Solvers!

A few months ago, when I was very new to Maxima, I posted my solutions to several multivariable optimization problems, and included a standard constrained optimization example for which the built in command solve failed.

Here’s the problem: In my multivariable calculus class, we’ve just encountered that problem again.  What my students and I discovered was that the order of the variables as specified by the user in the argument list caused the command to either succeed or fail, as you can see from the first two lines below: I wrote to the experts at maxima-discuss@lists.sourceforge.net.  They replied en masse very quickly and were very generous with their time and advice. They verified the problem, pointing out that solving systems of equations is really complicated, and that the solve program seeks to solve for the variables one at a time and things can go wrong if an inauspicious order of variables is followed.

The experts also tried our problem with the package to_poly_solve, and happily it succeeded without the hassle of needing to specify the variables in the right order.  For solve users who reference the solutions it produces as elements of a list, direct use of to_poly_solve presents a  challenge due to output in a format different from that of solve  — t0_poly_solve returns solutions as a set with %union.

I’ve written a little wrapper Ksolve.mac that is called like solve as illustrated above.

In Ksolve if an initial call to solve fails, the process automatically upgrades to to_poly_solve in several variations, and  (hopefully) returns solutions in the same format as solve.

Maxima for maxima (and minima) of functions z=f(x,y) Together with the 3D graphing capabilities of Maxima, we can bring  symbolic differentiation and the numerical solver to bear when we seek local extrema of a surface.

Here is a link to the html export of a wxMaxima session where I work on two examples from my multivariable calculus class.  And here is the wxMaxima session.

The MATH214 package for multivariable calculus

I’ve written a suite of Maxima commands for use in multivariable calculus class.  The package includes:

cross, dot, len,
unitT, unitN, unitB, curvature,
integrate2, integrate3,
integratePaths, integratePathv2, integratePathv3,
integrateSurf,
grad, div, curl

To use these commands:

2. extract (unzip) the contents into a directory you can find later. (I think macOS does this step behind the scenes, but in windows you’ll need right-click and “Extract All…”)
3. In wxMaxima, select File—Load Package…  then navigate to the directory in step 2. above and select MATH214.mac   The result should be an automatically generated input line similar to: Included in that package is my home-baked help utility.  After loading, help(MATH214) returns a list of functions in the package, and help(<function_name>) returns a description and usage example for any of the functions named in the list above.

Examples showing these commands being executed in wxMaxima can be found here.  In addition, the .zip file also contains these examples in the  wxMaxima session file math214_testfunctions.wxmx that you can load into your wxMaxima session.

It is only necessary to download and extract MATH214.zip once, but you will need to load the package (step 3 above) in each new wxMaxima session you’d like to use any of these commands.

Note that these commands duplicate existing Maxima functionality (like dot(x,y) and x.y) or perform similarly to other packages (like vect).  The purpose here is for the commands in MATH214 to have a calling syntax that closely follows the way we have defined these operations analytically in class, while avoiding the unfortunate namespace  conflicts between the existing packages vect and draw that has been recently documented.

Path Integrals

For my multivariable calculus class, I wanted an easy-to-call suite of symbolic integrators for path integrals of the form $\int_C f(x,y) dr$, $\int_C {\bf F}(x,y)\cdot d{\bf r} = \int_C \langle P(x,y), Q(x,y) \rangle\cdot \langle dx, dy \rangle$, or $\int_C {\bf F}(x,y,z)\cdot d{\bf r} = \int_C \langle P(x,y,z), Q(x,y,z), R(x,y,z) \rangle\cdot \langle dx, dy, dz \rangle$.

My overarching design idea was that the input arguments needed to look the way they do when I teach the course:

• a scalar field $f(x,y):R^2 \rightarrow R$ or a vector field ${\bf F}(x,y): R^2 \rightarrow R^2$ or ${\bf F}(x,y,z): R^3 \rightarrow R^3$
• a curve $C$ defined by a vector-valued function ${\bf r}(t): R \rightarrow R^n, a\le t \le b$ where $n=2,3$ as appropriate.

It took me a while to work out how to evaluate the integrand along the path within my function.  Things that worked fine on the command line failed when embedded into a batch file to which I passed functions as arguments.  I ended up using subst, one variable at a time.  I’d like to be able to do this in a single command which can detect whether we’re in 2 or 3 dimensions so that I don’t need separate commands.

For now, here’s what I came up with along with some illustrative examples taken from Paul’s online math notes, that show how to call these new commands I, I2 and I3.

/* path integral of a scalar integrand f(x,y) on path r(t) in R^2, t from a to b */
I(f,r,t,a,b):=block(
[f1,f2,dr,Iout],
f1:subst(x=r,f),
f2:subst(y=r,f1),
dr:sqrt(diff(r,t).diff(r,t)),
Iout: integrate(f2*dr,t,a,b),
Iout
);
/* path integral of a vector integrand F(x,y) on path r(t) in R^2, t from a to b */
I2(H,r,t,a,b):=block(
[H1,H2,I],
H1:subst(x=r,H),
H2:subst(y=r,H1),
I: integrate( H2.diff(r,t),t,a,b),
I
);
/* path integral of a vector integrand F(x,y,z) on path r(t) in R^3, t from a to b */
I3(H,r,t,a,b):=block(
[H1,H2,H3,I],
H1:subst(x=r,H),
H2:subst(y=r,H1),
H3:subst(z=r,H2),
I: integrate( H3.diff(r,t),t,a,b),
I
); Here’s an update:  a related maxima function for evaluating a complex integral $\int_\Gamma f(z) dz$

where $f: C \rightarrow C$ and the curve $\Gamma$ is given by $r: R \rightarrow C$.

