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QUESTION: 1

Gauss theorem uses which of the following operations?

Solution:

Answer: c

Explanation: The Gauss divergence theorem uses divergence operator to convert surface to volume integral. It is used to calculate the volume of the function enclosing the region given.

QUESTION: 2

Evaluate the surface integral ∫∫ (3x i + 2y j). dS, where S is the sphere given by x^{2} + y^{2} + z^{2}= 9.

Solution:

Answer: b

Explanation: We could parameterise surface and find surface integral, but it is wise to use divergence theorem to get faster results. The divergence theorem is given by ∫∫ F.dS = ∫∫∫ Div (F).dV

Div (3x i + 2y j) = 3 + 2 = 5. Now the volume integral will be ∫∫∫ 5.dV, where dV is the volume of the sphere 4πr^{3}/3 and r = 3units.Thus we get 180π.

QUESTION: 3

The Gauss divergence theorem converts

Solution:

Answer: d

Explanation: The divergence theorem for a function F is given by ∫∫ F.dS = ∫∫∫ Div (F).dV. Thus it converts surface to volume integral

QUESTION: 4

The divergence theorem for a surface consisting of a sphere is computed in which coordinate system?

Solution:

Answer: d

Explanation: Seeing the surface as sphere, we would immediately choose spherical system, but it is wrong. The divergence operation is performed in that coordinate system in which the function belongs to. It is independent of the surface region.

QUESTION: 5

Find the Gauss value for a position vector in Cartesian system from the origin to one unit in three dimensions.

Solution:

Answer: b

Explanation: The position vector in Cartesian system is given by R = x i + y j + z k. Div(R) = 1 + 1 + 1 = 3. By divergence theorem, ∫∫∫3.dV, where V is a cube with x = 0->1, y = 0->1 and z = 0->1. On integrating, we get 3 units.

QUESTION: 6

The divergence theorem value for the function x^{2} + y^{2} + z^{2} at a distance of one unit from the origin is

Solution:

Answer: d

Explanation: Div (F) = 2x + 2y + 2z. The triple integral of the divergence of the function is ∫∫∫(2x + 2y + 2z)dx dy dz, where x = 0->1, y = 0->1 and z = 0->1. On integrating, we get 3 units.

QUESTION: 7

If a function is described by F = (3x + z, y^{2} − sin x^{2}z, xz + ye^{x5}), then the divergence theorem value in the region 0<x<1, 0<y<3 and 0<z<2 will be

Solution:

Answer: c

Explanation: Div (F) = 3 + 2y + x. By divergence theorem, the triple integral of Div F in the region is ∫∫∫ (3 + 2y + x) dx dy dz. On integrating from x = 0->1, y = 0->3 and z = 0->2, we get 39 units.

QUESTION: 8

Find the divergence theorem value for the function given by (e^{z}, sin x, y^{2})

Solution:

Answer: b

Explanation: Since the divergence of the function is zero, the triple integral leads to zero. The Gauss theorem gives zero value.

QUESTION: 9

For a function given by F = 4x i + 7y j +z k, the divergence theorem evaluates to which of the values given, if the surface considered is a cone of radius 1/2π m and height 4π^{2} m.

Solution:

Answer: b

Explanation: Div (F) = 4 + 7 + 1 = 12. The divergence theorem gives ∫∫∫(12).dV, where dV is the volume of the cone πr^{3}h/3, where r = 1/2π m and h = 4π^{2} m. On substituting the radius and height in the triple integral, we get 2 units.

QUESTION: 10

Divergence theorem computes to zero for a solenoidal function. State True/False.

Solution:

Answer: a

Explanation: The divergence theorem is given by, ∫∫ F.dS = ∫∫∫ Div (F).dV, for a function F. If the function is solenoidal, its divergence will be zero. Thus the theorem computes to zero.

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