Vector Calculus - 5 - Mathematics MCQ

A vector normal to the surface S is given by



[Since, x² + y² + z² = 1 on S]
Now, we have
 ...(i) 
where R is the projection of S on the XY-plane.
The region R is bounded by X-axis, Y-axis and x² +y² = 1, z = 0.

So, on putting these values in Eq. (i), we get 


= 3/8

Correct Answer :- c

Explanation : u = yi + xyj

v = x²i + xy²j

u * v = [(i,j,k) (y,xy,0) (x², xy²,0)]

curl (u * v) = Δ * (u * v)

= [(i,j,k) (d/dx,d/dy.d/dz) (0,0,xy³-x³y)]

= i d/dx(xy³-x³y) +j(-d/dx(xy³-x³y)

= i(3xy² - x³) +j(3x²y - y³)

curl (u * v) = i(3xy² - x³) +j(y³ - 3x²y)

The equation of line BA is 

x + y = 2   ......(ii)
dx = - dy
Let  be the position vector in two- dimensional space then,

Then, the line integral over the line BA is

   ....(i)
C being unit circle around the, origin traversed once in the counter-clockwise direction. We know from Green’s theorem,

Comparing Eq. (i) with LHS of Eq. (ii), we have



So, 

We have, 



Similarly,

So, directional derivative of f indirection of vector   is nothing but the component of grad f in the direction of vector 

 (Putting y = x)

Derivative of f in the direction of 

Derivative of f in the direction of the point (8 , 8)

We have velocity vector 
So, the vorticity vector = Curl (Velocity vector

So, at (1, 1, 1) the vorticity vector

 Since 
or 
So, 


Since, the parametric equation of given circle a
x = cos θ,y = sin θ
or dx = - sin θ
dy - cos 0 d 0
So,

 If V be the volume enclosed by the surface S, then by Gauss’s divergence theorem


But V is the volume of a sphere of radius 3.
So, volume V  = 
Hence, 

Since,f(x) = (x - 8)^2/3 + 1
Or 
Slope of tangent at (0, 5) 
Or 
Slope of normal at (0, 5)
Or 
So, equation of normal at (0, 5) is 
y-5 = 3(x - 0)
Or y = 3x + 5

By Gauss’s Divergence theorem,

Surface, sphere of radius 3, centre at origin.
Here, 
By Gauss’s divergence theorem

Using this,

 Since, f(x, y)= xy
Now, we have to show that ( x d x + y dy) is exact so the value of the integral is independent of path

or integral = f(Q) - f(P)
= [xy]_(0,1) - [xy]_(1,0) = 0

Since, 

Since, 

Hence, 

Apply the divergence theorem

(Since,  is the position vector)