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<Input><Text-field prompt="> " style="Maple Input" layout="Normal">restart</Text-field>
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<Input><Text-field prompt="> " style="Maple Input" layout="Normal">Mat1Tools := module()
description "Tools used in the DTU course 01005 - Advanced Engineering Mathematics 1";
option package;
export
dot, prik, cross, kryds, vop, intervalsolve, paraplot, grad, srhs, slhs, condgf, vlen, div, jacobi, curl, rot, pint, flux, vsolve, tint, com, pnorm;
local columnMatrixToVector;
with(LinearAlgebra):
with(plots):
with(Integrator8):
columnMatrixToVector := proc(X)
if LinearAlgebra[ColumnDimension](X) = 1 and type(X, Matrix) then
return convert(X,Vector):
else
return X:
end if;
end proc:
dot := proc(x::Vector,y::Vector)
description "Scalar product, dot product of two vectors; Takes two vectors and returns the dot product of them. dot(<a,b>,<c,d>) -> a*c + b*d":
return VectorCalculus[DotProduct](columnMatrixToVector(x),columnMatrixToVector(y));
end proc:
prik:= dot:
cross := proc(x::Vector, y::Vector)
description "Cartesian product, cross product of two vectors. Takes two 3 dimensional vectors and returns the cross
product/normal vector of them. cross(<a,b,c>,<d,e,f>) -> <b*f - c*e, -a*f + c*d, a*e - b*d]>":
return convert(VectorCalculus[CrossProduct](columnMatrixToVector(x),columnMatrixToVector(y)),Vector);
end proc:
kryds := cross:
vlen := proc(v::Vector)
description "Computes the length of the given vector. I.e. vlen(<x,y>) -> sqrt(x^2 + y^2)":
return sqrt(dot(v, v));
end proc:
vop := proc(v)
description "Extract operands from a vector expression. I.e. vop(<1,2,3>) -> 1,2,3";
return op(convert(v,list));
end proc:
intervalsolve := Student[Calculus1][Roots]:
paraplot := proc(r::{procedure},range::{list(range)})
description "Parametric plot (paraplot) is a function which plots parametric equations. Takes the parametric equation and a list of ranges for the variables in the parametric equation.":
local var,i; var:=op(1,eval(r)):
if (numelems(range)=1) then
if (numelems(r(var))=2) then
return plot([vop(r(var)),var=range[1]],_rest); # Kurve i 2D
else
return plot3d(r(var),var=range[1],_rest) # Kurve i 3D
end if:
elif (numelems(range)=2) then
if (numelems(r(var))=2) then
return plot3d(<r(var),0>,var[1]=range[1],var[2]=range[2],orientation=[-90,0],lightmodel=none,_rest); # Plan i 2D
else
return plot3d(r(var),var[1]=range[1],var[2]=range[2],_rest); # Flade i 3D
end if:
elif (numelems([var])=3) then
return plots[display](Integrator8[sideFlader](r,[seq(vop(convert(range[i],list)),i=1..3)],[8,8,8]),_rest);
end if;
end proc:
grad:= proc(f::{procedure},x::{list})
description "Computes the gradient of the given function, with regard to the passed variables. I.e. grad(unapply(x+y,x,y),[x,y]) -> <1,1>.":
return convert(VectorCalculus[Gradient](f(vop(x)),x),Vector[column]);
end proc:
srhs := proc(X, type:=whattype(X))
description "Takes a list, vector, column matrix, or similar containing only equations and returns the right-hand side of the equations. slhs([x=0,y=10,z=15]) -> [0,10,15]":
local i, l:
l := convert(X, list):
return convert([seq(rhs(l[i]),i=1..numelems(l))],type);
end proc:
slhs := proc(X, type:=whattype(X))
description "Takes a list, vector, column matrix, or similar containing only equations and returns the left-hand side of the equations. slhs([x=0,y=10,z=15]) -> [x,y,z]":
local i, l:
l := convert(X, list):
return convert([seq(lhs(l[i]),i=1..numelems(l))],type);
end proc:
condgf := proc(V, $)
description "Takes a vector field in 2 or 3 dimensions as a list, vector, or similar and checks if it lives up to the mandatory criteria for being a gradient field. checkGradient(<y*cos(x*y), z+x*cos(x*y), y>) -> Condition satisfied.":
if numelems(V) > 3 then
return "The vectorfield must have a dimension of at most 3.";
elif numelems(V) < 2 then
return "The vectorfield must have a dimension of at least 2.";
elif numelems(V) = 2 then
if simplify(diff(V[1],y)) <> simplify(diff(V[2],x)) then
return "Not gradient field. ", Diff(V__1,y) <> Diff(V__2,x)
else
return "Condition satisfied."
