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\deflang1033\pard\ri4\plain\f3\fs72\cf0 Chapter  4  Multivariable calculus\plain\f3\fs36\cf0 
\par 
\par 4.1 Functions of several variables
\par 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}F := (x,y) -> sin(x)*cos(y)
\par {\pntext\f1\'b7\tab}Distance := (x,y,z) -> sqrt(x^2 + y^2 + z^2)
\par {\pntext\f1\'b7\tab}Temperature := (x,y,z,t) ->  
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    (4*PI*k*t)^(-3/2)*exp(-(x^2+y^2+z^2)/(4*k*t))
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}S2C := (r,t,u)-> (r*sin(t)*cos(u),r*sin(t)*sin(u), r*cos(t))
\par {\pntext\f1\'b7\tab}steps := (x,y) -> piecewise(
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1     [x > 0 and y > 0, 1],
\par     [x < 0 and y > 0, 2],
\par     [x < 0 and y < 0, 3],
\par     [x > 0 and y < 0, 4]
\par ):
\par \pard\ri4\plain\f4\fs36\cf1 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}F(PI/3,PI/3)
\par {\pntext\f1\'b7\tab}Distance(1,1,1)
\par {\pntext\f1\'b7\tab}Temperature(1,1,1,1)
\par {\pntext\f1\'b7\tab}steps(1,1)
\par {\pntext\f1\'b7\tab}steps(-2,3)
\par {\pntext\f1\'b7\tab}steps(0,0)
\par \pard\ri4\plain\f3\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}Df := plot::Inequality(x^2-2*y^2>=0, x=-5..5, y=-5..5,
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    Mesh=[100,100], FillColorFalse=RGB::Gray80
\par ):
\par plot(Df)
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0    
\par >>>>>>>>>> Time for Exercise 4.1
\par 
\par 4.2 Visualizing functions of several variables
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}plotfunc3d(F(x,y), x=-PI..PI, y=-PI..PI)
\par {\pntext\f1\'b7\tab}plotfunc3d(steps(x,y), x=-1..1, y=-1..1)
\par {\pntext\f1\'b7\tab}K := (x,y,t) -> sin(x)*sin(2*y)*sin(3*t):
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 plotfunc3d(K(x,y,0), x=-PI..PI, y=-PI..PI)
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0    
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}plotfunc3d(K(x,y,0.5), x=-PI..PI, y=-PI..PI,Scaling=Constrained)
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0   
\par 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}plotfunc3d(K(x,y,1), x=-PI..PI, y=-PI..PI,
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    Scaling=Constrained
\par )
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}f := (x,y,t) -> (x^2+y^2)*t^2:
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 S4D := plot::Function3d(
\par    f(x,y,t),x=-2..2,y=-2..2, t=0..3
\par ):
\par plot(S4D)
\par \pard\ri4\plain\f3\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}h := (x,y) -> 9*(x^2-y^2)*exp(-x^2-y^2):
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 a := 2.5: b:=2.5:
\par H := plot::Function3d(h(x,y), x=-a..a, y=-b..b):
\par plot(H)
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}DP := plot::Density(h(x,y), x=-a..a, y=-b..b,
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    Mesh=[50,50],
\par    FillColor=RGB::White, FillColor2=RGB::Indigo):
\par plot(DP)
\par \pard\ri4\plain\f3\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}CP := plot::Function3d(h(x,y), x=-a..a, y=-b..b,
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    ZContours=[n/5 $ n=-20..20],// show contours
\par    XLinesVisible=FALSE,        //hide mesh for x
\par    YLinesVisible=FALSE,    //hide mesh for y
\par    Filled=FALSE            //hide the surface
\par ):
\par plot(CP)
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0    
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}H := plot::Function3d(h(x,y),
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    x=-2.5..c, y=-2.5..c, c=-2.5..2.5
\par ):
\par plot(H)
\par \pard\ri4\plain\f3\fs36\cf0 
\par >>>>>>>>>> Time for Exercise 4.2 
\par 
\par 4.3 Limits of functions of several variables
\par 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}f := (x,y) -> sin(x*y)
\par {\pntext\f1\'b7\tab}limit(f(x,y), x=PI/2)
\par {\pntext\f1\'b7\tab}limit(%, y=PI/2)
\par {\pntext\f1\'b7\tab}float(%)
\par {\pntext\f1\'b7\tab}float(limit(limit(f(x,y), x=PI/2), y=PI/2))
