~v360 300 ~w175 2 889 864 0 11584 0 0 ~f? 16 14 12 ? 3 1 1 0 ? ? ? "Gotham-Book" ? ? ? 0 ? 1 1 "Times" 12 ? ? 15 0 c n 27 0 1 1 k 324 m"?n page ?p?a" ? 1 26177 26178 26115 26178 1 1 1 1 0 0 8405120 0 -1 1 0 -1 -1 -1 -1 -1 1 0 0 0 2 0 0 0 0 0 0 0 0 ? ? ? ? ? ? ? ~Q ]|Expr|[#b @`bb#_b#_b#_})!#!b'4["!!)$#!b'4`fb!0}" *~: ;2&c55)VectorCalculus| |`fb!D};4&c55*,F`f#}LiveMath`fb#@}&c55),K| |}& b!( b"0 b#8 b$@ b%H b&P!WW}`f }))#!b'4`fb!0};22D and 3D Measurements| |}& b!( b"0 b#8 b$@ b%H b&P!WW}! !! ! }| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 0 ~V?v0 (I)~p0 1 ~V?m0 (CornerD)~p0 1 ~V?m0 (CornerC)~p0 1 ~V?m0 (CornerB)~p0 1 ~V?m0 (CornerA)~p0 1 ~V?f0 (divField)~p0 1 ~V?f0 (unitnormal)~p0 1 ~V?v0 (flowtime)~p0 1 ~V?f0 (center)~p0 1 ~V?f0 (unittan)~p0 1 ~V?f0 (F)~p0 1 ~V?v0 (tt)~p0 1 ~V?v0 (thigh)~p0 1 ~V?v0 (tlow)~p0 1 ~V?f0 (UnitField)~p0 1 ~V?f0 (UnitFieldVector)~p0 1 ~V?f0 (CombinedElectricField)~p0 1 ~V?f0 (ElectricField)~p0 1 ~V?f0 (UnitTangent)~p0 1 ~V?f0 (DEField)~p0 1 ~V?f0 (NormalComponent)~p0 1 ~V?f0 (normal)~p0 1 ~V?f0 (TanComponent)~p0 1 ~V?f0 (Tangent)~p0 1 ~V?f0 (Field)~p0 1 ~V?f0 (m)~p0 1 ~V?f0 (n)~p0 1 ~V?v0 (dy)~p0 1 ~V?m0 (A)~p0 1 ~V?v0 (dx)~p0 1 ~V?v0 (df)~p0 1 ~V?m0 (pluckdata)~p0 1 ~V?v0 (PointB)~p0 1 ~V?m0 (PointA)~p0 1 ~V?m0 (CrossProduct)~p0 1 ~V?f0 (ThreeD'Zf)~p0 1 ~V?f0 ('ZDistSquare)~p0 1 ~V?f0 (DistSquared)~p0 1 ~V?f38 (Int)~p0 1 ~V?v0 (jump)~p0 1 ~V?f0 (Position)~p0 1 ~V?m0 (HottestPoint)~p0 1 ~V?m0 (Start)~p0 1 ~V?f0 ('ZTemp)~p0 1 ~V?f0 (Temp)~p0 1 ~V?v0 (ScaleFactor)~p0 1 ~V?v0 (CircleRadius)~p0 1 ~V?f0 (SpherePlotter)~p0 1 ~V?v0 (aWidth)~p0 1 ~V?v0 (aHeight)~p0 1 ~V?f0 ('Zg)~p0 1 ~V?f0 (g)~p0 1 ~V?f0 (FitterFromMma)~p0 1 ~V?f0 (BestFit)~p0 1 ~V?v0 (datalength)~p0 1 ~V?f0 (gradsqerror)~p0 1 ~V?f0 (sqerror)~p0 1 ~V?f0 (Fitter)~p0 1 ~V?f146 (ColumnsOf)~p0 1 ~V?m0 (data)~p0 1 ~V?v0 ('Dt)~p0 1 ~V?f0 (FindMinimum)~p0 1 ~V?v0 (N)~p0 1 ~V?m0 (DeepestDip)~p0 1 ~V?f0 ('Zf)~p0 1 ~V?f0 (zz')~p0 1 ~V?f0 (ww)~p0 1 ~V?f0 (znew)~p0 1 ~V?v0 (Normal)~p0 1 ~V?f0 (TanyCurve)~p0 1 ~V?f0 (TanxCurve)~p0 1 ~V?f0 (yCurve)~p0 1 ~V?f0 (xCurve)~p0 1 ~V?f0 (SurfacePoint)~p0 1 ~V?f0 (linearf)~p0 1 ~V?m0 (UnitTan)~p0 1 ~V?m0 (UnitGradNormal)~p0 1 ~V?v0 (Radius)~p0 1 ~V?m0 (q)~p0 1 ~V?v0 (scalefactor)~p0 1 ~V?m0 (Vector)~p0 1 ~V?f0 (Tan)~p0 1 ~V?f0 (gradf)~p0 1 ~V?f0 (yy')~p0 1 ~V?f0 (xx')~p0 1 ~V?m0 (NewTail)~p0 1 ~V?v0 (j)~p0 1 ~V?m0 (Matrix)~p0 1 ~V?f0 (tancompaccel)~p0 1 ~V?m0 (Q)~p0 1 ~V?f0 (accel)~p0 1 ~V?f0 (speed)~p0 1 ~V?f0 (vel)~p0 1 ~V?v0 (bestt)~p0 1 ~V?f0 (f)~p0 1 ~V?m0 (XpushalongY)~p0 1 ~V?v0 (length)~p0 1 ~V?f0 (tanline)~p0 1 ~V?m0 (Point)~p0 1 ~V?f0 (zz)~p0 1 ~V?f0 (P')~p0 1 ~V?f0 (yy)~p0 1 ~V?f0 (xx)~p0 1 ~V?f0 (P)~p0 1 ~V?m0 (X)~p0 1 ~V?v0 (s)~p0 1 ~Q ]|Expr|[#b @`bb#_b#_b#_})!# b$L" *~: ;.