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Find the value of divergence theorem for A = xy2 i + y3 j + y2z k for a cuboid given by 0<x<1, 0<y<1 and 0<z<1.
- a)1
- b)4/3
- c)5/3
- d)2
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
Find the value of divergence theorem for A = xy2i + y3j + y2z k for a ...
Answer: c
Explanation: A cuboid has six faces. ∫∫A.ds = ∫∫Ax=0 dy dz + ∫∫Ax=1 dy dz + ∫∫Ay=0 dx dz + ∫∫Ay=1 dx dz + ∫∫Az=0 dy dx + ∫∫Az=1 dy dx. Substituting A and integrating we get (1/3) + 1 + (1/3) = 5/3.
Explanation: A cuboid has six faces. ∫∫A.ds = ∫∫Ax=0 dy dz + ∫∫Ax=1 dy dz + ∫∫Ay=0 dx dz + ∫∫Ay=1 dx dz + ∫∫Az=0 dy dx + ∫∫Az=1 dy dx. Substituting A and integrating we get (1/3) + 1 + (1/3) = 5/3.
Most Upvoted Answer
Find the value of divergence theorem for A = xy2i + y3j + y2z k for a ...
The divergence theorem states that the flux of a vector field A through a closed surface is equal to the volume integral of the divergence of A over the region enclosed by the surface.
In this case, the vector field A is given by A = xy^2i + y^3j + y^2zk.
To find the divergence of A, we will take the divergence of each component of A and sum them up.
∇ · A = (∂/∂x)(xy^2) + (∂/∂y)(y^3) + (∂/∂z)(y^2z)
= y^2 + 3y^2 + 2yz
Now, let's evaluate the flux of A through a cuboid with sides defined by the points (0,0,0), (a,0,0), (a,b,0), (0,b,0), (0,0,c), (a,0,c), (a,b,c), and (0,b,c).
The flux of A through the closed surface of the cuboid is equal to the volume integral of the divergence of A over the region enclosed by the surface.
∬∬∬ (∇ · A) dV = ∫∫∫ (y^2 + 3y^2 + 2yz) dV
The limits of integration for x, y, and z are 0 to a, 0 to b, and 0 to c respectively.
∫∫∫ (y^2 + 3y^2 + 2yz) dV = ∫₀^a ∫₀^b ∫₀^c (y^2 + 3y^2 + 2yz) dz dy dx
Evaluating this triple integral will give us the value of the divergence theorem for the given vector field and cuboid.
In this case, the vector field A is given by A = xy^2i + y^3j + y^2zk.
To find the divergence of A, we will take the divergence of each component of A and sum them up.
∇ · A = (∂/∂x)(xy^2) + (∂/∂y)(y^3) + (∂/∂z)(y^2z)
= y^2 + 3y^2 + 2yz
Now, let's evaluate the flux of A through a cuboid with sides defined by the points (0,0,0), (a,0,0), (a,b,0), (0,b,0), (0,0,c), (a,0,c), (a,b,c), and (0,b,c).
The flux of A through the closed surface of the cuboid is equal to the volume integral of the divergence of A over the region enclosed by the surface.
∬∬∬ (∇ · A) dV = ∫∫∫ (y^2 + 3y^2 + 2yz) dV
The limits of integration for x, y, and z are 0 to a, 0 to b, and 0 to c respectively.
∫∫∫ (y^2 + 3y^2 + 2yz) dV = ∫₀^a ∫₀^b ∫₀^c (y^2 + 3y^2 + 2yz) dz dy dx
Evaluating this triple integral will give us the value of the divergence theorem for the given vector field and cuboid.
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