60 lines
1.8 KiB
TeX
60 lines
1.8 KiB
TeX
% surface integral
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\newpage
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\begin{center}
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\noindent
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\includegraphics[scale=0.5]{sailboat.png}
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\end{center}
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\bigskip
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\noindent
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A surface integral is like adding up all the wind on a sail.
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In other words, we want to compute
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$$\int\!\!\!\int({\bf F\cdot n})\,a\,dx\,dy$$
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where $({\bf F\cdot n})$ is the amount of wind normal to a tiny rectangle $a$
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of sail.
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Let $S$ be the surface of the sail parameterized by $x$ and $y$.
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(In this model, the $z$ direction points downwind.)
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By the properties of the cross product we have the following for the unit normal $\bf n$
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and the area $a$.
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$${\bf n}={ {{\partial S\over\partial x}\times{\partial S\over\partial y}}\over
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{\left|{\partial S\over\partial x}\times{\partial S\over\partial y}\right|}}\qquad
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a=\left|{\partial S\over\partial x}\times{\partial S\over\partial y}\right|$$
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Hence
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$$\int\!\!\!\int({\bf F\cdot n})\,a\,dx\,dy=\int\!\!\!\int{\bf F}\cdot
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\left({{\partial S\over\partial x}\times{\partial S\over\partial y}}\right)\,dx\,dy$$
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\newpage
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\noindent
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Evaluate the surface integral
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$$\int\!\!\!\int_S{\bf F\cdot n}\,d\sigma$$
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where ${\bf F}=xy^2z{\bf i}-2x^3{\bf j}+yz^2{\bf k}$, $S$ is the surface
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$z=1-x^2-y^2$, $x^2+y^2\le1$ and $\bf n$ is upper.\footnote{
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Kaplan, {\it Advanced Calculus,} p. 313.}
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\medskip
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\noindent
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Note that the surface intersects the $xy$ plane in a circle.
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By the right hand rule, crossing $x$ into $y$ yields $\bf n$ pointing upwards hence
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$${\bf n}\,d\sigma=\left({{\partial S\over\partial x}\times{\partial S\over\partial y}}\right)\,dx\,dy$$
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The following Eigenmath code computes the surface integral.
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The symbols $f$ and $h$ are used as temporary variables.
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\medskip
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\verb$z=1-x^2-y^2$
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\verb$F=(x*y^2*z,-2*x^3,y*z^2)$
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\verb$S=(x,y,z)$
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\verb$f=dot(F,cross(d(S,x),d(S,y)))$
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\verb$h=sqrt(1-x^2)$
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\verb$defint(f,y,-h,h,x,-1,1)$
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$${1\over48}\pi$$
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