&c55*Trigonometry| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 1 ~V?f276 (arccsc)~p0 2 ~V?f278 (arccot)~p0 2 ~V?f256 (sin)~p0 2 ~V?f257 (cos)~p0 2 ~V?f258 (tan)~p0 2 ~V?f261 (sec)~p0 2 ~V?f260 (csc)~p0 2 ~V?f262 (cot)~p0 2 ~V?f272 (arcsin)~p0 2 ~V?f273 (arccos)~p0 2 ~V?f274 (arctan)~p0 2 ~V?f277 (arcsec)~p0 2 ~Q ]|Expr|[#b @`bb#_b#_b#_})!# b$L" *~: ;bP7&c55*Hyperbolic| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 1 ~V?f309 (arcsech)~p0 2 ~V?f304 (arcsinh)~p0 2 ~V?f294 (coth)~p0 2 ~V?f310 (arccoth)~p0 2 ~V?f306 (arctanh)~p0 2 ~V?f293 (sech)~p0 2 ~V?f288 (sinh)~p0 2 ~V?f289 (cosh)~p0 2 ~V?f290 (tanh)~p0 2 ~V?f305 (arccosh)~p0 2 ~V?f308 (arccsch)~p0 2 ~V?f292 (csch)~p0 2 ~Q ]|Expr|[#b @`bb#_b#_b#_})%# b$L" *~: ;bP7&c55*Logarithms ,F Powers| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 1 ~V?f32 (log)~p0 2 ~V?f307 (ln)~p0 2 ~V?f291 (exp)~p0 2 ~V?c2 (e)~p0 2 ~Q ]|Expr|[#b @`bb#_b#_b#_})## b$L" *~: ;bP7&c55*Standard Rules| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 1 ~Q ]|Expr|[#b @`bb#_b#_b#_})%# b$L" *~: ;bP7&c55*Logarithms ,F Powers| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 2 ~Ht(?x^(-?y)):(1/?x^?y)~p0 3 ~Hs(exp(?z)):(e^?z)~p0 3 ~Hs(e^(ln(?x))):(?x)~p0 3 ~Hs(10^(log(?x))):(?x)~p0 3 ~Hs(?y^(log_?y(?x))):(?x)~p0 3 ~Hs(ln(e^?x)):(?x)~p0 3 ~Hs(log(10^?x)):(?x)~p0 3 ~Hs(log_?y(?y^?x)):(?x)~p0 3 ~He(ln(?u*?v)):(ln(?u)+ln(?v))~p0 3 ~He(log(?u*?v)):(log(?u)+log(?v))~p0 3 ~He(log_?y(?u*?v)):(log_?y(?u)+log_?y(?v))~p0 3 ~He(ln(?u/?v)):(ln(?u)-ln(?v))~p0 3 ~He(log(?u/?v)):(log(?u)-log(?v))~p0 3 ~He(log_?y(?u/?v)):(log_?y(?u)-log_?y(?v))~p0 3 ~He(ln(?u^?v)):(?v*ln(?u))~p0 3 ~He(log(?u^?v)):(?v*log(?u))~p0 3 ~He(log_?y(?u^?v)):(?v*log_?y(?u))~p0 3 ~He(ln(sqrt(?u))):(1/2*ln(?u))~p0 3 ~He(log(sqrt(?u))):(1/2*log(?u))~p0 3 ~He(log_?y(sqrt(?u))):(1/2*log_?y(?u))~p0 3 ~He(?z^(?x+?y)):(?z^?x*?z^?y)~p0 3 ~He(?z^(?x-?y)):(?z^?x*?z^(-?y))~p0 3 ~Q ]|Expr|[#b @`bb#_b#_b#_})!# b$L" *~: ;bP7&c55*Trigonometry| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 2 ~Q ]|Expr|[#b @`bb#_b#_b#_})+# b$L" *~: ;bP7&c55*Simplify ,M negation| | and common zeros}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Hs(sin(-?x)):(-sin(?x))~p0 4 ~Hs(cos(-?x)):(cos(?x))~p0 4 ~Hs(tan(-?x)):(-tan(?x))~p0 4 ~Hs(sin('p)):(0)~p0 4 ~Hs(sin(?n*'p)):(0)~p0 4 ~Hs(cos(1/2*'p)):(0)~p0 4 ~Hs(cos(?n/2*'p)):(0)~p0 4 ~Hs(-(cos(?x))^2-(sin(?x))^2):(-1)~p0 4 ~Hs(cos('p/2)):(0)~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})'# b$L" *~: ;bP7&c55*Transform to basic| | types}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Ht(tan(?x)):((sin(?x))/(cos(?x)))~p0 4 ~Ht(csc(?x)):(1/(sin(?x)))~p0 4 ~Ht(sin(?x)):(1/(csc(?x)))~p0 4 ~Ht(sec(?x)):(1/(cos(?x)))~p0 4 ~Ht(cos(?x)):(1/(sec(?x)))~p0 4 ~Ht(cot(?x)):((cos(?x))/(sin(?x)))~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})## b$L" *~: ;bP7&c55*Trig Addition| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Ht(cos(?x+?y)):(cos(?x)*cos(?y)-sin(?x)*sin(?y))~p0 4 ~Ht(sin(?x+?y)):(cos(?x)*sin(?y)+sin(?x)*cos(?y))~p0 4 ~Ht(cos(2*?x)):(2*(cos(?x))^2-1)~p0 4 ~Ht(sin(2*?x)):(2*cos(?x)*sin(?x))~p0 4 ~Ht(sin(?n*?x)):(cos((?n-1)*?x)*sin(?x)+cos(?x)*~ sin((?n-1)*?x))~p0 4 ~Ht(cos(?n*?x)):(cos(?x)*cos((?n-1)*?x)-sin(?x)*~ sin((?n-1)*?x))~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})/# b$L" *~: ;bP7&c55*Transform ,M into| | another flavor of trig function}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Ht((sin(?x))^2):(1-(cos(?x))^2)~p0 4 ~Ht((cos(?x))^2):(1-(sin(?x))^2)~p0 4 ~Ht((tan(?x))^2):((sec(?x))^2-1)~p0 4 ~Ht((sec(?x))^2):((tan(?x))^2+1)~p0 4 ~Ht((csc(?x))^2):((cot(?x))^2+1)~p0 4 ~Ht((cot(?x))^2):((csc(?x))^2-1)~p0 4 ~Hs((sin(?x))^2+(cos(?x))^2):(1)~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})b @# b%4" *~: ;bP7&c55*substituting | |z,]tan,Hx,O2,I into a rational function in sin,Hx,I and cos,H| |x,I}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Hs(cos(2*arctan(?z))):((1-?z^2)/(1+?z^2))~p0 4 ~Hs(sin(2*arctan(?z))):(2*?z/(1+?z^2))~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})## b$L" *~: ;bP7&c55*Other rules| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Ht((cos(?x))^2):(1/2*(cos(2*?x)+1))~p0 4 ~Ht((sin(?x))^2):(1/2*(-cos(2*?x)+1))~p0 4 ~Ht(cos(?x)*sin(?x)):(1/2*sin(2*?x))~p0 4 ~Hs(sin(arccos(?x))):(sqrt(-?x^2+1))~p0 4 ~Hs(cos(arcsin(?x))):(sqrt(-?x^2+1))~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})!# b!T" *~: ;bP7&c55*Hyperbolic| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 2 ~Q ]|Expr|[#b @`bb#_b#_b#_})+# b$L" *~: ;bP7&c55*Simplify ,M negation| | and common zeros}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Hs(sinh(-?x)):(-sinh(?x))~p0 4 ~Hs(cosh(-?x)):(cosh(?x))~p0 4 ~Hs(tanh(-?x)):(-tanh(?x))~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})'# b$L" *~: ;bP7&c55*Transform into | |other types}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Ht((sinh(?x))^2):((cosh(?x))^2-1)~p0 4 ~Ht((cosh(?x))^2):(1+(sinh(?x))^2)~p0 4 ~Ht((tanh(?x))^2):(1-(sech(?x))^2)~p0 4 ~Ht(sinh(?x)):((e^?x-e^(-?x))/2)~p0 4 ~Ht(cosh(?x)):((e^?x+e^(-?x))/2)~p0 4 ~Ht(tanh(?x)):((e^?x-e^(-?x))/(e^?x+e^(-?x)))~p0 4 ~Ht(tanh(?x)):((sinh(?x))/(cosh(?x)))~p0 4 ~Hs((cosh(?x))^2-(sinh(?x))^2):(1)~p0 4 ~Hs(-(cosh(?x))^2+(sinh(?x))^2):(-1)~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})%# b$L" *~: ;bP7&c55*Other hyperbolic| | rules}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 3 ~Ht((cosh(?x))^2):(1/2*(cosh(2*?x)+1))~p0 4 ~Ht((sinh(?x))^2):(1/2*(cosh(2*?x)-1))~p0 4 ~Ht(sinh(2*?x)):(2*cosh(?x)*sinh(?x))~p0 4 ~Ht(cosh(2*?x)):(2*(cosh(?x))^2-1)~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})!