\par {\pntext\f1\'b7\tab}g := (x,y) -> (1-x*y)/(1+x*y)
\par {\pntext\f1\'b7\tab}limit(limit(g(x,y), x=1), y=1)
\par {\pntext\f1\'b7\tab}limit(limit(g(x,y), y=1), x=1)
\par {\pntext\f1\'b7\tab}h := (x,y) -> (x^2-y^2)/(x^2+y^2)
\par {\pntext\f1\'b7\tab}limit(limit(h(x,y), x=0),y=0)
\par {\pntext\f1\'b7\tab}limit(limit(h(x,y), y=0), x=0)
\par {\pntext\f1\'b7\tab}H := subs(h(x,y), y=m*x)
\par {\pntext\f1\'b7\tab}H := simplify(H)
\par {\pntext\f1\'b7\tab}subs(H, m=1)
\par {\pntext\f1\'b7\tab}subs(H, m=2)
\par {\pntext\f1\'b7\tab}subs(final, m=3)
\par {\pntext\f1\'b7\tab}subs(H, m=4)
\par \pard\ri4\plain\f3\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}h := (x,y)->(x^2-y^2)/(x^2+y^2):
\par {\pntext\f1\'b7\tab}C := (x,m) -> [x, m*x, -(m^2-1)/(m^2+1)]:
\par {\pntext\f1\'b7\tab}C1 := plot::Curve3d(
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    subs(C(x,m),m=1), x=-1..1,
\par    UMesh=2, LineWidth=0.5
\par ):
\par C2 := plot::Curve3d(
\par    subs(C(x,m),m=2), x=-1..1,
\par    UMesh=2, LineWidth=1
\par ):
\par C3 := plot::Curve3d(
\par    subs(C(x,m),m=-1/2), x=-1..1,
\par    UMesh=2, LineWidth=1.5
\par ):
\par H := plot::Function3d(h(x,y), x=-1..1, y=-1..1):
\par plot(H, C1, C2, C3)
\par \pard\ri4\plain\f3\fs36\cf0 
\par >>>>>>>>>> Time for Exercise 4.3 
\par      
\par 4.4 Partial differentiation
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}h := (x,y) -> x*y/(x^2+y^2)
\par {\pntext\f1\'b7\tab}diff(h(x,y), x) // first derivative in x
\par {\pntext\f1\'b7\tab}diff(h(x,y), y) // first derivative in y
\par {\pntext\f1\'b7\tab}diff(h(x,y), x,y) // mixed derivative for x,y
\par {\pntext\f1\'b7\tab}diff(h(x,y), x,x) // second derivative in x
\par {\pntext\f1\'b7\tab}f := (x,y) -> 5-2*x^2-y^2:
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 TangentPlane := (a,b)->(
\par    subs(diff(f(x,y),x),x=a, y=b)*(x-a) +
\par    subs(diff(f(x,y),y),x=a, y=b)*(y-b) + f(a,b)
\par ):
\par \pard\ri4\plain\f4\fs36\cf1 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}plotfunc3d(
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    f(x,y),
\par    TangentPlane(1,1),
\par    TangentPlane(-2,-2)
\par )
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0    
\par >>>>>>>>>> Time for Exercise 4.4  
\par     
\par 4.5 Vector fields
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}z := (x,y)->sin(x)*cos(y)
\par {\pntext\f1\'b7\tab}F := diff(z(x,y),x):
\par {\pntext\f1\'b7\tab}G := diff(z(x,y),y):
\par {\pntext\f1\'b7\tab}V := plot::VectorField2d([F,G], x=-2..2, y=-2..2,
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    Mesh=[15,15]
\par ):
\par plot(V)
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0    
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}reset():
\par {\pntext\f1\'b7\tab}u := (x,y,z)->1-((x-1)^2+(y-1)^2+(z-1)^2):
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 F := diff(u(x,y,z),x):
\par G := diff(u(x,y,z),y):
\par H := diff(u(x,y,z),z):
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}V := plot::VectorField3d([F,G,H],
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    x=-2..2, y=-2..2, z=-2..2,
\par    Mesh=[10,10,10],
\par    PointSize=1
\par ):
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}P := plot::Point3d([1,1,1],
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    PointColor=RGB::Red,
\par    PointSize=3,
\par    PointStyle=FilledSquares
\par ):
\par plot(V,P)
\par \pard\ri4\plain\f3\fs36\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 
\par >>>>>>>>>> Time for Exercise 4.5
\par      
\par 4.6 Multiple integrals with MuPAD
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}reset():
\par \pard\ri4\plain\f3\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}int(int(3*x+7*y, x),y)
\par {\pntext\f1\'b7\tab}int(int(x^2+y^2, x),y)
\par {\pntext\f1\'b7\tab}int(int(int(x*y*z, x), y), z)
\par {\pntext\f1\'b7\tab}int(x*y*z, x)
\par {\pntext\f1\'b7\tab}int(%,y)
\par {\pntext\f1\'b7\tab}int(%,z)
\par {\pntext\f1\'b7\tab}int(int(x^2+y^2, x=0..1), y=0..1)
