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dharr

Dr. David Harrington

8482 Reputation

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21 years, 34 days
University of Victoria
Professor or university staff
Victoria, British Columbia, Canada

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Contact Dr. David Harrington
I am a retired professor of chemistry at the University of Victoria, BC, Canada. My research areas are electrochemistry and surface science. I have been a user of Maple since about 1990.

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These are answers submitted by dharr

values are zero...

dharr 8482
December 26 2020
0 2 
seq(fracdiff(v(t), t,alpha),alpha=-1..1.5,0.5);

gives (1/2)*t^2, .7522527780*t^(3/2), 0., 1.128379167*t^(1/2), 0., 0.

So the ones that are zero are just running along the x-axis and are hard to see.

play it...

dharr 8482
December 26 2020
1 2

Click on the plot, and then there will be some controls for the animaton - such as PLAY, Single/continuous cycle etc

pedal surface...

dharr 8482
December 24 2020
0 1

Nice worksheet. Some initial thoughts. I guess that a tangent plane can be defined for a point on a surface, and the normal from a pedal point to that plane is well-defined, so the set of all such points where the normal meets the tangent plane can be a pedal surface. The sphere for a point at the centre would be the same sphere, and I think that for any solid of revolution, the pedal surface would be the solid of revolution generated by rotating the pedal curve.

output for display...

dharr 8482
December 19 2020
1 1

If it is mainly for display then

lprint('fadc_vmon(vmon)'=fadc_vmon(vmon));

gives

fadc_vmon(vmon) = 1638.400000*vmon+200.

For digits to display see the tools -> options -> precision menu.

Note also Maple can convert to various computer languages - see the CodeGeneration package.

water thermo...

dharr 8482
December 12 2020
1 4

Like this? (Not sure what DeltaW is) [Edit: work calculated at end]
 

> 

restart;
> 

with(ThermophysicalData):with(CoolProp):
> 

P__atm:=101325;

101325

> 

T__boil:=Property(temperature, water, pressure = P__atm, Q = 0);
T__gas:=T__boil+0.0001;
T__liq:=T__boil-0.0001;

373.124295847687904

373.1243958

373.1241958

> 

Property(PhaseString, water, pressure = P__atm, temperature = T__liq);
Property(PhaseString, water, pressure = P__atm, temperature = T__gas);

"liquid"

"gas"

> 

Hgas:=Property(Hmolar, water, pressure = P__atm, temperature = T__gas);
Hliq:=Property(Hmolar, water, pressure = P__atm, temperature = T__liq);
Sgas:=Property(Smolar, water, pressure = P__atm, temperature = T__gas);
Sliq:=Property(Smolar, water, pressure = P__atm, temperature = T__liq);

48200.3815860304312

7549.42977094276921

132.491988478210232

23.5445083289082646

For gas -> liquid at the boiling point at 1 atm - delta G should be zero

> 

dH:=Hliq-Hgas;
dS:=Sliq-Sgas;
dG:=dH-T__boil*dS;

-40650.95182

-108.9474802

0.1e-4

Work to compress 1 mol under constant 1 atm pressure until all liquid.

Dmolar is molar density (mol m^-3), so molar volume is the reciprocal of this

> 

Vgas:=1/Property(Dmolar, water, pressure = P__atm, temperature = T__gas);
Vliq:=1/Property(Dmolar, water, pressure = P__atm, temperature = T__liq);

0.3014317605e-1

0.1879786899e-4

> 

W:=-P__atm*(Vliq-Vgas);

3052.352619

Neglecting liquid volume and treating gas as ideal

> 

Wideal:=Property(gasconstant,water)*T__boil;

3102.293958

> 

 


 

Download Thermo.mw

only one x value...

dharr 8482
December 03 2020
1 0

The equation has only one solution:  x=1/e. Implicit plot expects an equation in two variables, such as x+y=3.

