The velocity field of a fluid (in meters per second) is [ mathbf{F}=(3 y-4) mathbf{i}+e^{-y(z+1)} mathbf{j}+left(x^{2}+y^{2}ight) mathbf{k}

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The velocity field of a fluid (in meters per second) is

\[
\mathbf{F}=(3 y-4) \mathbf{i}+e^{-y(z+1)} \mathbf{j}+\left(x^{2}+y^{2}ight) \mathbf{k}
\]

(a) Estimate the flow rate (in cubic meters per second) through a small surface \(\mathcal{S}\) around the origin if \(\mathcal{S}\) encloses a region of volume \(0.01 \mathrm{~m}^{3}\).

(b) Estimate the circulation of \(\mathbf{F}\) about a circle in the \(x y\)-plane of radius \(r=0.1 \mathrm{~m}\) centered at the origin (oriented counterclockwise when viewed from above).


(c) Estimate the circulation of \(\mathbf{F}\) about a circle in the \(y z\)-plane of radius \(r=0.1 \mathrm{~m}\) centered at the origin (oriented counterclockwise when viewed from the positive \(x\)-axis).

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Calculus

ISBN: 9781319055844

4th Edition

Authors: Jon Rogawski, Colin Adams, Robert Franzosa

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