I prefer a scheme which appears "symmetric", ie. not to prefer one name over another, and not to have to patch up any term referenced by position or on a scheme not robust to renaming of all the variables in play, etc.

restart:

ee:=(-(1/2)*ib-(1/2)*ia+ic)*vc+(-(1/2)*ia-(1/2)*ic+ib)*vb+(-(1/2)*ic-(1/2)*ib+ia)*va:

rule:=ia+ib+ic=0:

eqs:=map(-rhs=-lhs,map2(isolate,rule,indets(rule,name)));

                            eqs := {ia + ib = -ic, ia + ic = -ib, ib + ic = -ia}

subs(eqs,map(simplify,ee,{ia+ib+ic=0}));

                                        3 vc ic   3 vb ib   3 va ia
                                        ------- + ------- + -------
                                           2         2         2

Obvious attempts using algsubs or simplify-with-side-relations seem to produce a result which, while equivalent to the desired target, contains four terms. This can be patched up by repairing just the offending pair of terms, it seems like "cheating" a little to me if this has to re-treat two of the terms while holding the others static. If all the names were changed to something else, breaking their lexicographic order, then the simplification scheme should ideally not need to be changed. Just some thoughts.

collect(simplify(ee, {rule}),[va,vb,vc]);      

                                   3 va ia   3 vb ib      /  3 ia   3 ib\
                                   ------- + ------- + vc |- ---- - ----|
                                      2         2         \   2      2  /

acer

There are a few differences between how title and caption (for 3D plots) are handled versus how legend (for 2D plots) are handled.

• title and caption can't be placed on the left or right
  
  
• plots:-display uses only the last caption title from a sequence of plot arguments, while it merges legends
  
  
• title and caption don't automatically get an graphic icon for the object, as legends do
  
  
• typeset cannot be applied to math expressions for titles and captions, as it can be for legends. But instead Typesetting:-Typeset can be applied

But you can construct a multi-line caption (or title) for a 3D plot which appears like a legend. We can build the graphic icons for the surface/curve, partly by using techniques as mentioned here.

restart;

with(Typesetting): with(plots):

clist := ["Cyan", "Orange"]:

P1 := plot3d(sin(x)*(1-y)^2, x=-10..10, y=-1..1, color=clist[1]):
P2 := plot3d(cos(x)*(1-y)^2, x=-10..10, y=-1..1, color=clist[2]):

lg := mrow(mn("        ",mathbackground=clist[1]),
                     mn("  "), Typesetting:-Typeset(sin(x)*(1-y)^2),
                     mn("\n        ",mathbackground=clist[2]),
                     mn("  "), Typesetting:-Typeset(cos(x)*(1-y)^2)):

display(P1, P2, title=lg);

display(P1, P2, caption=lg);

restart;

with(ColorTools): with(Typesetting): with(plots):

P1 := plot3d(sin(x)*(1-y)^2, x=-10..10, y=-1..1, color="SkyBlue"):
P2 := plot3d(cos(x)*(1-y)^2, x=-10..10, y=-1..1, color="PeachPuff"):
P3 := spacecurve([t,1/2,cos(t)+3], t=-10..10, thickness=2, color="Green"):
P4 := plot3d(sin(x)*(1-y)^2, x=-10..10, y=-1..-1/2, color="Red",
             style=point, grid=[30,10], symbol=solidsphere):

#
# Picking off the color could be done with a procedure, which
# naturally would be careful about the various formats that the
# COLOR substructure might take.
#
select(type, [op([1,..],P1)], 'specfunc(anything,{:-COLOR,:-COLOUR})')[1]:
cP1 := RGB24ToHex(ToRGB24(Color([op([2..],%)]))):
select(type, [op([1,..],P2)], 'specfunc(anything,{:-COLOR,:-COLOUR})')[1]:
cP2 := RGB24ToHex(ToRGB24(Color([op([2..],%)]))):
select(type, [op([1,..],P3)], 'specfunc(anything,{:-COLOR,:-COLOUR})')[1]:
cP3 := RGB24ToHex(ToRGB24(Color([op([2..],%)]))):
select(type, [op([1,..],P4)], 'specfunc(anything,{:-COLOR,:-COLOUR})')[1]:
cP4 := RGB24ToHex(ToRGB24(Color([op([2..],%)]))):
lg := mrow(

