If then the value of at (2, 2, 0) will be
If then the value of is
What is the value of constant b so that the vector
Since vector is solenoidal, therefore
or, [1 + 1 + b] = 0 or b = -2
Match List-I with List-ll and select the correct answer using the codes given below the lists:
A. Gauss’s divergence theorem
B. Stroke’s theorem
C. The divergence
D. The curl
Assertion (A): Vector differential operator is a vector quantity and it signifies that certain operations of a differentiation are to be carried out on the scalar function following it.
Reason (R): Vector differential operator posses properties similar to ordinary vectors
Vector differential operator (‘∇’) is not a vector quantity. Hence, assertion is a false statement.
Consider the following statements:
1. Divergence of a vector function at each point gives the rate per unit volume at which the physical entity is issuing from that point.
2. If a vector function ϕ represents temperature, then grad ϕ or ∇ϕ will represents rate of change of temperature with distance.
3. The curl of a vector function A gives the measure of the angular velocity at every point of the vector field.
All the given statements are correct.
Assertion (A): The Gauss’s divergence theorem permits us to express certain integrals by means of surface integrals.
Reason (R): Gauss’s divergence theorem states that “the surface integral of the curl of a vector field taken over any surface s is equal to the line integral of the vector field around the closed periphery (contour) of the surface.
Reason is a statement of stroke’s theorem not that of Gauss's divergence theorem.
If is any vector field in cartesian co-ordinates system, then
Let, be any vector field in cartesian co-ordinate system then, we can prove that
Also, Div. Curl
If and , then
If S is any closed surface enclosing a volume V and then the value of (is a unit vector) will be equal to
Assertion (A): The laplacian operator of a scalar function ϕ can be defined as “Gradient of the divergence of the scalar ϕ”
Reason (R): Laplacian operator may be a “scalar laplacian" or a “vector laplacian'’ depending upon whether it is operated with a scalar function or a vector, respectively.
Assertion is not true because the laplacian operator (∇2)of a scalar function ϕ can be defined as “Divergence of the gradient of the scalar ϕ”. i.e. ∇.∇ϕ
Match List-I (Terms) with List-II (Type) and select the correct answer using the codes given below the lists:
A. Curl= 0
B. Div = 0
C. Div grad (ϕ) = 0
D. Div div (ϕ) = 0
1. Laplace equation
4. Not defined
A B C D
(a) 2 3 1 4
(b) 4 1 3 2
(c) 2 1 3 4
(d) 4 3 1 2
Which of the following relations are not correct?
(A x B)2 = A2B2 - (A·B)2
If uF = ∇v, where u and v are scalar fields and F is a vector field, then F. curl F is equal to
A vector field which has a vanishing divergence is called as ____________
By the definition: A vector field whose divergence comes out to be zero or Vanishes is called as a Solenoidal Vector Field.
is a Solenoidal Vector field.
Which of the following statements is not true of a phasor?
A phasor is always a vector quantity.
Match List-I (Physical quantities) with List-ll (Dimensions) and select the correct answer using the codes given below the lists:
A. Electric potential
B. Magnetic flux
C. Magnetic field intensity
D. Magnetic flux density
A B C D
(a) 2 4 3 1
(b) 4 2 3 1
(c) 1 2 1 3
(d) 4 2 1 3
Which of the following statements is not true regarding vector algebra?
Option (c) is not correct because cross product of two unlike vectors is a third unit vector having positive sign for normal rotation and negative for reverse rotation while cross product of two like unit vectors is zero.
A rigid body is rotating with an angular velocity of ω where, and v is the line velocity. If is the position vector given by then the value of curl will be equal to
Taking the curl, we have:
If then which of the following relation will hold true?
= 1.1 + 1.1 + 1.1 = 1 + 1 + 1 = 3
Hence, both (a) and (b) will hold true.