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If V
is a vector field, prove that:
∇×(∇×V)=∇(∇⋅V)−(∇^2)V.
Given that U
is a function of x,y
and z
and A
a vector field, prove that:
∇×(UA)=(∇U)×A+U(∇×A).
Show that if f and g are two functions such that
f o g is onto, then g need not be onto.
a) Define Bijective function and Surjective function.
b) Find the limiting value of i) lim
𝑥→7
𝑥
2+2𝑥−63
𝑥−7
ii) lim
𝑥→∞
5𝑥−1
5𝑥+1
.
c) Test the continuity of following functions at x= -2 and x=3
𝑓(𝑥) = {
7𝑥 − 1 𝑖𝑓 𝑥 > 3
𝑥
2 − 8 𝑖𝑓 − 2 ≤ 𝑥 ≤ 3
8𝑥 + 3 𝑖𝑓 𝑥 < −2
If A and Bare vector fields, prove the following:
nabla(A* B)=(B* nabla)A+(A* nabla)B+B*( nabla* A)+A*( nabla* B) .
[SADT10] Let r=x hat i +y hat j +z hat k and r = ||r||
Show that:
nabla(lnr)= r r^ 2 .
and
nabla*(r^ n r)=0 .
[SADT9] The Laplacian of a function f of n variables x 1 ,x 2 ,*** x n denoted nabla^ 2 f is defined by
nabla^ 2 f(x 1 ,x 2 ,***,x n ):= partial^ 2 f partial x 1 ^ 2 + partial^ 2 f partial x 2 ^ 2 +***+ partial^ 2 f partial x n ^ 2
Now assume that f depends only on r where r=(x 1 ^ 2 +x 2 ^ 2 +***+x n ^ 2 )^ 1 2 i.e. f(x 1 ,x 2 ,***,x n )=g(r) for some function g. Show that, for x 1 ,x 2 ,***,x n ne0 ,
nabla^ 2 f= n-1 r g^ prime (r)+g^ prime prime (r)
[SADT3] For scalar functions u and v, show that
B=( nabla u)*( nabla v)
is solenoidal and that
A= 1 2 (u nabla v-v nabla u)
is a vector potential for B, i.e. B= nabla* A
The Laplacian of a function f
of n
variables x
1
,x
2
,⋯x
n
, denoted ∇
2
f
is defined by
∇
2
f(x
1
,x
2
,⋯,x
n
):=∂
2
f
∂x
2
1
+∂
2
f
∂x
2
2
+⋯+∂
2
f
∂x
2
n
Now assume that f
depends only on r
where r=(x
2
1
+x
2
2
+⋯+x
2
n
)
1
2
, i.e. f(x
1
,x
2
,⋯,x
n
)=g(r)
, for some function g
. Show that, for x
1
,x
2
,⋯,x
n
≠0
,
∇
2
f=n−1
r
g
′
(r)+g
′′
(r)
If A
and B
are vector fields, prove the following:
∇(A⋅B)=(B⋅∇)A+(A⋅∇)B+B×(∇×A)+A×(∇×B).
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