# b'4" *~: ;bP7&c55*Constants| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 1 ~V?c0 (C)~p0 2 ~V?c5 ('o)~p0 2 ~V?c1 ('p)~p0 2 ~V?c0 (c)~p0 2 ~V?c0 (b)~p0 2 ~V?c0 (a)~p0 2 ~V?c4 (i)~p0 2 ~V?c3 ('N)~p0 2 ~Q ]|Expr|[#b @`bb#_b#_b#_})!# b'4" *~: ;bP7&c55*Variables| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 1 ~V?v0 (h)~p0 2 ~V?v0 (v)~p0 2 ~V?v0 (u)~p0 2 ~V?v0 (w)~p0 2 ~V?v0 ('f)~p0 2 ~V?v0 ('r)~p0 2 ~V?v0 ('q)~p0 2 ~V?v0 (r)~p0 2 ~V?v0 (t)~p0 2 ~V?v0 (z)~p0 2 ~V?v0 (y)~p0 2 ~V?v0 (x)~p0 2 ~V?v0 (k)~p0 2 ~Q ]|Expr|[#b @`bb#_b#_b#_})!# b'4" *~: ;bP7&c55*Functions| |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1 1 ~V?f162 (max)~p0 2 ~V?f161 (min)~p0 2 ~V?f132 (FromSpherical)~p0 2 ~V?f130 (FromCylindrical)~p0 2 ~V?f128 (FromPolar)~p0 2 ~V?f144 (RowsOf)~p0 2 ~V?f146 (ColsOf)~p0 2 ~V?d16 (d)~p0 1 ~Q ]|Expr|[#b @`bb#_b#_b#_})!# b+-["! )7# b+-" *~: ;-&c55*Original| | Authors,Z Bill Davis,L Horacio Porta,L Jerry Uhl,L Bruce Carpenter| |}& b!( b"0 b#8 b$@ b%H b&P!WW})7# b+-Enhanced Version,Z Robert| | Curtis .E Distributor ,F Translator,Z MathMonkeys,L LLC| |}& b!( b"0 b#8 b$@ b%H b&P!WW}}}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0 0 ~Q ]|Expr|[#b @`bb#_b#_b#_})!# b'4[&! )!# b'4`fb#_}" *~: ;bP8&c55*`g| |pb))b'2| |!R^rJ!!!!)!V?@(!riZ/s8Lu=!([(i!([(i!!!!!!!iVt!'XH)!$2+@!"&]+| |!!!!)!V?A\!!!$!LX6#$!!!!)!V??l!!E9%!!!!i!!!!i!!!!1!$D7E!!iQ)| |!$D_-.B| |>Q\,BP)t6XJWbSIOcfg.RL]dW%tP7B&!7A'$R#KsJH16$JH1Z0JQ[P,JQ\"9| |!+peDJWPGENfX7/JI@#:JI@JG!$;1fJH16$JH1Z0JQ[P,JQ\"9!+peDJWPGE| |NfX7/JI@#:JI@JG!$;1fJH16$JH1Z0JQ[P,JQ\"9!+peDJWPGENfX7/JI@#:| |JI@JG!$;1fJH16$JH1Z0JQ[P,JQ\"9!+peDJWPGENfX7/JI@#:JI@JG!$;1f| |JH16$JH1Z0JQ[P,JQ\"9!+peDJWPGENfX7/JI@#:JI@JG!$;1fJH16$JH1Z0| |JQ[P,JQ\"9!+peDJWPGENfX7/JI@#:JI@JG!$;1pJH16$JH1Z0!F/cY>_-.2| |>`i7n=_Nt[JWbSIJWc.Y"-Ar^$\8h>%tP7R%fm%]!<3$!| |1''| |b'T(___| |3b#%!(.Zg6Vb| |H7JN+DBo#kD$&Yq7eU]TgW9@6?ge$R*I24D]`EZXM*!sgiXSeH\-!EjmJmEu| |-(7n+J20UfR,\B\F;?0e7VHoE!!!!j78?7R6=>BF| |9}| |}& b!( b"0 b#8 b$@ b%H b&P!WW}`f }),# b'4;bP:VC,N05 Flow Measurements| | by Integrals}& b!( b"0 b#8 b$@ b%H b&P!WW})'# b'4;0&c55&Basics| | ,M B1`fb"T};bP8&c55* }& b!( b"0 b#8 b$@ b%H b&P!WW}`f })!# b'4| |`fb#_}`g| |pb(_b'*| |!QkBB!!!!)!V?@(!riZ/s8Lu=!([(i!([(i!!!!!!!iVt!U=82!$2+@!"&]+| |!!!!)!V?A\!!!$!LX6#$!!!!)!V??l!!E9%!!!!i!!!!i!!!!1!$D7E!!iQ)| |!$D(n5u_NKmF3it_E\)i_7pqY+;dE5mK%/LBH^D2(1(EX| |@@9-lX'Srt[fA+>kN?uZQeFM'J&3l,j#Rj+EIc)OLYH%k`B?P| |DS7h+BVX.Is4if3Fm9kI5TB%dEBRWW37&,3>=W!r!!!!!8OZBBY!QNJ| |9}| |}& b!( b"0 b#8 b$@ b%H b&P!WW}`f }) # b'4| |}& b!( b"0 b#8 b$@ b%H b&P!WW}`fb!,})!# b'4`f }[!!!_ !! }| |}& b!( b"0 b#8 b$@ b%H b&P!WW}}}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0 0 ~Q ]|Expr|[#b @`bb#_b#_b#_})## b%$" *~: ;bP8&c55*Graphics Primitives| |}& b!( b"0 b#8 b$@ b%H b&P WW}]|[~p1 0 ~S17 ? 