\par \pard\ri4\plain\f3\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}f := (x,y) -> (3*x^2 + 5*y^2)
\par {\pntext\f1\'b7\tab}int(f(x,y), y=-sin(x)..sin(x))
\par {\pntext\f1\'b7\tab}int(%, x=0..PI)
\par {\pntext\f1\'b7\tab}int(int(f(x,y), y=-sin(x)..sin(x)),x=0..PI)
\par {\pntext\f1\'b7\tab}V := (x^2+y^2+z^2=x)
\par {\pntext\f1\'b7\tab}x := r*sin(t)*cos(u):
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 y := r*sin(t)*sin(u):
\par z := r*cos(t):
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}simplify(V) assuming r>0
\par \pard\ri4\plain\f3\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}plot::Spherical([cos(u)*sin(t), u, t],
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    u=0..2*PI, t=0..PI
\par ):
\par plot(%)
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0    
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}F := x/sqrt(x^2+y^2+z^2)
\par {\pntext\f1\'b7\tab}x := r*sin(t)*cos(u):
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 y := r*sin(t)*sin(u):
\par z := r*cos(t):
\par \pard\ri4\plain\f3\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}G := x/sqrt(x^2+y^2+z^2)
\par {\pntext\f1\'b7\tab}G := simplify(G) assuming r>0
\par {\pntext\f1\'b7\tab}Jacobian := matrix(3,3,
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1   [[diff(x,r), diff(x,t),diff(x,u)],
\par    [diff(y,r), diff(y,t),diff(y,u)],
\par    [diff(z,r), diff(z,t),diff(z,u)]]
\par )
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}Jacobian := simplify(linalg::det(Jacobian))
\par {\pntext\f1\'b7\tab}int(
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    int(
\par       int(G*Jacobian, r=0..cos(u)*sin(t)),
\par    u=0..2*PI),
\par t=0..PI)
\par \pard\ri4\plain\f3\fs36\cf0 
\par >>>>>>>>>> Time for Exercise 4.6   
\par 
\par 4.7 Visualizing and calculating volumes
\par 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}reset():
\par {\pntext\f1\'b7\tab}SURF := z = a*(x^2+y^2):
\par {\pntext\f1\'b7\tab}TopCircle := subs(SURF, z=h, x=R*cos(t), y=R*sin(t))
\par {\pntext\f1\'b7\tab}TopCircle := simplify(TopCircle)
\par {\pntext\f1\'b7\tab}a := 1:
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 h := 9:
\par R := sqrt(h/a):
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}TopCurveEq := t -> [R*cos(t),R*sin(t),h]
\par {\pntext\f1\'b7\tab}Cylinder := plot::Sweep(TopCurveEq(t), t=0..2*PI,
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    FillColor=[0,1,0,0.2],
\par    FillColorType=Flat
\par ):
\par \pard\ri4\plain\f4\fs36\cf1 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}Surface := plot::Cylindrical(
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    [sqrt(z/a),t,z], t=0..2*PI, z=0..h
\par ):
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}BaseCircle := plot::Circle3d(R, [0,0,0], [0,0,1],
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    FillColor=RGB::Red,
\par    Filled=TRUE
\par ):
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}plot(Cylinder, Surface, BaseCircle,
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1   Scaling=Unconstrained
\par )
\par \pard\ri4\plain\f3\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}reset(): // this is important, why?
\par {\pntext\f1\'b7\tab}R := sqrt(h/a):
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 Vol := PI*R^2*h-int(int(a*r^2*r, r=0..R), t=0..2*PI)
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}LargePar := subs(Vol, a=1, h=9)
\par {\pntext\f1\'b7\tab}SmallPar := subs(Vol, a=1, h=4)
\par {\pntext\f1\'b7\tab}FinalVolume := LargePar - SmallPar
\par \pard\ri4\plain\f3\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}reset():
\par {\pntext\f1\'b7\tab}z := x -> x^2:
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 A := plot::ZRotate(z(x), x=4..9):
\par B := plot::Curve3d([x,0,x^2], x=4..9):
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}plot(A, B, Scaling=Unconstrained)
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0    
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}int(PI*(sqrt(u))^2, u=4..9)
\par \pard\ri4\plain\f3\fs36\cf0 
\par >>>>>>>>>> Time for Exercise 4.7 
\par }