Mat1Tools is a Maple library consisting of procedures used throughout the Advanced Engineering Mathematics 1 (01005) DTU course.
Big thanks to Jonathan Højlev for providing the foundation for this package through Maple intro evenings and his Template
- Ensure you have the Prerequisites installed
- Download the package: Mat1Tools.mla
- Place the package in Maples lib folder
- The folder can be found by running the following command in Maple:
FileTools:-JoinPath(["lib"],base=mapledir); - This might require elevated privilege (Admin)
- The folder can be found by running the following command in Maple:
- Restart Maple
To import the package into a document or worksheet simply write: with(Mat1Tools)
and the Procedures will be available to you.
Index:
- paraplot
- dot / prik
- cross / kryds
- intervalsolve
- grad
- vop
- slhs
- srhs
- condgf
- vlen
- div
- jacobi
- curl / rot
- pint
- flux
- vsolve
- com
- tint
- hessian
- pnorm
- plotnorm
paraplot(r, intervals, plotOpts)
Description: Plots parametric equations.
Parameters:
r: function / parametric equationintervals: list of rangesplotOpts: plot options
Example:
r:=(u,v) -> <u,exp(-u/3)*sin(u)*v>;
intervals := [0..2*Pi,0..1];
paraplot(r,intervals,color="#FF0004")
For more examples see the showcase.
dot(x, y) / prik(x, y)
Parameters:
x: Vectory: Vector
Example:
x := <a,b,c>;
y := <d,e,f>;
dot(x,y)
> ad + be + cf
For more examples see the showcase.
cross(x, y) / kryds(x, y)
Parameters:
x: 3D Vectory: 3D Vector
Example:
x := <a,b,c>;
y := <d,e,f>;
cross(x,y);
> <b*f - c*e,
-a*f + c*d,
a*e - b*d>
For more examples see the showcase.
intervalsolve(equation, variable_range)
Description: Solves for solutions to the provided equation in the provided variable range.
Parameters:
equation: Equationvariable_range: Named range
Example:
lign := sin(x)= 1;
intervalsolve(lign,x=0..4*Pi);
> [Pi/2, (5*Pi)/2]
For more examples see the showcase.
grad(f, vars)
Description: Computes the gradient of a function regarding the list of variables passed.
Parameters:
f: Procedurevars: List of variable names
Example:
f := (x,y) -> x^2 + y^2;
grad(f,[x,y]);
> <1,1>
For more examples see the showcase.
vop(v)
Description: Vector operands (vop) extracts the elements of a vector.
Parameter:
v: Vector
Example:
v := <a,b,c>;
vop(v);
> a,b,c
For more examples see the showcase.
slhs(X, type)
Description: Sequential left hand side (slhs) takes a list, vector, column matrix, or similar containing only equations and returns the left-hand side of the equations.
Parameters:
X: List, vector, column matrix, or similartype: The type of the returned element. Standard is the same type as X
Example:
slhs(<x=0,y=10,z=15>)
> <x, y, z>
For more examples see the showcase.
srhs(X, type)
Description: Sequential right hand side (srhs) takes a list, vector, column matrix, or similar containing only equations and returns the right-hand side of the equations.
Parameters:
X: List, vector, column matrix, or similartype: The type of the returned element. Standard is the same type as X
Example:
srhs(<x=0,y=10,z=15>)
> <0, 10, 15>
For more examples see the showcase.
condgf(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.
Parameters:
V: List, vector, or similar
Example:
condgf(<x, y>)
> "Condition satisfied."
For more examples see the showcase.
vlen(v)
Description: Vector length (vlen) computes the length of the passed vector.
Parameters:
v: Vector
Example:
v := <x,y>:
vlen(v);
> sqrt(x^2 + y^2)
For more examples see the showcase.
div(V)
Description: Creates a procedure/function for the divergence of the passed vector field.
Parameter:
V: Vector field procedure
Example:
V := unapply(<z*x,y,y^2>,[x,y,z]):
div(V);
> (x,y,z) -> z + 1
For more examples see the showcase.
jacobi(r)
Description: Computes the jacobi function of a parametric equation variable of up to 3 variables.
Parameter:
r: parametric equation (Procedure/Function)
Example:
r := unapply(<6*cos(u),6*sin(u),u>,u):
r(u);
jacobi(r);
> sqrt(37)
For more examples see the showcase.
curl(V) / rot(V)
Description: Creates a rotation procedure/function for the given vector field.