           mn("\n           ",mathbackground=cP1, size=10),
           mn("  "), Typesetting:-Typeset(sin(x)*(1-y)^2),

           mn("\n           ",mathbackground=cP2, size=10),
           mn("  "), Typesetting:-Typeset(cos(x)*(1-y)^2),

           mn("\n     "), mn(cat(" "," "$42," "),mathbackground=cP3, size=3),
           mn("  "), Typesetting:-Typeset([t,0.5,cos(t)+3]),

           mn(cat("\n","· "$3),mathcolor=cP4, size=14),
           mn("  "), Typesetting:-Typeset(sin(x)*(1-y)^2)

           ):

plots:-display(P1, P2, P3, P4, caption=lg);

It should even be possible to to write procedures which 1) construct such captions from what is found in a PLOT3D structure (though any math expression would have to be passed separately), and 2) merge captions from separate 3D plots into a multi-line caption, while also merging the plots.

If desired the results from those calls to Typesetting:-Typeset could get size modification, to use a smaller math font. size, I suspect.

acer

Download tempdens.mw

acer

See the help topic updates/Maple2016/compatibility in your Maple 2016.

Download formalsum2016.mw

acer

(It turns out that I mostly just had to look for the file where I'd done pretty much this, earlier.)

If the IterativeMaps commands also give you grief you could also implement it (slower) using plot3d or densityplot.

As mentioned in the code below, increasing `maxiter` will reduce the areas of nonconvergence (black areas) for such univariate polynomials.

I used Maple 2015.2 on 64bit Linux.

Download newtbasin0.mw

And here it is for N=600 and maxiter=200. Using a logarithm for adjusting the saturation layer might be a way to get a nicer impression of the convergence rate.

acer

c:=INTERVAL(1.41421356167 .. 1.41421356308):

uo,low:=op(op(c));                          

                           uo, low := 1.41421356167, 1.41421356308

Or,

up,low:=op([1,..],c);

                           up, low := 1.41421356167, 1.41421356308

acer

You can use the add and the if command together.

Note the single backticks (left-quotes) around the word if.

V := Vector[row]( 10, i -> i^2 );

                 V := [1, 4, 9, 16, 25, 36, 49, 64, 81, 100]

add( `if`( v>30, v, 0), v=V );

                                     330

add( map( v -> `if`( v>30, v, 0 ), V ) );

                                     330

add( piecewise( v>30, v, 0 ), v=V );

                                     330

M := Matrix( 5, 5, (i,j) -> (i+j) );

                                 [2  3  4  5   6]
                                 [              ]
                                 [3  4  5  6   7]
                                 [              ]
                            M := [4  5  6  7   8]
                                 [              ]
                                 [5  6  7  8   9]
                                 [              ]
                                 [6  7  8  9  10]

add( `if`( M[i,3] > 5, M[i,3], 0), i=1..5 );

                                     21

add( `if`( v > 5, v, 0), v = M[..,3] );

                                     21

Is that the kind of thing you mean?

acer

Download factorfun.mw

acer

If the following code is saved to a Maple initialization file then a new entry for calling the RationalFunctionPlot command should appear in the right context-menu, under the existing "Plots" submenu, if the expression is of type `ratpoly`.

You can of course adjust this. The `caption` might not be set to the empty string. Or several such new menu items could be added, in their own subsubmenu, to provided various common choices for the command's options.

