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If φ = xy^{2}z^{3} then match ListI with ListII and select the correct answer using the codes given below the lists:
Here
Also,
And,
The unit vector which points from z = h on the zaxis towards (r, φ, 0) in cylindrical coordinates as shown below is given by
Let the unit vector be given by
Now,
∴ Unit vector,
If the vector V given below is irrotational, then the values of a, b and c will be respectively
Since the given vector is irrotational, therefore curl V = 0 or,
Since, therefore, a = 4, b = 2, and c = 1
Match ListI (Vector Identities) with Listll. (Equivalent expression) and select the correct answer using the codes given below the lists:
The vector differential operator, in spherical coordinate system is given by
Assertion (A): Divergence of a vector function at each point gives the rate per unit volume at which the physical entity is issuing from that point.
Reason (R): If some physical entity is generated or absorbed within a certain region of the field, then that region is known as source or sink respectively and if there are no sources or sinks in the field, the net outflow of the incompressible physical entity over any part of the region is zero. However, the net outflow is said to be positive, if the total strength of the sources are greater than the total strength of sink and viceversa.
Both assertion and reason are true and reason is the correct explanation of assertion. Reason is the physical interpretation of divergence.
The vector directed from (2,  4,1) to (0, 2,0) in Cartesian coordinates is given by
The vector is given as
What is the value of where
Here, s is the surface bounded by x = 0, x = 1, y = o, y = 1, z = 0, z =1 and are unit vectors along x, yand z axes respectively.
By divergence theorem,
Since, the surface s is bounded by x = 0, 1; y = 0, 1 and z = 0, 1 so, putting the limits, we have
The vector field will be irrotational,
Now,
Hence, is irrotationai.
The vector field will be solenoidal,
Here,
Hence, is solenoidal.
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