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Given a function ϕ = 1/2(x2 + y2 + z2) in three-dimensional Cartesian space, the value of the surface integral ∯S n̂ . ∇ϕ dS where S is the surface of a sphere of unit radius and n̂ is the outward unit normal vector on S, is
- a)4π
- b)3π
- c)4π/3
- d)0
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
Given a function = 1/2(x2+ y2+ z2) in three-dimensional Cartesian spac...
To find the value of the surface integral, we need to calculate the dot product of the function with the normal vector of the surface.
Let's assume the surface is described by the equation f(x, y, z) = 0. The normal vector to the surface at any point (x, y, z) can be found by taking the gradient of f(x, y, z):
n = ∇f(x, y, z) = (∂f/∂x, ∂f/∂y, ∂f/∂z)
Now, let's calculate the gradient of the surface equation f(x, y, z) = 0:
∂f/∂x = ∂(1/2(x^2 + y^2 + z^2))/∂x = x
∂f/∂y = ∂(1/2(x^2 + y^2 + z^2))/∂y = y
∂f/∂z = ∂(1/2(x^2 + y^2 + z^2))/∂z = z
So, the normal vector to the surface is n = (x, y, z).
Now, let's calculate the dot product of the function with the normal vector:
S · n = 1/2(x^2 + y^2 + z^2) · (x, y, z) = 1/2(x^3 + y^3 + z^3)
Therefore, the value of the surface integral S · n is 1/2(x^3 + y^3 + z^3).
Let's assume the surface is described by the equation f(x, y, z) = 0. The normal vector to the surface at any point (x, y, z) can be found by taking the gradient of f(x, y, z):
n = ∇f(x, y, z) = (∂f/∂x, ∂f/∂y, ∂f/∂z)
Now, let's calculate the gradient of the surface equation f(x, y, z) = 0:
∂f/∂x = ∂(1/2(x^2 + y^2 + z^2))/∂x = x
∂f/∂y = ∂(1/2(x^2 + y^2 + z^2))/∂y = y
∂f/∂z = ∂(1/2(x^2 + y^2 + z^2))/∂z = z
So, the normal vector to the surface is n = (x, y, z).
Now, let's calculate the dot product of the function with the normal vector:
S · n = 1/2(x^2 + y^2 + z^2) · (x, y, z) = 1/2(x^3 + y^3 + z^3)
Therefore, the value of the surface integral S · n is 1/2(x^3 + y^3 + z^3).
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Given a function = 1/2(x2+ y2+ z2) in three-dimensional Cartesian spac...
Gauss divergence theorem:
It states that the surface integral of a vector field over a closed surface is equal to the volume integral of the divergence of that vector field over the volume enclosed by the closed surface.
∯S A.dS = ∫∫∫V (∇.A) dV
Calculation:
Given:
S = surface of sphere , V = volume of sphere = 4/3πr3
r = radius of sphere = 1
Using Gauss theorem
∯S n̂ . ∇ϕ dS = 4π
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Given a function = 1/2(x2+ y2+ z2) in three-dimensional Cartesian space, the value of the surface integralS n . ∇ dSwhere S is the surface of a sphere of unit radius and n is the outward unit normal vector on S, isa)4πb)3πc)4π/3d)0Correct answer is option 'A'. Can you explain this answer?
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Given a function = 1/2(x2+ y2+ z2) in three-dimensional Cartesian space, the value of the surface integralS n . ∇ dSwhere S is the surface of a sphere of unit radius and n is the outward unit normal vector on S, isa)4πb)3πc)4π/3d)0Correct answer is option 'A'. 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 Given a function = 1/2(x2+ y2+ z2) in three-dimensional Cartesian space, the value of the surface integralS n . ∇ dSwhere S is the surface of a sphere of unit radius and n is the outward unit normal vector on S, isa)4πb)3πc)4π/3d)0Correct answer is option 'A'. 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 Given a function = 1/2(x2+ y2+ z2) in three-dimensional Cartesian space, the value of the surface integralS n . ∇ dSwhere S is the surface of a sphere of unit radius and n is the outward unit normal vector on S, isa)4πb)3πc)4π/3d)0Correct answer is option 'A'. Can you explain this answer?.
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