161. Let Φ be an arbitrary smooth real valued scalar function and V be an arbitrary smooth vector valued function in a three-dimensional space. Which one of the following is an identify?

1. 1. Curl(Φ V) = ∇(ΦDivV)

   
   
   

   2. DivV = 0

   
   
   

   3. Div Curl V = 0

   
   
   

   4. Div(Φ V) = ΦDiv V

   
   
   

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1. View Hint View Answer Discuss in Forum

   Div Curl V = 0

   Correct Option: C

   Div Curl V = 0

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162. The surface integral
     
     x² + y² + z² = 9 is _______.

1. 1. 213

   
   
   

   2. 232

   
   
   

   3. 222

   
   
   

   4. 216

   
   
   

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1. View Hint View Answer Discuss in Forum

   By Gauss divergence theorem,
   ∫_s F.n ds = &int_v divF dV
   Here F = 9xi - 3yj
   div F = 9 - 3 = 6
   

   Correct Option: D

   By Gauss divergence theorem,
   ∫_s F.n ds = &int_v divF dV
   Here F = 9xi - 3yj
   div F = 9 - 3 = 6
   

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163. Curl of vector V(x, y, z) = 2x2i + 3z2j + y3k at x = y = z = 1 is

1. 1. – 3i

   
   
   

   2. 3i

   
   
   

   3. 3i – 4j

   
   
   

   4. 3i – 6k

   
   
   

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1. View Hint View Answer Discuss in Forum

   
   
   = i[3y² – 6z] – j[0] + k[0 + 0]
   At x = 1, y = 1, z = 1
   Curl = – 3i

   Correct Option: B

   
   
   = i[3y² – 6z] – j[0] + k[0 + 0]
   At x = 1, y = 1, z = 1
   Curl = – 3i

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164. Divergence of the vector field
     x²zî + xyĵ - yz²k̂ at (1, -1, 1) is

1. 1. 0

   
   
   

   2. 3

   
   
   

   3. 5

   
   
   

   4. 6

   
   
   

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1. View Hint View Answer Discuss in Forum

   Given F = x² aî + xyĵ - yz²k̂
   div F = ∇ . F
   

   =	δ	(x2z) +	δ	(xy) +	δ	(-yz2)
   	δx		δy		δz

   
   div F at (1, – 1, 1) = 2 + 1 + 2 = 5

   Correct Option: C

   Given F = x² aî + xyĵ - yz²k̂
   div F = ∇ . F
   

   =	δ	(x2z) +	δ	(xy) +	δ	(-yz2)
   	δx		δy		δz

   
   div F at (1, – 1, 1) = 2 + 1 + 2 = 5

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165. Curl of vector F = x²z²î + 2xy²ĵ + 2y²z³k̂ is

1. 1. (4yz³ + 2xy²)î + 2x²zĵ - 2y²z³k̂

   
   
   

   2. (4yz³ + 2xy²)î - 2x²zĵ - 2y²zk̂

   
   
   

   3. 2xz²î - 4xyzĵ + 6y²z²k̂

   
   
   

   4. 2xz²î + 4xyzĵ + 6y²z²k̂

   
   
   

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1. View Hint View Answer Discuss in Forum

   Given F = x²z²i - 2xy²j + 2y²z³k
   
   = i(4yz³ + 2xy²) - j(0 - 2x²z) + k(-2y² z - 0)
   = (4yz² + 2xy² + 2x² zj) - 2y² zk

   Correct Option: A

   Given F = x²z²i - 2xy²j + 2y²z³k
   
   = i(4yz³ + 2xy²) - j(0 - 2x²z) + k(-2y² z - 0)
   = (4yz² + 2xy² + 2x² zj) - 2y² zk

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