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Find the Gauss value for a position vector in Cartesian system from the origin to one unit in three dimensions.
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.
Answer: b
See lessExplanation: 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.
The divergence theorem for a surface consisting of a sphere is computed in which coordinate system?
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.
Answer: d
See lessExplanation: 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.
The Gauss divergence theorem converts
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.
Answer: d
See lessExplanation: The divergence theorem for a function F is given by ∫∫ F.dS = ∫∫∫ Div (F).dV. Thus it converts surface to volume integral.
Evaluate the surface integral ∫∫ (3x i + 2y j). dS, where S is the sphere given by x2 + y2 + z2 = 9.
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 oRead more
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 4πr3 /3 and r = 3units.Thus we get 180π.
See lessGauss theorem uses which of the following operations?
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.
Answer: c
See lessExplanation: 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.
Find the area of a right angled triangle with sides of 90 degree unit and the functions described by L = cos y and M = sin x
Answer: d Explanation: dM/dx = cos x and dL/dy = -sin y ∫∫(dM/dx – dL/dy)dx dy = ∫∫ (cos x + sin y)dx dy. On integrating with x = 0->90 and y = 0- >90, we get area of right angled triangle as -180 units (taken in clockwise direction). Since area cannot be negative, we take 180 units.
Answer: d
Explanation: dM/dx = cos x and dL/dy = -sin y
∫∫(dM/dx – dL/dy)dx dy = ∫∫ (cos x + sin y)dx dy. On integrating with x = 0->90 and y = 0-
>90, we get area of right angled triangle as -180 units (taken in clockwise direction).
Since area cannot be negative, we take 180 units.
See lessThe Shoelace formula is a shortcut for the Green’s theorem. State True/False.
Answer: a Explanation: The Shoelace theorem is used to find the area of polygon using cross multiples. This can be verified by dividing the polygon into triangles. It is a special case of Green’s theorem.
Answer: a
See lessExplanation: The Shoelace theorem is used to find the area of polygon using cross
multiples. This can be verified by dividing the polygon into triangles. It is a special case of Green’s theorem.
The Green’s theorem can be related to which of the following theorems mathematically?
Answer: b Explanation: The Green’s theorem is a special case of the Kelvin- Stokes theorem, when applied to a region in the x-y plane. It is a widely used theorem in mathematics and physics.
Answer: b
See lessExplanation: The Green’s theorem is a special case of the Kelvin- Stokes theorem, when applied to a region in the x-y plane. It is a widely used theorem in mathematics and physics.
Applications of Green’s theorem are meant to be in
Answer: b Explanation: Since Green’s theorem converts line integral to surface integral, we get the value as two dimensional. In other words the functions are variable with respect to x,y, which is two dimensional.
Answer: b
See lessExplanation: Since Green’s theorem converts line integral to surface integral, we get the value as two dimensional. In other words the functions are variable with respect to x,y, which is two dimensional.
If two functions A and B are discrete, their Green’s value for a region of circle of radius a in the positive quadrant is
Answer: d Explanation: Green’s theorem is valid only for continuous functions. Since the given functions are discrete, the theorem is invalid or does not exist
Answer: d
See lessExplanation: Green’s theorem is valid only for continuous functions. Since the given
functions are discrete, the theorem is invalid or does not exist