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Maximum value of directional derivative of f= x^{2}yz at the point (1, 4,1) is
If C is a smooth curve in R3 from (–1, 0, 1) to (1, 1, –1), then the value of is
Correct Answer : D
Explanation : The Stoke’s theorem states that “the surface integral of the curl of a function over a surface bounded by a closed surface is equal to the line integral of the particular vector function around that surface.”
Stokes theorem says that ∫F·dr = ∬curl(F)·n ds.
The relation between the line integral and the surface integral is
If Φ(x, y, z) = 3x^{2}y  y^{3}z^{2 } then the value of grad Φ at the point (1, 2 , 1 ) is
If Φ is a differentiable scalar point function, then the value of curl grad Φ is
A particle moves along the curve Acceleration of the particle in the direction of the motion is
Using stokes' theorem evaluate the line integral, where L is the intersection of x^{2} + y^{2} + z^{2} = 1 and x + y = 0 traversed in the clockwise direction when viewed from the point (1, 1, 0)
By stoke’s theorem
where L is the intersection of x^{2} + y^{2} + z^{2 }= 1 and x + y = 0.
The outward unit normal vector
The divergence theorem states that the surface integral of the normal component of a vector point function “F” over a closed surface “S” is equal to the volume integral of the divergence of F⃗ taken over the volume “V” enclosed by the surface S. Thus, the divergence theorem is symbolically denoted as :
Let is a solution of the Laplace equation then the vector field is
⇒ Divergence is nonzero and curl is zero,
Hence is not solenoidal but irrotational.
Given that
Therefore,
Given that
Given that
and
Therefore,
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