:-__AugmentCM := :-ContextMenu:-CurrentContext:-Copy():
:-__AugmentCM:-Entries:-Add("Rational Function",
            ":-Student:-Precalculus:-RationalFunctionPlot(%EXPR,':-caption'=\"\")",
            ':-ratpoly',
            ':-operator'=:-Typesetting:-mover(Typesetting:-mo("→"),
                         :-Typesetting:-mi("Rational Function Plot")),
            ':-category'="Category 1",
            ':-helpstring'="Rational Function Plot",
            ':-submenu'=["Plots"]
           ):
:-ContextMenu:-Install(:-__AugmentCM);

Or you could adjust this code to invoke instead the Student:-Precalculus:-RationalFunctionTutor command. This command is already available via the main menubar choice Tools->Tutors->Precalculus but invoking it from there won't automatically pass in the desired expression. But if this command is added to the context-menu then the right-clicked expression will be passed in and used.

See the help page with topic worksheet,reference,initialization for documentation on how to create an initialization file.

If you want all the students to get this enhancement automatically for a network installation of Maple then you just add it to the (possibly new) file <MAPLE>/lib/maple.ini where <MAPLE> is the location of the Maple installation. Or do it for each machine, if Maple is installed locally on machines. Test it first, of course.

acer

h := x -> f(g(x)):

D(h)(0);

                             D(f)(g(0)) D(g)(0)

eval( D(h)(0), D(g)(0)=0 );

                                      0

Even if you prefer `diff` over `D` it still evaluates to zero, for the same reason.

eval( eval(diff(h(x),x),x=0), eval(diff(g(x),x),x=0)=0 );

                                      0

acer

Download collectfactor.mw

acer

This is covered in the subsubsubsection "2-dimensional case"->"Interpolation"->"Uniform data grid"->"Interpolating procedure" of the help page available via ?examples,Interpolation_and_Smoothing .

The basic methodology is simple, see `f` below. For that I'd suggest using Vectors/Array/Matrices and square-bracket indexing and not lists and `op`.

Using float[8] datatype helps with resources. There is overhead in getting a pair of scalars into the indexable containers (Arrays, lists, etc) that ArrayInterpolation accepts. It's designed to be fast for dealing with large amounts of interpolating points all at once. Repeated use to process individual points will be less efficient than processing them all at once. But perhaps you won't know up front what they all are. There is an efficiency section in that example worksheet for the 1D interpolation case's efficiency, and I follow that is the more efficienct `F` below.

Download 2dinterp.mw

acer

Apart from increasing the grid sizing to get a smoother 3D plot (as Doug suggested), you can also make the 2D contourplot look a little better by using a different choice of custom contour values.

Download sz2012_4m_K.mw

[edited] The 2D colorbar facsimile (faked, as a legend) can look a little nicer if the `.` used to force the nonempty strings (color blobs) are also colored.

sz2012_4m_KK.mw

The GUI seems unwilling to typeset colored 2D Math if the mathcolor (foreground color) and the mathbackground (background color) are exactly the same, on my Linux at least. A small adjustment of either, so they are not identical, seems to work. Proper care should be taken to make the adjusted RGB24 color value be valid with entries all in 0-255 (which I didn't do). A further possible refinement is that intsead of atomic-identifiers like `#mrow(...)` one could use Typsettting:-mrow, etc, so that size=posint qualifier could be used to also modify the typeset size. (I didn't do that either.)

Download altcolorbar.mw

This legend-as-colorbar trick could be made into a convenient procedure. It doesn't have to be used with just contourplots. It might also be used with densityplots, or 2D pointplots, or contour plots implemented via implicitplot calls.

acer

The problem is that when you save a Units:-Unit beast to a .mla archive and then access it in a new session the attributes (that commands in Units:-Standard expect to see on it) have been stripped off. That makes the command error out.

This can be illustrated by example:

Download unitmla1.mw

Now let's restart and access that saved value. The same call to Units:-Standard:-`*` now emits an error message. This happens because the internal code of that command is expecting those attributes, as printed above.

Download unitmla2.mw

I'm not sure what the most practical workaround to this problem might be. The following reinstantiates the Units:-Unit call and produces the value anew, with the attributes. I don't know whether other, related problems might arise though.

Download unitmla3.mw

This behaviour appears for me in Maple 13, 17, and 2015 (which are all I tested). I shall submit a bug report.

acer

Download diffat.mw

acer