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TopCat
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Homework Statement
Given \textbf{F} = x\textbf{i} + y\textbf{j} + z\textbf{k}, what is the flux of \textbf{F} through the cylinder x^2 + y^2 =1 bounded by the planes z=0, x+y+z=2.
By Gauss' Theorem, \int\int_{S}\textbf{F}\cdot d\textbf{S} = \int\int\int_{V}(\nabla\cdot \textbf{F})dV
But \nabla\cdot \textbf{F}=3, so the flux through the surface equals 3 times the enclosed volume. Using cylindric coordinates to calculate the volume from the integrals or using the fact the the volume is half that of a cylinder of radius 4, the volume is 2\pi and that gives 6\pi as the flux.
However, on a multiple choice test I just took, the answers offered were 0, \pi, 2\pi, 4\pi, and 10\pi. Where did I make a mistake?
Given \textbf{F} = x\textbf{i} + y\textbf{j} + z\textbf{k}, what is the flux of \textbf{F} through the cylinder x^2 + y^2 =1 bounded by the planes z=0, x+y+z=2.
The Attempt at a Solution
By Gauss' Theorem, \int\int_{S}\textbf{F}\cdot d\textbf{S} = \int\int\int_{V}(\nabla\cdot \textbf{F})dV
But \nabla\cdot \textbf{F}=3, so the flux through the surface equals 3 times the enclosed volume. Using cylindric coordinates to calculate the volume from the integrals or using the fact the the volume is half that of a cylinder of radius 4, the volume is 2\pi and that gives 6\pi as the flux.
However, on a multiple choice test I just took, the answers offered were 0, \pi, 2\pi, 4\pi, and 10\pi. Where did I make a mistake?
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