Vector Calculus - 2 - Mathematics MCQ

Correct Answer :- c

Explanation : y = 2x²

dy/dx = 4x 

So F. dr

∫(x = 0 to 1) [3x(2x²)dx . (2x²)² d(2x²)]

∫(x = 0 to 1) (6x² - 16x⁵)dx

= -7/6

We know that in Green’s theorem,

On comparing, we get M = -y² and N = xy

We have,



Thus, directional derivative of f in the direction of   the point P(1, 1, 1),

Problem is to find unit normal vector to the surface of the sphere
f = x² + y² + z² - 1 = 0
We know unit normal vector  is given by



 

Since, From Green’s theorem, we have 

(A is the region of the figure)

On changing in polar coordinates, we get
 

or u = - y²x, v = 2yx²



In two-dim ensional flow, equation of continuity

Fluid is incompressible at this point.


Fluid flow is rotational.
Thus, fluid flow at  is rotational and incompressible.

Given, F(x, y) = In(x² + z²) = log (y + z²)

Coordinate of point p is (4, 0).




or  

Given,

Vectors  are coplanar, if

Since, On comparing  with  we get M = x² + xy and 
N = x² + y²
So, from Green’s theorem

Since, We have 
On comparing Eq. (r) with 

Since div(curl⇀v)=0, the net rate of flow in vector field curl⇀v\) at any point is zero. Taking the curl of vector field ⇀F eliminates whatever divergence was present in ⇀F

If P (x,y,z) is a variable point in a three- dimensional space, O the origin, i, j and k the unit vectors along the x-axis, y-axis and z-axis respectively, then the vector OP given by x i + y j +z k is called the position vector of the point P and is denoted by r.

Divergence of any vector f = f1i+ f2j +f3k denoted by div f is defined as the scalar ( delta f1)/(delta x) + (delta f2)/(delta y) + (delta f3)/(delta z)

where the delta s denote partial derivatives.

Using this definition we find that div r = delta (x)/ delta x +delta (y)/delta y + delta (z)/ delta z = 1 + 1 +1 = 3.

Hence, divergence of a position vector = div r = 3.