Parameter:
V: Vector field (Procedure/Function)
Example:
V := unapply(<z*x,y,y^2>,[x,y,z]):
curl(V);
> (x,y,z) -> <2*y, x, 0>
For more examples see the showcase.
pint(r,intervals,f=1)
Description: Parametric equation integration (pint) integrates the parametric equation given in the given intervals, optionally with the provided weight function.
Parameters:
r: Parametric equation (Procedure/Function)intervals: List of rangesf: Weight function (Procedure/Function). Default = 1
Example:
r := unapply(<u*cos(v),u*sin(v),u^2>,[u,v]):
r(u,v);
intervals := [0..2,0..Pi/2];
pint(r,intervals);
> (17*Pi*sqrt(17))/24 - Pi/24
# With weight function
f:=(x,y,z)->z^2;
pint(r,intervals,f);
> (7769*Pi*sqrt(17))/1680 - Pi/1680
For more examples see the showcase.
flux(V, r, intervals)
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.
Parameters:
V: Vector field (Procedure/Function)r: Parametric equation (Procedure/Function)intervals: List of ranges
Surface example:
V := unapply(<z*x,y,y^2>,[x,y,z]):
V(x,y,z);
r := unapply(<u*cos(v),u*sin(v),u^2>,[u,v]):
r(u,v);
intervals := [0..2,0..Pi/2];
flux(V,r,intervals);
> -(19*Pi)/3
Object example:
V := unapply(<x,y,z>,[x,y,z]):
V(x,y,z);
r := unapply(<u,v,w>,[u,v,w]):
r(u,v,w)
flux(V,r,[0..1,0..1,0..1]);
> 3
For more examples see the showcase.
vsolve(veq, vars)
Description: Vector solve solves vector equations, where each side of the operand is a vector, for the specified variables
Parameters:
veq: Vector equationvars: A list/set of variable names
Example:
veq := <a+b, a - b> = <4, 10>;
vsolve(veq,{a,b});
> {a = 7, b = -3}
For more examples see the showcase.
com(f, r, intervals)
Description: Determines the center of mass (COM) of the parameterized object defined by the the weight function f and the parametric equation r.
Parameters:
f: Weight function (Procedure/Function)r: Parametric equation (Procedure/Function)intervals: List of ranges
Example:
f := (x,y,z) -> 1;
T := unapply(<u,(1-u)*v>,[u,v]):
com(f,T,[0..1,0..1])
> <1/3 , 1/3>
For more examples see the showcase.
tint(V, r, intervals)
Description: Determines the tangential line integral of a vector field V along a given line with the parametric equation r.
Parameters:
V: Vector field (Procedure/Function)r: Parametric equation (Procedure/Function)interval: Integrator range
Example:
V := (x,y) -> <-y,x>:
r := u -> <cos(u), sin(u)>:
interval := 0..2*Pi:
tint(V,r,interval)
> 2*Pi
For more examples see the showcase.
hessian(f)
Description: Creates a procedure/function for the hessianmatrix of a given function.
Parameters:
f: Function (Procedure/Function)
Example:
f := (x,y) -> x^2 + y^2 + x*y:
hessian(f)
> <2, 1; 1, 2>
For more examples see the showcase.
pnorm(r)
Description: Determines the normal vector of the parameterized surface given as r.
Parameters:
r: Parametric equation for surface (Procedure/Function)
Example:
r := (u,v) -> <cos(u), sin(u), v>:
pnorm(r)
> <cos(u), sin(u), 0>
For more examples see the showcase.
plotnorm(f, limits)
Description: Plots the normal vector of the parameterized surface given as r.
Parameters:
r: Parametric equation for surface (Procedure/Function)limits: List of ranges
Example:
r := (u,v) -> <cos(u), sin(u), v>:
plotnorm(r, [0..2*Pi,0..2])
For more examples see the showcase.
- Download the package source code: Mat1Tools.mw
- Change/Add procedures
- Add a show case to Procedure showcases
- Add a function overview to README.md
- Create pull request
- Provide a proper description of the addition or modification, so that testers don't have to test everything (Unit tests pending).
You'll of cause be able to use your own modification right away by replacing the existing Mat1Tools.mla file in your Maple lib folder (See Installation)