end if;
else
if simplify(diff(V[1],y)) <> simplify(diff(V[2],x)) then
return "Not gradient field. ", Diff(V__1,y) <> Diff(V__2,x)
elif simplify(diff(V[1],z)) <> simplify(diff(V[3],x)) then
return "Not gradient field. ", Diff(V__1,z) <> Diff(V__3,x)
elif simplify(diff(V[2],z)) <> simplify(diff(V[3],y)) then
return "Not gradient field. ", Diff(V__2,z) <> Diff(V__3,y)
else
return "Condition satisfied."
end if;
end if;
end proc:
div := proc(V::{Vector, procedure})
description "Creates a procedure/function for the divergence of the passed vector field.":
local v;
if type(V, procedure) then
v := V(op(1, eval(V)));
else
v := V;
end if;
if numelems(v) = 2 then
VectorCalculus[SetCoordinates]('cartesian'[x,y]);
unapply(VectorCalculus[Divergence](VectorCalculus[VectorField](v)), [x, y]);
else
VectorCalculus[SetCoordinates]('cartesian'[x,y,z]);
unapply(VectorCalculus[Divergence](VectorCalculus[VectorField](v)), [x, y, z]);
end if;
end proc:
jacobi := proc(r::{procedure})
description "Computes the jacobi function of a parametric equation variable of up to 3 variables":
local i, var;
var := op(1, eval(r));
r := unapply(columnMatrixToVector(r(var)),var):
if numelems([var]) = 1 then
return simplify(vlen(diff(r(var), var))) assuming var::real;
elif numelems([var]) = 2 and numelems(r(var)) = 2 then
return simplify(abs(VectorCalculus[Jacobian](r(var), [var], determinant)[2])) assuming var[1]::real, var[2]::real;
elif numelems([var]) = 2 then
return simplify(vlen(cross(diff(r(var), var[1]), diff(r(var), var[2])))) assuming var[1]::real, var[2]::real;
elif numelems([var]) = 3 then
return simplify(abs(VectorCalculus[Jacobian](r(var), [var], determinant)[2])) assuming var[1]::real, var[2]::real, var[3]::real;
else
return "Incorrect number of dimensions";
end if;
end proc:
curl := proc(V::{Vector, procedure})
description "Creates a rotation procedure/function for the given vector field.":
local v, var;
VectorCalculus:-BasisFormat(false);
if type(V, procedure) then
v := V(op(1, VectorCalculus:-eval(V)));
var := [op(1, VectorCalculus:-eval(V))];
return unapply(VectorCalculus:-Curl(Student[VectorCalculus][VectorCalculus:-VectorField](v)), var);
else
v := V;
return VectorCalculus:-Curl(Student[VectorCalculus][VectorCalculus:-VectorField](v));
end if;
end proc:
rot := curl:
pint := proc(r::{procedure}, integrateRange::{list(range)}, f::{procedure} := 1)
description "Paramatric equation integration (pint) integrates the parametric equation given in the given intervals, optionally with the provided weight function.":
local var, i;
var := [op(1, eval(r))];
return simplify(int(f(vop(r(vop(var))))*jacobi(r), seq(var[i] = integrateRange[i], i = 1 .. numelems(integrateRange))));
end proc:
flux:= proc(V::{procedure}, r::{procedure}, integrateRange::{list(range)})
description "Computes the flux of the provided 3D vector field throughout the provided parameterized surface or solid object (assuming it's closed) in the given intervals.":
local var, i;
var := op(1, eval(r));
if numelems([var]) = 2 then:
return int(dot(convert(V(vop(r(var))), Vector), pnorm(r)), var[1]=integrateRange[1], var[2]=integrateRange[2]);
elif numelems([var]) = 3 then:
# Gauss divergence theorem
return int(
div(V)(vop(r(var)))*jacobi(r),
var[1]=integrateRange[1], var[2]=integrateRange[2], var[3]=integrateRange[3]);
else:
return "Can only determine flux of surfaces. (Too many variables received)";
end if:
end proc:
vsolve := proc(veq, vars)
description "Solves the given vector equation for the variables defined in the second argument.":
