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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 4π2 m.
- a)1
Correct answer is option ''. Can you explain this answer?
Most Upvoted Answer
For a function given by F = 4x i + 7y j +z k, the divergence theorem e...
Explanation:
Given Function:
The function given is F = 4x i + 7y j + z k.
Surface Considered:
The surface considered is a cone with a radius of 1/2π m and a height of 4π2 m.
Divergence Theorem:
The divergence theorem states that the flux of a vector field through a closed surface is equal to the volume integral of the divergence of the vector field over the region enclosed by the surface.
Calculation:
- The divergence of the given function F is ∇ ⋅ F = ∂(4x)/∂x + ∂(7y)/∂y + ∂z/∂z = 4 + 7 + 1 = 12.
- The volume enclosed by the cone can be calculated using the formula V = (1/3)πr^2h, where r is the radius and h is the height of the cone. Substituting the values, V = (1/3)π(1/2π)^2(4π^2) = 2π.
- Therefore, by the divergence theorem, the flux of the vector field through the cone is equal to the volume integral of the divergence of the vector field over the cone, which is 12 * 2π = 24π.
Final Answer:
Therefore, the divergence theorem evaluates to a value of 24π for the given function and cone surface.
Given Function:
The function given is F = 4x i + 7y j + z k.
Surface Considered:
The surface considered is a cone with a radius of 1/2π m and a height of 4π2 m.
Divergence Theorem:
The divergence theorem states that the flux of a vector field through a closed surface is equal to the volume integral of the divergence of the vector field over the region enclosed by the surface.
Calculation:
- The divergence of the given function F is ∇ ⋅ F = ∂(4x)/∂x + ∂(7y)/∂y + ∂z/∂z = 4 + 7 + 1 = 12.
- The volume enclosed by the cone can be calculated using the formula V = (1/3)πr^2h, where r is the radius and h is the height of the cone. Substituting the values, V = (1/3)π(1/2π)^2(4π^2) = 2π.
- Therefore, by the divergence theorem, the flux of the vector field through the cone is equal to the volume integral of the divergence of the vector field over the cone, which is 12 * 2π = 24π.
Final Answer:
Therefore, the divergence theorem evaluates to a value of 24π for the given function and cone surface.
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Community Answer
For a function given by F = 4x i + 7y j +z k, the divergence theorem e...
Div (F) = 4 + 7 + 1 = 12. The divergence theorem gives ∫∫∫(12).dV, where dV is the volume of the cone πr3h/3, where r = 1/2π m and h = 4π2 m. On substituting the radius and height in the triple integral,
we get 2 units.
we get 2 units.
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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 4π2 m.a)1Correct answer is option ''. Can you explain this answer?
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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 4π2 m.a)1Correct answer is option ''. Can you explain this answer? for Civil Engineering (CE) 2025 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about 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 4π2 m.a)1Correct answer is option ''. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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 4π2 m.a)1Correct answer is option ''. Can you explain this answer?.
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