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then the value of where C is the curve in the XYplane, y = 2x^{2} from (0,0) to (1,2) is
Correct Answer : c
Explanation : y = 2x^{2}
dy/dx = 4x
So F. dr
∫(x = 0 to 1) [3x(2x^{2})dx . (2x^{2})^{2} d(2x^{2})]
∫(x = 0 to 1) (6x^{2}  16x^{5})dx
= 7/6
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 = y^{2} and N = xy
The directional derivative of f(x, y, z) = x^{2} + y^{2} + z^{2} at the point (1, 1, 1) in the direction
We have,
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 x^{2} + y^{2} + z^{2} = 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 = x^{2} + y^{2} + z^{2}  1 = 0
We know unit normal vector is given by
Apply Green’s theorem the value of where C is the boundary of the area enclosed by the Xaxis and the upper half of the circle x^{2} + y^{2} = a^{2} is
Since, From Green’s theorem, we have
(A is the region of the figure)
On changing in polar coordinates, we get
or u =  y^{2}x, v = 2yx^{2}
In twodim 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 Zdirections are respectively. Which one of the following is the directional derivative of the function F(x, z) = In (x^{2} + z^{2}) at the point P(4, 0), in the direction of
Given, F(x, y) = In(x^{2} + z^{2}) = log (y + z^{2})
Coordinate of point p is (4, 0).
or
For a scalar function f(x, y, z) = x^{2} + 3y^{2} + 2z^{2}, the gradient at the point P(1, 2, 1) is
The value of α for which the following three vectors are coplanar is
Given,
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 = x^{2} + xy and
N = x^{2} + y^{2}
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
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 xaxis, yaxis and zaxis 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.
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