\mnb150ÿ{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fmodern\fprq1 Lucida Sans Typewriter;}{\f3\fswiss\fprq2 Trebuchet MS;}{\f4\fswiss\fprq2 System;}} {\colortbl\red0\green0\blue0;\red255\green0\blue0;} \deflang1033\pard\ri4\plain\f3\fs36\cf0 Chapter 4 Multivariable calculus\plain\f3\fs28\cf0 \ \ 4.1 Functions of several variables\ \ \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}F := (x,y) -> sin(x)*cos(y)\ Distance := (x,y,z) -> sqrt(x^2 + y^2 + z^2)\ Temperature := (x,y,z,t) -> (4*PI*k*t)^(-3/2)*exp(-(x^2+y^2+z^2)/(4*k*t))\ S2C := (r,t,u)-> (r*sin(t)*cos(u),r*sin(t)*sin(u), r*cos(t))\ steps := (x,y) -> piecewise(\ \pard\li600\ri1\fi-300\plain\f2\fs28\cf1 [x > 0 and y > 0, 1],\ [x < 0 and y > 0, 2],\ [x < 0 and y < 0, 3],\ [x > 0 and y < 0, 4]\ ):\ \pard\ri4\plain\f2\fs28\cf1 \ \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}F(PI/3,PI/3)\ Distance(1,1,1)\ Temperature(1,1,1,1)\ steps(1,1)\ steps(-2,3)\ steps(0,0)\ \pard\ri4\plain\f3\fs28\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}Df := plot::Inequality(x^2-2*y^2>=0, x=-5..5, y=-5..5,\ \pard\li600\ri1\fi-300\plain\f2\fs28\cf1 Mesh=[100,100], FillColorFalse=RGB::Gray80\ ):\ plot(Df)\ \pard\ri4\plain\f2\fs28\cf1 \plain\f3\fs28\cf0 \ >>>>>>>>>> Time for Exercise 4.1 \par \ 4.2 Visualizing functions of several variables\ \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}plotfunc3d(F(x,y), x=-PI..PI, y=-PI..PI)\ plotfunc3d(steps(x,y), x=-1..1, y=-1..1)\ K := (x,y,t) -> sin(x)*sin(2*y)*sin(3*t):\ \pard\li600\ri1\fi-300\plain\f2\fs28\cf1 plotfunc3d(K(x,y,0), x=-PI..PI, y=-PI..PI)\ \pard\ri4\plain\f2\fs28\cf1 \plain\f3\fs28\cf0 \ \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}plotfunc3d(K(x,y,0.5), x=-PI..PI, y=-PI..PI,Scaling=Constrained)\ \pard\ri4\plain\f2\fs28\cf1 \plain\f3\fs28\cf0 \par \ \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}plotfunc3d(K(x,y,1), x=-PI..PI, y=-PI..PI,\ \pard\li600\ri1\fi-300\plain\f2\fs28\cf1 Scaling=Constrained\ )\ \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}f := (x,y,t) -> (x^2+y^2)*t^2:\ \pard\li600\ri1\fi-300\plain\f2\fs28\cf1 S4D := plot::Function3d(\ f(x,y,t),x=-2..2,y=-2..2, t=0..3\ ):\ plot(S4D)\ \pard\ri4\plain\f3\fs28\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}h := (x,y) -> 9*(x^2-y^2)*exp(-x^2-y^2):\ \pard\li600\ri1\fi-300\plain\f2\fs28\cf1 a := 2.5: b:=2.5:\ H := plot::Function3d(h(x,y), x=-a..a, y=-b..b):\ plot(H)\ \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}DP := plot::Density(h(x,y), x=-a..a, y=-b..b,\ \pard\li600\ri1\fi-300\plain\f2\fs28\cf1 Mesh=[50,50],\ FillColor=RGB::White, FillColor2=RGB::Indigo):\ plot(DP)\ \pard\ri4\plain\f3\fs28\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}CP := plot::Function3d(h(x,y), x=-a..a, y=-b..b,\ \pard\li600\ri1\fi-300\plain\f2\fs28\cf1 ZContours=[n/5 $ n=-20..20],// show contours\ XLinesVisible=FALSE, //hide mesh for x\ YLinesVisible=FALSE, //hide mesh for y\ Filled=FALSE //hide the surface\ ):\ plot(CP)\ \pard\ri4\plain\f2\fs28\cf1 \plain\f3\fs28\cf0 \ \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}H := plot::Function3d(h(x,y),\ \pard\li600\ri1\fi-300\plain\f2\fs28\cf1 x=-2.5..c, y=-2.5..c, c=-2.5..2.5\ ):\ plot(H)\ \pard\ri4\plain\f3\fs28\cf0 \par >>>>>>>>>> Time for Exercise 4.2 \par \ 4.3 Limits of functions of several variables\ \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}f := (x,y) -> sin(x*y)\ limit(f(x,y), x=PI/2)\ limit(%, y=PI/2)\ float(%)\ float(limit(limit(f(x,y), x=PI/2), y=PI/2))\ g := (x,y) -> (1-x*y)/(1+x*y)\ limit(limit(g(x,y), x=1), y=1)\ limit(limit(g(x,y), y=1), x=1)\ h := (x,y) -> (x^2-y^2)/(x^2+y^2)\ limit(limit(h(x,y), x=0),y=0)\ limit(limit(h(x,y), y=0), x=0)\ H := subs(h(x,y), y=m*x)\ H := simplify(H)\ subs(H, m=1)\ subs(H, m=2)\ subs(final, m=3)\ subs(H, m=4)\ \pard\ri4\plain\f3\fs28\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}h := (x,y)->(x^2-y^2)/(x^2+y^2):\ C := (x,m) -> [x, m*x, -(m^2-1)/(m^2+1)]:\ C1 := plot::Curve3d(\ \pard\li600\ri1\fi-300\plain\f2\fs28\cf1 subs(C(x,m),m=1), x=-1..1,\ UMesh=2, LineWidth=0.5\ ):\ C2 := plot::Curve3d(\ subs(C(x,m),m=2), x=-1..1,\ UMesh=2, LineWidth=1\ ):\ C3 := plot::Curve3d(\ subs(C(x,m),m=-1/2), x=-1..1,\ UMesh=2, LineWidth=1.5\ ):\ H := plot::Function3d(h(x,y), x=-1..1, y=-1..1):\ plot(H, C1, C2, C3)\ \pard\ri4\plain\f3\fs28\cf0 \par >>>>>>>>>> Time for Exercise 4.3 \ \ 4.4 Partial differentiation\ \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}h := (x,y) -> x*y/(x^2+y^2)\ diff(h(x,y), x) // first derivative in x\ diff(h(x,y), y) // first derivative in y\ diff(h(x,y), x,y) // mixed derivative for x,y\ diff(h(x,y), x,x) // second derivative in x\ f := (x,y) -> 5-2*x^2-y^2:\ \pard\li600\ri1\fi-300\plain\f2\fs28\cf1 TangentPlane := (a,b)->(\ subs(diff(f(x,y),x),x=a, y=b)*(x-a) +\ subs(diff(f(x,y),y),x=a, y=b)*(y-b) + f(a,b)\ ):\ \pard\ri4\plain\f2\fs28\cf1 \ \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}plotfunc3d(\ \pard\li600\ri1\fi-300\plain\f2\fs28\cf1 f(x,y),\ TangentPlane(1,1),\ TangentPlane(-2,-2)\ )\ \pard\ri4\plain\f2\fs28\cf1 \plain\f3\fs28\cf0 \ >>>>>>>>>> Time for Exercise 4.4 \par \ 4.5 Vector fields\ \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}z := (x,y)->sin(x)*cos(y)\ F := diff(z(x,y),x):\ G := diff(z(x,y),y):\ V := plot::VectorField2d([F,G], x=-2..2, y=-2..2,\ \pard\li600\ri1\fi-300\plain\f2\fs28\cf1 Mesh=[15,15]\ ):\ plot(V)\ \pard\ri4\plain\f2\fs28\cf1 \plain\f3\fs28\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}reset():\ u := (x,y,z)->1-((x-1)^2+(y-1)^2+(z-1)^2):\ \pard\li600\ri1\fi-300\plain\f2\fs28\cf1 F := diff(u(x,y,z),x):\ G := diff(u(x,y,z),y):\ H := diff(u(x,y,z),z):\ \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}V := plot::VectorField3d([F,G,H],\ \pard\li600\ri1\fi-300\plain\f2\fs28\cf1 x=-2..2, y=-2..2, z=-2..2,\ Mesh=[10,10,10],\ PointSize=1\ ):\ \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}P := plot::Point3d([1,1,1],\ \pard\li600\ri1\fi-300\plain\f2\fs28\cf1 PointColor=RGB::Red,\ PointSize=3,\ PointStyle=FilledSquares\ ):\ plot(V,P)\ \pard\ri4\plain\f3\fs28\cf0 \par Technical comment: \par ================= \par The 3D vector field class in MuPAD uses a dot instead of an arrow. In other \par words, the dot replaces the arrow tip. This way the picture is much clear.