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Q. 16

Expert-verifiedFound in: Page 1119

Book edition
1st

Author(s)
Peter Kohn, Laura Taalman

Pages
1155 pages

ISBN
9781429241861

Use what you know about averages to propose a formula for the average rate of flux of a vector field F(x, y ,z) through a smooth surface S in the direction of n.

$\mathrm{Hence},\mathrm{the}\mathrm{average}rateoffluxofavectorfield\mathrm{f}(\mathrm{x},\mathrm{y},\mathrm{z})\mathrm{on}\mathrm{a}\mathrm{smooth}\mathrm{surface}\mathrm{S}\mathrm{is},\phantom{\rule{0ex}{0ex}}{\mathbf{F}}_{av}\left(S\right)=\frac{{\int}_{S}\u200a\mathbf{F}(x,y,z)\cdot \mathbf{n}dS}{{\int}_{S}\u200a1dS}$

The given function is f(x, y, z) .

$\mathrm{Surface}\mathrm{integrals}\mathrm{of}\mathrm{multivariate}\mathrm{functions}\mathrm{f}(\mathrm{x},\mathrm{y},\mathrm{z})\mathrm{is},\phantom{\rule{0ex}{0ex}}\int \mathrm{f}(\mathrm{x},\mathrm{y},\mathrm{z})n.\mathrm{dS}.\phantom{\rule{0ex}{0ex}}\mathrm{The}\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{smooth}\mathrm{surface}\mathrm{S}\mathrm{is},\phantom{\rule{0ex}{0ex}}{\int}_{\mathrm{S}}1\mathrm{dS}\phantom{\rule{0ex}{0ex}}\mathrm{Hence},\mathrm{the}\mathrm{average}rateoffluxofavectorfield\mathrm{f}(\mathrm{x},\mathrm{y},\mathrm{z})\mathrm{on}\mathrm{a}\mathrm{smooth}\mathrm{surface}\mathrm{S}\mathrm{is},\phantom{\rule{0ex}{0ex}}{\mathbf{F}}_{av}\left(S\right)=\frac{{\int}_{S}\u200a\mathbf{F}(x,y,z)\cdot \mathbf{n}dS}{{\int}_{S}\u200a1dS}$

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