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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Finding the surface area" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "restart;with(plots): with
(linalg):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Problem: Find the su
rface area of a sphere of radius a:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 71 "x:=(u,v)->a*sin(u)*cos(v);y:=(u,v)->a*sin(u)*sin(v);z
:=(u,v)->a*cos(u);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "plot3
d(subs(a=1,[x(u,v),y(u,v),z(u,v)]),u=0..Pi,v=0..2*Pi,axes=boxed);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "r:=(u,v)->[x(u,v),y(u,v),z(u
,v)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "r_u:=(u,v)->diff(r
(u,v),u);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "r_v:=(u,v)->di
ff(r(u,v),v);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "cross:=cro
ssprod(r_u(u,v),r_v(u,v));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
44 "F0:=sqrt(dotprod(cross,cross,`orthogonal`));" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 17 "F:=simplify(F0); " }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 30 "Int(Int(F,u=0..Pi),v=0..2*Pi);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 30 "int(int(F,u=0..Pi),v=0..2*Pi);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 34 "restart;with(plots): with(linalg):" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Problem: Find the surface
 area of an ellipsoid." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "x
:=(u,v)->a*cos(u)*sin(v);y:=(u,v)->b*sin(u)*sin(v);z:=(u,v)->c*cos(v);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "plot3d(subs(a=1,b=1,c=2
,[x(u,v),y(u,v),z(u,v)]),u=0..2*Pi,v=0..Pi,axes=boxed,scaling=constrai
ned);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "r:=(u,v)->[x(u,v),
y(u,v),z(u,v)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "r_u:=(u,
v)->diff(r(u,v),u);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "r_v:
=(u,v)->diff(r(u,v),v);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "
cross:=crossprod(r_u(u,v),r_v(u,v));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "F0:=sqrt(dotprod(cross,cross,`orthogonal`));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "F:=simplify(F0); " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Int(Int(F,u=0..Pi),v=0..2*Pi
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "int(int(subs(a=1,b=1,
c=2,F),u=0..Pi),v=0..2*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
55 "evalf(int(int(subs(a=1,b=1,c=2,F),u=0..Pi),v=0..2*Pi));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf(ellipsoid(1,1,2));" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "15 1 0" 47 }
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