\ \par >>>>>>>>>> Time for Exercise 4.5\ \ 4.6 Multiple integrals with MuPAD\ \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}reset(): \par \pard\ri4\plain\f3\fs28\cf0 \ \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}int(int(3*x+7*y, x),y)\ int(int(x^2+y^2, x),y)\ int(int(int(x*y*z, x), y), z)\ int(x*y*z, x)\ int(%,y)\ int(%,z)\ int(int(x^2+y^2, x=0..1), y=0..1)\ \pard\ri4\plain\f3\fs28\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}f := (x,y) -> (3*x^2 + 5*y^2)\ int(f(x,y), y=-sin(x)..sin(x))\ int(%, x=0..PI)\ int(int(f(x,y), y=-sin(x)..sin(x)),x=0..PI)\ V := (x^2+y^2+z^2=x)\ x := r*sin(t)*cos(u):\ \pard\li600\ri1\fi-300\plain\f2\fs28\cf1 y := r*sin(t)*sin(u):\ z := r*cos(t):\ \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}simplify(V) assuming r>0\ \pard\ri4\plain\f3\fs28\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}plot::Spherical([cos(u)*sin(t), u, t],\ \pard\li600\ri1\fi-300\plain\f2\fs28\cf1 u=0..2*PI, t=0..PI\ ):\ plot(%)\ \pard\ri4\plain\f2\fs28\cf1 \plain\f3\fs28\cf0 \ \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}F := x/sqrt(x^2+y^2+z^2)\ x := r*sin(t)*cos(u):\ \pard\li600\ri1\fi-300\plain\f2\fs28\cf1 y := r*sin(t)*sin(u):\ z := r*cos(t):\ \pard\ri4\plain\f3\fs28\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}G := x/sqrt(x^2+y^2+z^2)\ G := simplify(G) assuming r>0\ Jacobian := matrix(3,3,\ \pard\li600\ri1\fi-300\plain\f2\fs28\cf1 [[diff(x,r), diff(x,t),diff(x,u)],\ [diff(y,r), diff(y,t),diff(y,u)],\ [diff(z,r), diff(z,t),diff(z,u)]]\ )\ \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}Jacobian := simplify(linalg::det(Jacobian))\ int(\ \pard\li600\ri1\fi-300\plain\f2\fs28\cf1 int(\ int(G*Jacobian, r=0..cos(u)*sin(t)),\ u=0..2*PI),\ t=0..PI)\ \pard\ri4\plain\f3\fs28\cf0 \par >>>>>>>>>> Time for Exercise 4.6 \ \par 4.7 Visualizing and calculating volumes\ \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}reset(): \par {\pntext\f1\'b7\tab}SURF := z = a*(x^2+y^2):\ TopCircle := subs(SURF, z=h, x=R*cos(t), y=R*sin(t))\ TopCircle := simplify(TopCircle)\ a := 1:\ \pard\li600\ri1\fi-300\plain\f2\fs28\cf1 h := 9:\ R := sqrt(h/a):\ \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}TopCurveEq := t -> [R*cos(t),R*sin(t),h]\ Cylinder := plot::Sweep(TopCurveEq(t), t=0..2*PI,\ \pard\li600\ri1\fi-300\plain\f2\fs28\cf1 FillColor=[0,1,0,0.2],\ FillColorType=Flat\ ):\ \pard\ri4\plain\f2\fs28\cf1 \ \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}Surface := plot::Cylindrical(\ \pard\li600\ri1\fi-300\plain\f2\fs28\cf1 [sqrt(z/a),t,z], t=0..2*PI, z=0..h\ ):\ \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}BaseCircle := plot::Circle3d(R, [0,0,0], [0,0,1],\ \pard\li600\ri1\fi-300\plain\f2\fs28\cf1 FillColor=RGB::Red,\ Filled=TRUE\ ):\ \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}plot(Cylinder, Surface, BaseCircle,\ \pard\li600\ri1\fi-300\plain\f2\fs28\cf1 Scaling=Unconstrained\ ) \par \pard\ri4\plain\f3\fs28\cf0 \ \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}reset(): // this is important, why?\ R := sqrt(h/a):\ \pard\li600\ri1\fi-300\plain\f2\fs28\cf1 Vol := PI*R^2*h-int(int(a*r^2*r, r=0..R), t=0..2*PI)\ \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}LargePar := subs(Vol, a=1, h=9)\ SmallPar := subs(Vol, a=1, h=4)\ FinalVolume := LargePar - SmallPar\ \pard\ri4\plain\f3\fs28\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}reset():\ z := x -> x^2:\ \pard\li600\ri1\fi-300\plain\f2\fs28\cf1 A := plot::ZRotate(z(x), x=4..9):\ B := plot::Curve3d([x,0,x^2], x=4..9):\ \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}plot(A, B, Scaling=Unconstrained)\ \pard\ri4\plain\f2\fs28\cf1 \plain\f3\fs28\cf0 \ \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f2\fs28\cf1 {\pntext\f1\'b7\tab}int(PI*(sqrt(u))^2, u=4..9)\ \pard\ri4\plain\f3\fs28\cf0 \par >>>>>>>>>> Time for Exercise 4.7 \par }