255 ? ? (?):(?):(10)~p0 1 ~F1 1 3 5 ? ? 0 ? ? (?):(?):(?)~p0 1 ~L1 0 ? (?):(?)~p0 1 ~D2 5 1 16777215 ? ? (?):(?):(?):(?)~p0 1 ~O3 ? ? 1:1:16711680:1:(?):(?):(?):(?)~p0 1 ~G1 1 208 208 0 1 5 4 10 (-3...3):(-3...3):(?=~ 0...2*'p):('p/5):(10)~Q ]|Expr|[#b @`bb#_b#_b#_})%# b$ " *~: ;bP8&c55*Graph Building | |Blocks}& b!( b"0 b#8 b$@ b%H b&P WW}]|[~p1 0 ~R8405120 1 (x,y):(y=bottom...top):(x=left...right):(~ 0)~p0 1 ~R8405120 1 (x,y):(x=left...right):(y=bottom...~ top):(0)~p0 1 ~X2 8405120 (max(left,min(0,right)),y):(y=bottom...~ top):(y)~p0 1 ~X1 8405120 (x,max(bottom,min(0,top))):(x=left...~ right):(x)~p0 1 ~V?c64 (left)~p0 1 ~V?c65 (right)~p0 1 ~V?c66 (bottom)~p0 1 ~V?c67 (top)~p0 1 ~t~p0 1 ~Z1 1 208 208 0 0.90731 0.42048 0.00000 0.04288 -0.09253 -0.99478 -0.41829 0.90257 -0.10198 1 4 4 5 10 0 0 ? ? (~ -3...3):(-3...3):(-9...9):(?=0...2*'p):('p/5):(~ 10)~Q ]|Expr|[#b @`bb#_b#_b#_})%# b$ " *~: ;bP8&c55*Graph Building | |Blocks}& b!( b"0 b#8 b$@ b%H b&P WW}]|[~p1 0 ~R8405120 1 (x,y,bottom):(y=south...north):(x=~ west...east):(0)~p0 1 ~R8405120 1 (x,y,bottom):(x=west...east):(y=south...~ north):(0)~p0 1 ~X1 8405120 (max(west,min(0,east)),max(south,min(~ 0,north)),z):(z=bottom...top):(z)~p0 1 ~X1 8405120 (max(west,min(0,east)),y,max(bottom,~ min(0,top))):(y=south...north):(y)~p0 1 ~X1 8405120 (x,max(south,min(0,north)),max(bottom,~ min(0,top))):(x=west...east):(x)~p0 1 ~V?c69 (west)~p0 1 ~V?c68 (east)~p0 1 ~V?c70 (south)~p0 1 ~V?c71 (north)~p0 1 ~V?c66 (bottom)~p0 1 ~V?c67 (top)~p0 1 ~t~p0 1 ~Ie(x,s,t,z,y)~p0 0 ~Q ]|Expr|[#b @`bb#_b#_b#_})!# b'4["!!(6`f0}" *~: ;0&c55)B,N1,I | |Measuring flow across a curve with the integral`f }& }() | | `f#} " *~: ;0& measures the net| | amount of fluid flowing across this curve per unit of flow time| |,N} !! }}& b!( b"0 b#8 b$@ b%H b&P WW}]|[~p0 2 ~Q ]|Expr|[#b @`bb#_b#_b#_})!# b%$`fb#@}" *~: ;bP8&c55)Answer| |}& b!( b"0 b#8 b$@ b%H b&P WW}]|[~p1 2 ~Q ]|Expr|[#b @`bb#_b#_b#_})!# b'4[b A!!(8" *~: ;0& Fixing a particular| |`f#} t`f },L you know that in a short`f#} flowtime`f } approximately}| |(5 `f#} flowtime units of fluid flow across| | the a small segment of the curve of length`f#} ds`f } }(¢ered| | at`f#} " *~: ;0& tells you that| | the net flow of this vector field across the curve above is`f#} }| |`f }(*`f#}0,N628319`f } gallons per second,N} !! }| |}& b!( b"0 b#8 b$@ b%H b&P WW}]|[~p0 3 ~Q ]|Expr|[#b @`bb#_b#_b#_})"# b%$`f0}" *~: ;bP8&c55)Additional&c55*,Z| |}& b!( b"0 b#8 b$@ b%H b&P WW}]|[~p1 3 ~Q ]|Expr|[#b @`bb#_b#_b#_})!# b)E[#!!)b ^# b)E" *~: ;bP8&c55*It| | is beautiful how Bill ,F Jerry reduce a very complicated concept| | down to such simplicity,Z the integral is computing the area| | of a parallelogram using a vector dot product,N| |}& b!( b"0 b#8 b$@ b%H b&P WW}) # b)E}& b!