74 lines
1.7 KiB
TeX
74 lines
1.7 KiB
TeX
\subsection{Surface area}
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Let $S$ be a surface parameterized by $x$ and $y$.
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That is, let $S=(x,y,z)$ where $z=f(x,y)$.
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The tangent lines at a point on $S$ form a tiny parallelogram.
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The area $a$ of the parallelogram is given by the magnitude of the cross product.
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$$a=\left|{\partial S\over\partial x}\times{\partial S\over\partial y}\right|$$
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By summing over all the parallelograms we obtain the total surface area $A$.
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Hence
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$$A=\int\!\!\!\int dA=\int\!\!\!\int a\,dx\,dy$$
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The following example computes the surface area of a unit disk
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parallel to the $xy$ plane.
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\medskip
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\verb$z=2$
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\verb$S=(x,y,z)$
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\verb$a=abs(cross(d(S,x),d(S,y)))$
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\verb$defint(a,y,-sqrt(1-x^2),sqrt(1-x^2),x,-1,1)$
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$$\pi$$
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\medskip
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\noindent
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The result is $\pi$, the area of a unit circle, which is what we expect.
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The following example computes the surface area of $z=x^2+2y$ over
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a unit square.
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\medskip
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\verb$z=x^2+2y$
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\verb$S=(x,y,z)$
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\verb$a=abs(cross(d(S,x),d(S,y)))$
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\verb$defint(a,x,0,1,y,0,1)$
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$${3\over2}+{5\over8}\log(5)$$
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\medskip
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\noindent
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As a practical matter, $f(x,y)$ must be very simple in order
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for Eigenmath to solve the double integral.
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\newpage
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\noindent
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Find the area of the spiral ramp defined by\footnote{
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Williamson and Trotter, {\it Multivariable Mathematics,} p. 598.}
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$$S=\left(\matrix{u\cos v\cr u\sin v\cr v}\right),\qquad 0\le u\le1,\qquad 0\le v\le3\pi$$
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In this example, the coordinates $x$, $y$ and $z$ are all
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functions of an independent parameter space.
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\medskip
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\verb$x=u*cos(v)$
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\verb$y=u*sin(v)$
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\verb$z=v$
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\verb$S=(x,y,z)$
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\verb$a=abs(cross(d(S,u),d(S,v)))$
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\verb$defint(a,u,0,1,v,0,3pi)$
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$${3\over2}\pi\log(1+2^{1/2})+{3\pi\over2^{1/2}}$$
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\verb$float$
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$$10.8177$$
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