then the value of where C is the curve in the XY-plane, y = 2x2 from (0,0) to (1,2) is
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
Value of the integral where C is the square cut from the first quadrant by the lines x = 1 and y = 1 will be (use Green’s theorem to change the line integral into double integral)
We know that in Green’s theorem,
On comparing, we get M = -y2 and N = xy
The divergence of the vector field
The directional derivative of f(x, y, z) = x2 + y2 + z2 at the point (1, 1, 1) in the direction
Thus, directional derivative of f in the direction of the point P(1, 1, 1),
The unit normal vector to the surface of the sphere x2 + y2 + z2 = 1 at the point and are unit normal vectors in the Cartesian coordinate system)
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
If over the path shown in the figure is
Apply Green’s theorem the value of where C is the boundary of the area enclosed by the X-axis and the upper half of the circle x2 + y2 = a2 is
Since, From Green’s theorem, we have
(A is the region of the figure)
On changing in polar coordinates, we get
A fluid element has a velocity The motion at (x, y) =
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.
Unit vectors in X and Z-directions are respectively. Which one of the following is the directional derivative of the function F(x, z) = In (x2 + z2) at the point P(4, 0), in the direction of
Given, F(x, y) = In(x2 + z2) = log (y + z2)
Coordinate of point p is (4, 0).
For a scalar function f(x, y, z) = x2 + 3y2 + 2z2, the gradient at the point P(1, 2, -1) is
then the value of div Curl is
The value of α for which the following three vectors are coplanar is
Vectors are coplanar, if
Apply Green’ s theorem the value of where C is the square formed by the lines y = ±1, x = ±1 is
Since, On comparing with we get M = x2 + xy and
N = x2 + y2
So, from Green’s theorem
The divergence of the vector field at a point (1, 1, 1) is equal to
Since, We have
On comparing Eq. (r) with
is the reciprocal system to the vectors then the value of
where are constant vectors then is equal to
, then the value of div curl is
The value of div will be
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.
Divergence operators is defined for