Identities and Equations
We will cover following topics
Vector Identities
∇ operator applied once to point functions
i) If f is a scalar point function and →G is a vector point function, then ∇⋅(f→G)=∇f⋅→G+f(∇⋅→G).
ii) If f is a scalar point function and →G is a vector point function, then ∇×(f→G)=∇f×→G+f(∇×G).
iii) If →F and →G are vector point functions, then ∇(→F⋅→G)=(→F⋅∇)→G+(→G⋅∇)→F+→F×(∇×→G)+→G×(∇×→F)
iv) If →F and →G are vector point functions then ∇⋅(→F×→G)=→G⋅(∇×→F)−→F⋅(∇×→G) i.e., div →F×→G=→G⋅Curt→F−→F⋅Curt→G
v) If →F and →G are vector product functions, then ∇×(→F×→G)=→F(∇⋅→G)−→G(∇⋅→F)+(→G⋅∇)→F−(→F⋅∇)→G
PYQs
Vector Identities
1) Let →v=v1ˆi+v2ˆj+v3ˆk. Show that curl(curl→v)=grad(div→v)−∇2→v
[2018, 12M]
2) Calculate ∇2(rn) and find its expression in terms of r and n, r being the distance of any point (x,y,z) from the origin, n being a constant and ∇2 being the Laplace operator.
[2013, 10M]
3) If u and v are two scalar fields and →f is a vector field, such that u→f=gradv, find the value of →f⋅curl→f.
[2011, 10M]
4) Prove that div(f→V)=f(div¯V)+(grad.f)→V, where f is a scalar function.
[2010, 20M]
5) Prove that ∇2f(r)=d2fdr2+2rdfdr where r=(x2+y2+z2)1/2. Hence find f(x) such that ∇2f(r)=0.
[2008, 15M]
6) Show that div(gradrn)=n(n+1)rn−2, where r=√x2+y2+z2.
[2009, 12M]
7) Show that curl (k×grad1r)+grad(kgrad1r)=0 where r is the distance from the origin and k is the unit vector in the direction OZ.
[2005, 15M]
8) Prove the identity ∇(¯A⋅¯B)=(¯B.¯A)¯A+(¯A⋅∇)¯B+¯B×(∇ׯA)+¯A×(∇ׯB).
[2004, 15M]
9) Derive the identity ∭V(ϕ∇2ψ−ψ∇2ϕ)dV=∬S(ϕ∇ψ−ψ∇ϕ)ˆndS where V is the volume bounded by the closed surface S.
[2004, 15M]
10) Show that if a′, b′ and c′ are the reciprocals of the non-coplanar vectors a, b and c, then any vector r may be expressed as r=(r.a′)a+(r.c′)c.
[2003, 12M]
11) Prove the identity ∇A2=2(A.∇)A+2A×(∇×A) where ∇=ˆi∂∂x+ˆj∂∂y+ˆk∂∂z.
[2003, 15M]
12) Let ¯R be the unit vector along the vector ¯r(t). Show that ¯RׯdRdt=¯rr2ׯdrdt where r=|r|.
[2002, 15M]
13) Show that (curl¯v)=grad(div¯v)−∇2v.
[2002, 15M]
14) Show that curla×rr3=−ar3+3rr5(a⋅r), where a is constant vector.
[2001, 12M]
Vector Equations
1) Prove that the vector →a=3→i+ˆj−2ˆk, →b=−ˆi+3ˆj+4ˆk, →c=4ˆi−2ˆj−6ˆk can from the sides of a triangle. Find the length of the medians of the triangle.
[2016, 10M]
2) If →A=x2yz→i−2xz3→j+xz2→k, →B=2z→i+y→j−x2→k, find the value of ∂2∂x∂y(→A+→B) at (1,0,−2).
[2012, 12M]
3) If ¯A=2ˆi+ˆk, ¯B=ˆi+ˆj+ˆk, ¯C=4ˆi−3ˆj−7ˆk, determine a vector ¯R satisfying the vector equation ¯RׯB=¯CׯB and ¯R⋅¯A=0.
[2006, 15M]
4) Let D be a closed and bounded region having boundary S. Further, let f is a scalar function having second partial derivatives defined on it. Show that ∬s( fgradf )ˆnds=∭v[|gradf|2+f∇2f]dv. Hence ∬sf( grad f)⋅ˆnds or otherwise evaluate for f=2x+y+2z over s:x2+y2+z2=4.
[2002, 15M]