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Q. 16

Expert-verified
Found in: Page 1119

### Calculus

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.

$\mathrm{Hence},\mathrm{the}\mathrm{average}rateoffluxofavectorfield\mathrm{f}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)\mathrm{on}\mathrm{a}\mathrm{smooth}\mathrm{surface}\mathrm{S}\mathrm{is},\phantom{\rule{0ex}{0ex}}{\mathbf{F}}_{av}\left(S\right)=\frac{{\int }_{S} \mathbf{F}\left(x,y,z\right)\cdot \mathbf{n}dS}{{\int }_{S} 1dS}$

See the step by step solution

## Step 1. Given

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

## Step 2. Finding the average rate of flux of a vector field through smooth surface

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