Multivariable Procedures 

Function: calcplot procedures: dxdyplot, dydxplot - plot a 2D region described in Cartesian coordinates 

Calling Sequence:  

dxdyplot x = g(y) .. k(y), y = a .. b, opts 

dydxploty = h(x) .. m(x), x = a .. b, opts 

Parameters:  

The variable names x and y must be used. 

g(y), k(y)- expressions in the variable y. 

h(x), m(x) - expressions in the variable  

a, b, c - symbolic constants, such as Pi, integers, or real floating point numbers. 

opts -a sequence of plot options. 

Description: 

  • The dxdyplot, dydxplot procedures plot regions in the plane with the given arguments as boundary curves
 

  • The procedures draw three line segments that pass through the interior of the region
 

  • The syntax is the same as if int procedure were invoked for a double integral over the region
 

  • The procedures return plot structures, accept plot options, and may be passed to the display procedure from the plots package
 

Examples: 

> restart: with(calcplot):with(student):
 

 

> dxdyplot(x=0..sqrt(y),y=0..1);
 

Plot_2d
 

> A:=Doubleint(1,x=0..sqrt(y),y=0..1):
A=value(A);
 

Int(Int(1, x = 0 .. `*`(`^`(y, `/`(1, 2)))), y = 0 .. 1) = `/`(2, 3)
 

> dydxplot(y=x^2..1,x=0..1);
 

Plot_2d
 

> A:=Doubleint(1,y=x^2..1,x=0..1):
A=value(A);
 

Int(Int(1, y = `*`(`^`(x, 2)) .. 1), x = 0 .. 1) = `/`(2, 3)
 

Function: calcplot procedures: dxdydzplot, dxdzdyplot, dydxdzplot, dydzdxplot, dzdxdyplot, dzdydxplot - plot the boundary of a solid region described in Cartesian coordinates. 

Calling Sequence:  

dxdydzplotx = f(y, z) .. g(y, z), y = h(z) .. k(z), z = a .. b, opts 

dxdzdyplotx = f(y, z) .. g(y, z), z = h(y) .. k(y), y = a .. b, opts 

dydxdzploty = f(x, z) .. g(x, z), x = h(z) .. k(z), z = a .. b, opts 

dydzdxploty = f(x, z) .. g(x, z), z = h(x) .. k(x), x = a .. b, opts 

dzdxdyplotz = f(x, y) .. g(x, y), x = h(y) .. k(y), y = a .. b, opts 

dzdydxplotz = f(x, y) .. g(x, y), y = h(x) .. k(x), x = a .. b, opts 

Parameters:  

The variable names and z  must be used. 

f, g      - two-variable expressions.  

h, k     - one-variable expressions. 

a, b, c - symbolic constants, such as Pi, integers, or real floating point numbers. 

opts    - a sequence of plot3d opt ions. 

Description: 

  • All six procedures plot regions in space with the given arguments providing ranges for the variables
 

  • The syntax is the same as if the int procedure were invoked for a triple integral over the region in Cartesian coordinates
 

  • Each procedure returns a plot3d structure, accepts plot3d options, and may be passed to the display procedure from the plots package
 

Examples: 

> restart: with(calcplot):with(student):
 

> x1:=sqrt(9-y^2):
 

> plt:=dzdxdyplot(z=0..y/2,x=-x1..x1,y=0..3,labels=[x,y,z],labelfont=[times,bold,14],orientation=[-46,71],grid=[45,45],style=patchnogrid,lightmodel=light2,scaling=unconstrained,axes=frame,transparency=0.5):
 

> plt;
 

Plot_2d
 

> V:=Tripleint(1,z=0..y/2,x=-x1..x1,y=0..3):
 

> V=value(V);
 

Int(Int(Int(1, z = 0 .. `+`(`*`(`/`(1, 2), `*`(y)))), x = `+`(`-`(`*`(`^`(`+`(9, `-`(`*`(`^`(y, 2)))), `/`(1, 2))))) .. `*`(`^`(`+`(9, `-`(`*`(`^`(y, 2)))), `/`(1, 2)))), y = 0 .. 3) = 9
 

Function: calcplot[regionplot2d] -plots images of regions under arbitrary coordinate transformations 

Calling Sequence:  

regionplot2dv = h(u) .. k(u), u = a .. b, [x(u, v), y(u, v)], opts 

Parameters:  

g(u), k(u)  - expressions in the variable u. 

[x(u, v), y(u, v)] - a list of expressions in the variables u and v. 

a, b, c  - symbolic constants, such as Pi, integers, or real floating point numbers. 

opts - a sequence of plot options . 

Description: 

  • The regionplot2d procedure plots the image of a region in the uv-plane of the form `and`(`<=`(h(u), v), `<=`(v, k(u))), `and`(`<=`(a, u), `<=`(u, b)) under a coordinate transformation given by the list [x(u, v), y(u, v)]
 

  • The procedure also draws four segments passing through the interior of the region
 

  • The variable order is the same as if the int procedure were invoked for a double integral over the region
 

  • The procedure returns a plot structure, accepts plot options, and may be passed to the display procedure from the plots package
 

Examples: 

> restart: with(calcplot):MathMaple:-ini(): _EnvExplicit:=true:
 

a) Find the centroid of the first quadrant region in the xy-plane that is bounded by the curves
and `*`(`^`(x, 3), `*`(y)) = 4. 

Solution 

a) 

> eq1:=u=x*y: eq2:=v=x^3*y:eq1,eq2;
 

u = `*`(x, `*`(y)), v = `*`(`^`(x, 3), `*`(y))
 

Solving for x and y, we obtain 

> sol:=solve([%],[x,y]):%;
 

[[x = `/`(`*`(v), `*`(`^`(`*`(v, `*`(u)), `/`(1, 2)))), y = `/`(`*`(`^`(`*`(v, `*`(u)), `/`(1, 2)), `*`(u)), `*`(v))], [x = `+`(`-`(`/`(`*`(v), `*`(`^`(`*`(v, `*`(u)), `/`(1, 2)))))), y = `+`(`-`(`/`(...
 

We select the positive values. 

> x:=unapply(subs(sol[1],x),u,v):y:=unapply(subs(sol[1],y),u,v):
 

regionplot2d plots the image of a region in the uv-plane. 

> plt:=regionplot2d(u=2..4,v=1..5, [x(u,v),y(u,v)],color=green,scaling=unconstrained,labels=[x,y],title="Region R in the xy-plane",titlefont=[times,bold,16]):
 

> plt;
 

Plot_2d
 

We calculate the Jacobian 

> Jm,J:=VectorCalculus:-Jacobian([x(u,v),y(u,v)],[u,v],'determinant'):%;
 

Matrix(%id = 153278896), `+`(`-`(`/`(`*`(`/`(1, 2)), `*`(v))))
 

> A:=Doubleint(abs(J),u=2..4,v=1..5):A=value(A);
 

Int(Int(`+`(`/`(`*`(`/`(1, 2)), `*`(abs(v)))), u = 2 .. 4), v = 1 .. 5) = ln(5)