Month: June 2018

A MATLAB-like find() function for Maxima

In MATLAB, you can find the indices of array entries that match a user-supplied condition with the function find()

Here’s a similar function in Maxima.  This makes liberal use of the functions sublist_indices(), lambda(),  and several functions from the stringproc package: sequal(), eval_string(), and simplode().  Oh, and also the shameless hack of using ascii(32) as the space character 🙂

Here are some examples.  The code is included at the bottom.

find

find(exp):=block(
 oo:op(exp),
 if sequal(oo,"=") or sequal(oo,">") or sequal(oo,">=") or sequal(oo,"<") or sequal(oo,"<=") or sequal(oo,"#") then 
   sublist_indices(first(exp),lambda([x1],eval_string(simplode([x1,op(exp),last(exp)]))))
 else 
   if sequal(oo,"and") or sequal(oo,"or") then(
     e1:first(exp),
     e2:second(exp),
     sublist_indices(first(e1),lambda([x1],
       eval_string(simplode([x1,op(e1),last(e1),ascii(32),oo,ascii(32),x1,op(e2),last(e2)]))))
   )
)$
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Two little list utilities for MATLAB-like array indexing

I grew up with MATLAB, where extracting a subset of a vector V was easy as feeding a vector of indices into a vector.  For example entries i thru j could be had with V(i:j) and an more complicated index scheme could be accomplished with a vector of indices i_index and then V(i_index).

In Maxima, first(), last(), firstn(), rest(), and most generally makelist() allows for all that and more but with a little more cumbersome calling protocols.  Here are one-liners that achieve something like the two canonical MATLAB examples above:

indexby

A maxima function to replicate MATLAB diff() and an efficiency comparison

I needed a function to compute a list containing the difference between adjacent entries in another list, as diff() does for an array in MATLAB.

I first wrote  something using makelist() and found it very slow for large lists.  I then rewrote the function using rest() and found it much faster:

differ

 

Surface Integrals, Triple Integrals, and the Divergence Theorem of Gauss in Maxima

In earlier posts, I describe the Package of Maxima functions MATH214 for use in my multivariable calculus class, with applications to Greens  Theorem and Stokes Theorem.

Here we show how the  surface integral function integrateSurf() and triple integration  function integrate3()  (together with the divergence function div() )work on a Gauss’s Theorem example:

Gauss2

We integrate the parabolic surface and the circular base surface separately, and show their sum is equal to the triple integral of the divergence.

Gauss1

The functions above are included in the MATH214 package, but I list them below as well:

integrateSurf(F,S,uu,aa,bb,vv,cc,dd):=block(
 [F2],
 F2:psubst([x=S[1],y=S[2],z=S[3]],F),
 integrate(integrate(trigsimp(F2.cross(diff(S,uu),diff(S,vv))),uu,aa,bb),vv,cc,dd));

integrate3(F,xx,aa,bb,yy,cc,dd,zz,ee,ff):=block(
 integrate(integrate(integrate(F,xx,aa,bb),yy,cc,dd),zz,ee,ff));

div(f,x,y,z):=diff(f[1],x)+diff(f[2],y)+diff(f[3],z)$

cross(_u,_v):=[_u[2]*_v[3]-_u[3]*_v[2],_u[3]*_v[1]-_u[1]*_v[3],_u[1]*_v[2]-_u[2]*_v[1]]$

 

Path Integrals in the Plane, Double Integrals, and Greens Theorem in Maxima

In an earlier post I described the Maxima package MATH214 for use in my multivariable calculus class.  I’ve posted examples with applications to Gauss’s  Theorem and Stokes Theorem.

Here we take the double integration routine integrate2() and the 2D path integral integratePathv2() for a spin with a Green’s Theorem example from Stewart’s Calculus Concepts and Contexts:

Greens2

Greens1

And of course polar coordinates are nice too:

Greens3.PNG

The two functions used above are included in the MATH214 package, but I list them below as well.

integratePathv2(H,r,t,a,b):=block(
[H2],
H2:psubst([x=r[1],y=r[2]],H),
 integrate( trigsimp(H2.diff(r,t)),t,a,b)
);

integrate2(F,xx,aa,bb,yy,cc,dd):=block(
 integrate(integrate(F,xx,aa,bb),yy,cc,dd));

 

 

Surface Integrals and Stokes Theorem in Maxima

 

In an earlier post I detailed the Maxima functions contained  the MATH214 package for use in my multivariable calculus class.  The package at that link has now been updated with some further integration utilities:  integrate2() and integrate3() for double and triple integrals and integrateSurf() for surface integrals of vector fields in 3D.  I’ve posted examples with applications to Green’s  Theorem and Gauss’s Theorem.

Here’s a test drive of the surface integration function using a Stokes theorem example I found on the web:

Verify Stokes theorem for the surface S described by the paraboloid z=16-x^2-y^2 for z>=0

and the vector field

F =3yi+4zj-6xk

First the path integral of the vector field around the circular boundary of the surface using integratePathv3() from the MATH214 package

Stokes1

And also the surface integral using integrateSurf().  Notice that in the order of integration we specify in integrateSurf(),  (first  y then x) the surface normal vector computed with cross() in that same variable order points inward—the negative orientation.  We reverse the direction with an extra negative inside the surface integral.

Stokes2

Although they are included in the MATH214 package, here are the  functions used above:

integrateSurf(F,S,uu,aa,bb,vv,cc,dd):=block(
 [F2],
 F2:psubst([x=S[1],y=S[2],z=S[3]],F),
 integrate(integrate(trigsimp(F2.cross(diff(S,uu),diff(S,vv))),uu,aa,bb),vv,cc,dd));

cross(_u,_v):=[_u[2]*_v[3]-_u[3]*_v[2],_u[3]*_v[1]-_u[1]*_v[3],_u[1]*_v[2]-_u[2]*_v[1]]$

curl(f,x,y,z):=[ diff(f[3],y)-diff(f[2],z),curl(f,x,y,z):=[ diff(f[3],y)-diff(f[2],z),                     diff(f[1],z)-diff(f[3],x),   diff(f[2],x)-diff(f[1],y) ]$
integratePathv3(H,r,t,a,b):=block(
[H3],
H3:psubst([x=r[1],y=r[2],z=r[3]],H),
 integrate( trigsimp(H3.diff(r,t)),t,a,b)
);