( b"0 b#8 b$@ b%H b&P WW}| |)b!R# b)EAlthough this is a wonderful explanation,L once you | |study it once or twice,L you then get into the habit of just | |using the flow,Macross integral to compute,N What I like to | |do is always look at the fields and try to get a sense of what| | we are computing in the flow,Macross integral,N| |}& b!( b"0 b#8 b$@ b%H b&P WW}! !! ! }| |}& b!( b"0 b#8 b$@ b%H b&P WW}]|[~p0 4 ~T~Q ]|Expr|[#b @`bb#_b"Vb D})+# b(K" *~: ;bP8&c55*Flow Across Visualizer| |,Z Ellipse Region}& b!( b"0 b#8 b$@ b%H b&P WW}]|[~p0 0 ~A(m(?x,?y)=0.80000000000000004*sin(?y))~p0 0 ~d~A(n(?x,?y)=0.40000000000000002*?x+0.10000000000000001*~ ?y)~p0 0 ~d~A(Field(?x,?y)=(m(?x,?y),n(?x,?y)))~p0 0 ~d~A(Field(?x,?y)=(0.80000000000000004*sin(?y),~ 0.40000000000000002*?x+0.10000000000000001*?y))~p0 1 ~sb/_!!! } b00(!! c#T"_c/__c/__! "" } ^ _~Q ]|Expr|[#b @`bb#_b"Vb D})&# b%$" *~: ;bP8&c55*Curve Parameterization| |,Z Ellipse}& b!( b"0 b#8 b$@ b%H b&P WW}]|[~p0 0 ~A(xx(?t)=2*cos(?t))~p0 0 ~d~A(yy(?t)=sin(?t))~p0 0 ~d~Q ]|Expr|[#b @`bb#_b"Vb D}),# b(K" *~: ;bP8&c55*Derivatives to | |give the tangent vectors,Z}& b!( b"0 b#8 b$@ b%H b&P WW}]|[~p0 0 ~A(xx'(?t)=EvaluateAt(Derivative(t)*xx(t)):(t=~ ?t))~p0 0 ~A(xx'(?t)=-2*sin(?t))~p0 1 ~d~sb/_!!! } b00&!! c#T"_c/__c/__!" } ^ _~A(~ yy'(?t)=EvaluateAt(Derivative(t)*yy(t)):(t=?t))~p0 0 ~A(yy'(?t)=cos(?t))~p0 1 ~d~sb/_!!! } b00&!! c#T"_c/__c/__!" } ^ _~Q ]|Expr|[#b @`bb#_b"Vb D})%# b%$" *~: ;bP8&c55*Full Vector Field| |}& b!( b"0 b#8 b$@ b%H b&P WW}]|[~p1 0 ~G1 1 353 670 3 2 5 4 10 (-4...8):(-6...6):(c=~ 0...2*'p):(1/25*'p):(50)~Q ]|Expr|[#b @`bb#_b#_b#_})%# b$ " *~: ;bP8&c55*Graph Building | |Blocks}& b!( b"0 b#8 b$@ b%H b&P WW}]|[~p1 0 ~A(c=2.7646015351590183)~p0 1 ~d~R8405120 1 (x,y):(y=bottom...top):(x=left...~ right):(0)~p0 1 ~R8405120 1 (x,y):(x=left...right):(y=bottom...~ top):(0)~p0 1 ~X2 8405120 (max(left,min(0,right)),y):(y=bottom...~ top):(y)~p0 1 ~X1 8405120 (x,max(bottom,min(0,top))):(x=left...~ right):(x)~p0 1 ~V?c64 (left)~p0 1 ~V?c65 (right)~p0 1 ~V?c66 (bottom)~p0 1 ~V?c67 (top)~p0 1 ~F1 1 4 5 ? ? 16744448 ? ? (x,y):(m(x,y),n(x,y)):(~ (x=left...right)*(y=bottom...top))~p0 0 ~L9 255 ? (xx(t),yy(t)):(t=0...2*'p)~p0 0 ~S21 ? 10027263 ? ? (xx(c),yy(c)):(?):(7)~p0 0 ~F1 1 3 5 ? ? 21760 ? ? (xx(c),yy(c)):(yy'(c),~ -xx'(c)):(t=0...2*'p)~p0 0 ~t~p0 1 ~Q ]|Expr|[#b @`bb#_b"Vb D})'# b%$" *~: ;bP8&c55*Full Unit Vector| | Field}& b!( b"0 b#8 b$@ b%H b&P WW}]|[~p0 0 ~G1 1 353 670 2 2 5 4 10 (-7...9):(-8...8):(c=~ 0...2*'p):(1/25*'p):(50)~Q ]|Expr|[#b @`bb#_b#_b#_})%# b$ " *~: ;bP8&c55*Graph Building | |Blocks}& b!( b"0 b#8 b$@ b%H b&P WW}]|[~p1 0 ~A(c=6.0318578948924033)~p0 1 ~d~R8405120 1 (x,y):(y=bottom...top):(x=left...~ right):(0)~p0 1 ~R8405120 1 (x,y):(x=left...right):(y=bottom...~ top):(0)~p0 1 ~X2 8405120 (max(left,min(0,right)),y):(y=bottom...