−1), (1,3,4), (2,1,−2) respectively. Find the distance from P to the plane OQR, where ′O′ is the origin.

2) Find the angle between the tangents to the curve →r=t2ˆi−2tˆj+t3ˆk at the points t=1 and t=2.

3) Find div F and curl F, where F=∇(x3+y3+z3−3xyz).

1983

1) Prove that curl (curl F ) = grad div F−∇2F.