Vector Calculus - 8
20 Questions MCQ Test Topic-wise Tests & Solved Examples for IIT JAM Mathematics | Vector Calculus - 8
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then div curl 
is equal toIf Φ is a differentiable scalar function, then div grad Φ is equal to
then 
isCorrect Answer :- a
Explanation : curl F = ∇ × F
= (∂/∂x, ∂/∂y, ∂/∂z) × (F1, F2, F3)
(∂F3/∂y − ∂F2/∂z , ∂F1/∂z − ∂F3/∂x , ∂F2/∂x − ∂F1/∂y)
= (∂z/∂y − ∂y/∂z , ∂x/∂z − ∂z/∂x, ∂y/∂x − ∂x/∂y)
= (0, 0, 0)
Note that since this curl is 0, the radial vector field
F(x, y, z) = (x, y, z) is irrotational.
along the circle x2 + y2 = 1 is 
isDirectional derivative of ψ(x,y,z) = xy2 + 4xyz + z2 at the point (1, 2, 3) in the direction of 
is
and 
are two vectors then the value of 
will be
is
then the value of div 
will beLet W be the region bounded by the planes x = 0, y = 0, y = 3, z = 0 and x + 2z = 6. Let S be the boundary of this region. Using gauss’ divergence theorem, evaluate, where
and ň is the outward unit normal vector to S.
By gauss divergence theorem
If 
is a differentiable vector point function, then the value of div curl 
is
Let 
and 
Q. The unit vector perpendicular to the plane containing 
and 
is
Given that
and 
Therefore, The unit vector perpendicular to the plane containing vector 
and 
is
Let us consider the scalar point function f(x y, z) =x2 + y2 + z2
Q. The grad of f(x, y, z) is
Grad f=
Let us consider the scalar point function f(x y, z) =x2 + y2 + z2
Q. The directional derivative of f(x, y, z) at the point P(1, 1, 1) along 
is
(Grad f)(1,1,1)= 
Now, the directional derivative o f f at P(1,1,1) along 
is
Let 
where a, b and c are constants and S is the surface of unit sphere.
Q. The value of 
is
By Gauss divergence theorem,
Let where a, b and c are constants and S is the surface of unit sphere.
Q. The value of 
is
By Gauss divergence theorem,
Since, V is enclosed by a sphere of unit radius. Thereofore
a + b + c = 1
and 
is
We take, 
= 1 - 2 + 1 = 0
So, B is normal to A.
If 
and curve C is the arc of the curve y = x3 from (0,0) to (2,8), then the value of 
Since, C is the curve y = x3 from (0,0) to (2,8)
So, let x = t ⇒ y = t3
If 
is the position vector of any point on C, then
or 
or 
At (0, 0) ⇒ t = x = 0 and at (2, 8) ⇒ t = 2
So, 
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