~ top):(y)~p0 1 ~X1 8405120 (x,max(bottom,min(0,top))):(x=left...~ right):(x)~p0 1 ~V?c64 (left)~p0 1 ~V?c65 (right)~p0 1 ~V?c66 (bottom)~p0 1 ~V?c67 (top)~p0 1 ~L9 255 ? (xx(t),yy(t)):(t=0...2*'p)~p0 0 ~S21 ? 10027263 ? ? (xx(c),yy(c)):(?):(7)~p0 0 ~F1 1 4 5 ? ? 16744448 ? ? (x,y):((m(x,y),n(x,~ y))/abs(m(x,y),n(x,y))):((x=left...right)*(y=bottom...~ top))~p0 0 ~F1 1 3 5 ? ? 21760 ? ? (xx(c),yy(c)):(yy'(c),~ -xx'(c)):(t=0...2*'p)~p0 0 ~t~p0 1 ~Q ]|Expr|[#b @`bb#_b"Vb D})*# b%$" *~: ;bP8&c55*Unit Vector Field| | On Curve }& b!( b"0 b#8 b$@ b%H b&P WW}]|[~p0 0 ~G1 1 353 670 2 2 5 4 10 (-7...9):(-8...8):(c=~ 0...2*'p):(1/25*'p):(50)~Q ]|Expr|[#b @`bb#_b#_b#_})%# b$ " *~: ;bP8&c55*Graph Building | |Blocks}& b!( b"0 b#8 b$@ b%H b&P WW}]|[~p1 0 ~A(c=2.7646015351590183)~p0 1 ~d~R8405120 1 (x,y):(y=bottom...top):(x=left...~ right):(0)~p0 1 ~R8405120 1 (x,y):(x=left...right):(y=bottom...~ top):(0)~p0 1 ~X2 8405120 (max(left,min(0,right)),y):(y=bottom...~ top):(y)~p0 1 ~X1 8405120 (x,max(bottom,min(0,top))):(x=left...~ right):(x)~p0 1 ~V?c64 (left)~p0 1 ~V?c65 (right)~p0 1 ~V?c66 (bottom)~p0 1 ~V?c67 (top)~p0 1 ~L9 255 ? (xx(t),yy(t)):(t=0...2*'p)~p0 0 ~S21 ? 10027263 ? ? (xx(c),yy(c)):(?):(7)~p0 0 ~F1 1 4 5 ? ? 16744448 ? ? (xx(c),yy(c)):((m(xx(~ c),yy(c)),n(xx(c),yy(c)))/abs(m(xx(c),yy(c)),n(~ xx(c),yy(c)))):(t=0...2*'p)~p0 0 ~F1 1 3 5 ? ? 21760 ? ? (xx(c),yy(c)):(yy'(c),~ -xx'(c)):(t=0...2*'p)~p0 0 ~t~p0 1 ~Q ]|Expr|[#b @`bb#_b"Vb D})-# b%$" *~: ;bP8&c55*Flow Across,Z | |Normal "!Symbol^:!&c( .W: &c55* Field 'b!4| |}& b!( b"0 b#8 b$@ b%H b&P WW}]|[~p1 0 ~G1 1 356 468 1 1 5 4 10 (-2...6):(-64...72):(~ ?=0...2*'p):('p/5):(10)~Q ]|Expr|[#b @`bb#_b#_b#_})%# b$ " *~: ;bP8&c55*Graph Building | |Blocks}& b!( b"0 b#8 b$@ b%H b&P WW}]|[~p1 0 ~R8405120 1 (x,y):(y=bottom...top):(x=left...right):(~ 0)~p0 1 ~R8405120 1 (x,y):(x=left...right):(y=bottom...~ top):(0)~p0 1 ~X2 8405120 (max(left,min(0,right)),y):(y=bottom...~ top):(y)~p0 1 ~X1 8405120 (x,max(bottom,min(0,top))):(x=left...~ right):(x)~p0 1 ~V?c64 (left)~p0 1 ~V?c65 (right)~p0 1 ~V?c66 (bottom)~p0 1 ~V?c67 (top)~p0 1 ~L8 255 ? (t,(yy'(t),-xx'(t))*@(m(xx(t),yy(t)),~ n(xx(t),yy(t)))):(t=0...2*'p)~p0 0 ~t~p0 1 ~Q ]|Expr|[#b @`bb#_b"Vb D})'# b%$" *~: ;bP8&c55*Flow Across Integrals| |'b!4'b!4}& b!( b"0 b#8 b$@ b%H b&P WW}]|[~p0 0 ~A(I=Integral(((yy'(t),-xx'(t))*@(m(xx(t),yy(t)),~ n(xx(t),yy(t))))*d*t):(0):(2*'p))~p0 1 ~A(I=Integral(((cos(t),-(-2*sin(t)))*@(0.80000000000000004*~ sin(yy(t)),0.40000000000000002*xx(t)+0.10000000000000001*~ yy(t)))*d*t):(0):(2*'p))~p0 2 ~sb/^!!! } b002!! c#T"!c&H#_c/__c/__! " # $ % &" } ^ _~ ~A(I=0.62831852741252181)~p0 3 ~sb/_!!! } $&!! c#T"!c&H#_c/__c/__ } ^ _~A(I=~ Integral((-n(xx(t),yy(t))*xx'(t)+m(xx(t),yy(t))*~ yy'(t))*d*t):(0):(2*'p))~p0 1 ~A(I=Integral((-(0.40000000000000002*xx(t)+0.10000000000000001*~ yy(t))*xx'(t)+(0.80000000000000004*sin(yy(t)))*~ yy'(t))*d*t):(0):(2*'p))~p0 2 ~sb/_! ! } b00( ! c#T"_c/__c/__! "" } ^ _~A(~ I=Integral((-(0.40000000000000002*(2*cos(t))+0.10000000000000001*~ sin(t))*(-2*sin(t))+(0.80000000000000004*sin(sin(~ t)))*cos(t))*d*t):(0):(2*'p))~p0 3 ~sb/_! ! } b00, ! c#T"_c/__c/__! " # $" } ^ _~ ~A(I=0.62831852741252181)~p0 4 ~sb/_!! } $&! c#T"!c&H#_c/__c/__ } ^ _~Q ]|Expr|[#b @`bb#_b#_b#_})b N# b%$" *~: ;bP8&c55*The fight between| | the field and the normal vectors is positive,L indicating that| | the flow across is from inside to outside,N | |}& b!( b"0 b#8 b$@ b%H b&P WW}]|[~p0 1 ~t~p0 4 ~Q ]|Expr|[#b @`bb#_b#_b#_})!# b%$" *~: ;bP8&c55* | |}& b!( b"0 b#8 b$@ b%H b&P WW}]|[~p0 1 ~Q ]|Expr|[#b @`bb#_b#_b#_}))# b%$`fb#@}" *~: ;bP8&c55)B,N1,Na,N| |ii,I }& b!( b"0 b#8 b$@ b%H b&P WW}]|[~p0 1 ~Q ]|Expr|[#b @`bb#_b#_b#_})!# b'4[#!!(+" *~: ;0& Take another look| | at the integral}() `f#} " *~: ;0& This tells you| | that the net flow of this vector field across this curve is }| |(0about`f#} 0,N63`f } fluid units per time unit,N}(>Why does | |this also tell you that the net flow is from inside to outside| |,_} !! }}& b!( b"0 b#8 b$@ b%H b&P WW}]|[~p0 2 ~Q ]|Expr|[#b @`bb#_b#_b#_})!# b%$`fb#@}" *~: ;bP8&c55)Answer| |}& b!( b"0 b#8 b$@ b%H b&P WW}]|[~p1 2 ~Q ]|Expr|[#b 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3 322 -1 1 322 -1 2 322 -1 ~c3 4 327 -1 3 327 -1 1 327 -1 2 327 -1 ~c2 10 327 -1 9 327 -1 5 327 -1 ~c2 12 327 -1 11 327 -1 6 327 -1 ~c15 14 327 -1 13 327 -1 5 327 -1 103 -1 6 327 -1 102 -1 1 327 -1 26 -1 2 327 -1 27 -1 3 327 -1 25 -1 12 327 -1 84 -1 10 327 -1 85 -1 ~c3 4 331 -1 3 331 -1 1 331 -1 2 331 -1 ~c2 10 331 -1 9 331 -1 6 331 -1 ~c2 12 331 -1 11 331 -1 7 331 -1 ~c7 23 331 -1 22 331 -1 1 331 -1 2 331 -1 6 331 -1 7 331 -1 10 331 -1 12 331 -1 ~c3 24 331 -1 23 331 -1 7 331 -1 6 331 -1 ~c3 26 331 -1 25 331 -1 1 331 -1 2 331 -1 ~c5 27 331 -1 26 331 -1 6 331 -1 7 331 -1 10 331 -1 12 331 -1 ~c1 28 331 -1 27 331 -1 ~c3 4 334 -1 3 334 -1 1 334 -1 2 334 -1 ~c3 4 340 -1 3 340 -1 1 340 -1 2 340 -1 ~c2 11 340 -1 10 340 -1 5 340 -1 ~c2 13 340 -1 12 340 -1 6 340 -1 ~c15 20 340 -1 19 340 -1 5 340 -1 103 -1 6 340 -1 102 -1 1 340 -1 26 -1 2 340 -1 27 -1 3 340 -1 25 -1 13 340 -1 84 -1 11 340 -1 85 -1 ~c3 4 342 -1 3 342 -1 1 342 -1 2 342 -1 ~c2 8 342 -1 7 342 -1 5 342 -1 ~c2 10 342 -1 9 342 -1 6 342 -1 ~c3 4 346 -1 3 346 -1 1 346 -1 2 346 -1 ~c2 10 346 -1 9 346 -1 6 346 -1 ~c2 12 346 -1 11 346 -1 7 346 -1 ~c7 23 346 -1 22 346 -1 1 346 -1 2 346 -1 6 346 -1 7 346 -1 10 346 -1 12 346 -1 ~c3 24 346 -1 23 346 -1 7 346 -1 6 346 -1 ~c3 26 346 -1 25 346 -1 1 346 -1 2 346 -1 ~c5 27 346 -1 26 346 -1 6 346 -1 7 346 -1 10 346 -1 12 346 -1 ~c1 28 346 -1 27 346 -1 ~e