Answer :
Final answer:
The flux of the vector field v = ri+yj+k through the surface of a sphere can be calculated using the divergence theorem and spherical coordinates. The flux is found to be 4π, which corresponds to option E in the question.
Explanation:
The question asks us to calculate the flux of the vector field v = ri+yj+k through the surface of a sphere. To do this, we can use the divergence theorem, which states that the flux of a vector field through a closed surface is equal to the volume integral of the divergence of the field over the region enclosed by the surface.
The divergence of the vector field v = ri+yj+k is given by div(v) = ∂r/∂r + ∂y/∂y + ∂/∂z = 3. Since the surface in question is a sphere with equation r² + y² + 2z² = 1, we can use spherical coordinates to express the surface as r = 1, θ = 0 to π, and φ = 0 to 2π.
Using the divergence theorem, the flux of the vector field through the sphere is given by ∬∬s div(v) dA, where dA = r²sinθ dθdφ is the surface element in spherical coordinates. Evaluating this double integral gives us the flux as ∬∬s 3r²sinθ dθdφ. Integrating over θ and φ gives us the flux as 4π. Therefore, the answer is option E.
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