Vector Calculus - 3 - IIT JAM MCQ

Let T(x, y), z) = xy² + 2z – x²z² be the temperature at a point (x, y, z). Temperature increase most rapidly in the direction of gradient i.e., ∇ T

So, temperature decreases most rapidly in the direction of .

By Guass is divergence theorem,
,
where V is the volume enclosed by S

since the volume V enclosed by a sphere of unit radius is equal to (4/3)π(1³)  = (4/3)π .

Work done by ,

Correct Answer :- B

Explanation : Straight line x = 2t, y = t, z = 3t 0 ≤ t ≤ 1

Work done = ∫F · dr

=  ∫(0 to 1) F·dr/dt * dt

= ∫(0 to 1) F·(dr/dt)dt

= ∫(0 to 1)[3(2t)²i + (2.2t.3t − t)j + 3tk] · [2i + j + 3k]dt

= ∫(0 to 1)[24t² + 12t² − t + 9t]dt

= [8t³ + 4t³(1/2t²) + 9/2t²](0 to 1)

= 8 + 4 + 4

= 16

Green's theorem gives the relationship between a line integral around a simple closed curve, C, in a plane and a double integral over the plane region R bounded by C. It is a special two-dimensional case of the more general Stokes' theorem.

Stokes’ Theorem gives the relationship between a line integral around a simple closed curve, C, in space, and a surface integral over a piecewise, smooth surface.

Green’s theorem in its “curl form”.

where F = P(x,y) i + Q(x,y) j and dr = dx i + dy j

is as follows: (curl form of Green’s Theorem)

c∫ F(x,y) . dr = c∫ F . T ds = r∫ ∫ curl F dA

where curl F is the z-component of curl F = curl F . k

for stoke’s theorem

Correct Answer :- a

Explanation :
a * (b * c) + b * (c * a) + c * (a * b) = 0

L.H.S.
a * (b * c) + b * (c * a) + c * (a * b)
= a * ( a) + b * (b) + c * ( c)
[∵ b * c = a ; c * a = b and a * b = c ]
= a * a + b * b + c * c
= 0 + 0 + 0 

[∵ a * a= 0;  b * b = 0 ; c * c= 0]

= 0 + 0 + 0
= 0
= R.H.S.

a * (b * c) + b * (c * a) + c * (a * b) = 0,

Hence proved

⇒ – (ax + y + a) + 1 + x + y = 0
⇒ (– a + 1)x + (– a + 1) = 0
⇒ (– a + 1) (1 + x) = 0
⇒ a = 1 (because x is not constant).