/* path integral of a complex integrand f(z): C --> C, on path z(t): R --> C, t from a to b */
IC(f,r,t,a,b):=block(
[f1,dz,Iout],
f1:subst(z=r,f),
dz:diff(r,t),
Iout: integrate(f1*dz,t,a,b),
Iout
); :=, ”(), define and div, grad, curl

I recently posted about : and :=  for defining functional expressions.  I’m starting to enjoy these emoji-like constructions 😉

This is another  colon-equals post.  This time for defining functions involving the maxima differentiation command diff.

Notice below that if we define a function with :=, the naive use of :=diff doesn’t produce a derivative with the expected results upon evaluation. In fact, it’s a good thing that :=diff works like that.  The error with fp(3) above comes from the fact that we’ve actually defined an operator that differentiates the function with respect to the argument we pass…in the case above, differentiating with respect to the symbol u makes sense, while differentiating with respect to the constant 3 doesn’t.

So how to make the derivative function do what we want?  Two ways, that are subtly different, in ways I’m not completely sure of.  More about that when I learn more :-).

First is define, Also you can use    ”()       quote-quote with parens around the whole right hand side: I used define to write functions for vector valued 3D curves in an earlier post.   In figuring this out, I also learned that the :=diff form is really useful.  Below are three little functions in which I use :=diff to define the vector calculus operators grad, div and curl.  Notice that we pass the function f as an argument, and the :=diff form allows Maxima to differentiate them behind the scenes and return the results of the grad, div, and curl operators as you’d expect. These versions of div, grad and curl behave differently, and for me more as expected, than the functions of those names included in the Maxima vect package.  You can download the .mac file here.

/* Three Maxima functions for the multivariable calculus operators  grad, div, and curl
TheMaximaList.org, 2016
*/

grad(f,x,y,z):=[diff(f,x),diff(f,y),diff(f,z)]$div(f,x,y,z):=diff(f,x)+diff(f,y)+diff(f,z)$

curl(f,x,y,z):=[ diff(f,y)-diff(f,z),
diff(f,z)-diff(f,x),
diff(f,x)-diff(f,y) ]$Here is a screenshot showing how to call these functions: Curvature, T, N, & B A classic topic in multivariable calculus involves the study of a vector valued function ${\bf r}(t)=\langle x(t),y(t),z(t) \rangle$ using the three canonical unit vectors — the tangent vector ${\bf T}(t)$, the normal vector ${\bf N}(t)$, and the Binormal vector ${\bf B}(t)$ — and the scalar curvature $\kappa(t)$. Here are maxima functions that compute these, called unitT, unitN, unitB, and curvature. For a vector valued function ${\bf r}(t)$, these are called as T(t):=unitT(r(t),t); You can download the .mac file here. /* unitT computes the unit tangent vector for a vector valued function of a scalar variable r(t)=[x(t),y(t),z(t)] */ unitT(r,t):= block([rp,rpn,T], define(rp(t),diff(r,t)), define(rpn(t),sqrt(rp(t).rp(t))), define(T(t),rp(t)/rpn(t)), trigsimp(T(t)) )$

/* unitN computes the unit normal vector for a vector valued function
of a scalar variable r(t)=[x(t),y(t),z(t)]
unitN requires unitT */

unitN(r,t):=
block([T,Tp,Tpn,N],
define(T(t),unitT(r,t)),
define(Tp(t),diff(T(t),t)),
define(Tpn(t),sqrt(Tp(t).Tp(t))),
define(N(t),Tp(t)/Tpn(t)),
trigsimp(N(t))
)$/* unitB computes the unit normal vector for a vector valued function of a scalar variable r(t)=[x(t),y(t),z(t)] unitB requires unitT and unitN */ unitB(r,t):= block([T,N,B], define(T(t),unitT(r,t)), define(N(t),unitN(r,t)), define(B(t),[T(t)*N(t)-T(t)*N(t), T(t)*N(t)-T(t)*N(t), T(t)*N(t)-T(t)*N(t)]), trigsimp(B(t)) )$

/* curvature computes the curvature
curvature requires unitT */

curvature(r,t):=
block([T,Tdot,rdot,K],
define(T(t),unitT(r,t)),
define(Tdot(t),diff(T(t),t)),
define(rdot(t),diff(r,t)),
define(K(t),trigsimp(sqrt(Tdot(t).Tdot(t)/rdot(t).rdot(t)))),
K(t)
)\$

A surface, a tangent plane, a normal line

I set out to do a visualization that is basic to the multivariable calculus classroom:  plot a surface, a tangent plane, and a normal line.  I didn’t do it on the first try…or the second.

I had to learn some things about the draw3d functionality in Maxima and wxMaxima.  Here’s a great resource by Wilhelm Haager.  In the end I’m delighted at the amazing flexibility of that command, but still scratching my head a little about how this would appear to an otherwise uninitiated beginner to Maxima.

Here’s the html export of my session.