local i, s, l, r, subeq:
l := lhs(veq):
r := rhs(veq):
subeq := []:
if not numelems(l) = numelems(r) then
return "Left and right side of equation must be of equal size";
end if;
# Create sub equations
for i from 1 to numelems(l) do
subeq := [op(subeq),l[i] = r[i]]:
end do:
s := solve(subeq,vars):
if not s = NULL then
return s;
else
return "No solutions found";
end if;
end proc:
tint := proc(V::{procedure}, r::{procedure}, integrateRange::{range})
description "Calculates the tangential line integral of a vector field along a given line.":
local var;
var := op(1,eval(r));
return simplify(int(dot(V(vop(r(var))), diff(r(var),var)),var=integrateRange))
end proc:
com := proc(f::{procedure}, r::{procedure}, integrateRange::{list(range)})
description "Determines the center of mass (COM) given the weight function f, the parametric equation r and it's variable intervals.":
local var, i, jac, coordinate__funcs, coordinate_integrals, mass_func, coordinate_func, mass:
var := [op(1, eval(r))]:
jac := jacobi(r):
mass := pint(r,integrateRange,f):
mass_func := f(vop(r(vop(var)))):
coordinate__funcs := seq(r(vop(var))[i],i=1..numelems(var)):
coordinate_integrals := []:
for coordinate_func in coordinate__funcs do:
coordinate_integrals := [op(coordinate_integrals), int(mass_func*coordinate_func*jac, seq(var[i] = integrateRange[i], i = 1 .. numelems(integrateRange)))]:
end do:
1/mass*convert(coordinate_integrals,Vector);
end proc:
pnorm := proc(r::{procedure})
description "Determines the normalvektor of the parameterized surface given as r";
local var:
var := op(1,eval(r)):
if not numelems([var]) = 2 then:
return "Parametric equation must be a surface"
else:
return simplify(cross(
diff(r(var),var[1]),
diff(r(var),var[2]))) assuming (var[1])::real assuming (var[2])::real;
end if:
end proc:
end module;</Text-field>
</Input>
<Output><Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">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</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L4" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="> " style="Maple Input" layout="Normal">packageName := "Mat1Tools.mla":</Text-field>
</Input>
</Group>
<Section collapsed="false" isCollapsible="true" drawButton="true" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title><Text-field style="Heading 1" layout="Heading 1">maple/lib dir</Text-field></Title>
<Group hide-input="false" labelreference="L8" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="> " style="Maple Input" layout="Normal">savepath := FileTools:-JoinPath(["lib",packageName],base=mapledir): # Maple/lib</Text-field>
</Input>
</Group>
</Section>
<Section collapsed="false" isCollapsible="true" drawButton="true" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title><Text-field style="Heading 1" layout="Heading 1">Current dir</Text-field></Title>
<Group hide-input="false" labelreference="L9" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="> " style="Maple Input" layout="Normal">savepath := cat(currentdir(),kernelopts(dirsep),packageName): # Current dir</Text-field>
</Input>
</Group>
</Section>
<Group labelreference="L3" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="> " style="Maple Input" layout="Normal">savelib('Mat1Tools',savepath);</Text-field>
</Input>
</Group>
<Group labelreference="L10" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input><Text-field prompt="> " style="Maple Input" layout="Normal"></Text-field>
</Input>
</Group>
</Worksheet>