Edit: if you really wanted the whole vertical line you can get it, but I would usually use a parametric plot for this.
 

> 

eq:=(1+ln(x))/x=0;

(1+ln(x))/x = 0

> 

plots:-implicitplot(eq,x=0..5,y=0..5);

> 

 


 

Download implicitplot.mw

any font on your system...

dharr 8482
November 28 2020
3 4

You can use any font on your system. I don't really understand all the different Computer Modern fonts, but they seem to work. [Edit: Thanks to @acer the update. Image below is from export as .pdf ]

> 

P1 := plot(x^2):
P2 := plot(x^2,
           axesfont=[cmr12,roman,16],
           labelfont=[cmr16,roman,24]):
> 

plots:-display(Array([ P1, P2 ]));

 

 

 

 

Download Latexfonts.mw

pencil of conics tangent to 4 lines...

dharr 8482
November 22 2020
0 3


 

> 

restart;
> 

with(plots):local D;

Choose four points. We want a family of conics tangent to lines passing through these four points. First generate the lines, using y=m*(x-x0)+y0

> 

pts:=[[1,3/2],[-1/2,1],[-1,-1],[4/3,-3/4]]; #pts in quadrants 1,2,3,4
slopes:=[seq(((dx,dy)->dy/dx)((pts[i]-pts[(i mod 4)+1])[]),i=1..4)]:
lines:=zip((pt,slope)->y=slope*(x-pt[1])+pt[2],pts,slopes);
plotpts:=plot(pts,style=point,colour=red,symbol=solidcircle,symbolsize=15):
plotlines:=plot(rhs~(lines),style=line,colour=blue):
display(plotpts,plotlines,view=[-2..2,-2..2]);

[[1, 3/2], [-1/2, 1], [-1, -1], [4/3, -3/4]]

[y = (1/3)*x+7/6, y = 4*x+3, y = (3/28)*x-25/28, y = -(27/4)*x+33/4]

General conic

> 

gen:=A*x^2+B*x*y+C*y^2+D*x+E*y+1=0;

A*x^2+B*x*y+C*y^2+D*x+E*y+1 = 0

Slope dy/dx at an arbitrary point x,y on the conic.

> 

conslope:=implicitdiff(gen,y,x);

-(2*A*x+B*y+D)/(B*x+2*C*y+E)

So for a point (x[i],y[i]) on a conic to be tangent to line i we require the following three equations to hold for (x,y)=(x[i],y[i]): the eqn of lines[i], conic eqn, slope of tangent = slope of line

> 

eqns:={seq(eval({lines[i],gen,slopes[i]=conslope},{x=x[i],y=y[i]})[],i=1..4)};

{4 = -(2*A*x[2]+B*y[2]+D)/(B*x[2]+2*C*y[2]+E), -27/4 = -(2*A*x[4]+B*y[4]+D)/(B*x[4]+2*C*y[4]+E), 1/3 = -(2*A*x[1]+B*y[1]+D)/(B*x[1]+2*C*y[1]+E), 3/28 = -(2*A*x[3]+B*y[3]+D)/(B*x[3]+2*C*y[3]+E), y[1] = (1/3)*x[1]+7/6, y[2] = 4*x[2]+3, y[3] = (3/28)*x[3]-25/28, y[4] = -(27/4)*x[4]+33/4, A*x[1]^2+B*x[1]*y[1]+C*y[1]^2+D*x[1]+E*y[1]+1 = 0, A*x[2]^2+B*x[2]*y[2]+C*y[2]^2+D*x[2]+E*y[2]+1 = 0, A*x[3]^2+B*x[3]*y[3]+C*y[3]^2+D*x[3]+E*y[3]+1 = 0, A*x[4]^2+B*x[4]*y[4]+C*y[4]^2+D*x[4]+E*y[4]+1 = 0}

So we have 12 equations and 13 variables: 8 for the coordinates of the four points and the 5 conic parameters. We choose x[1], the x coordinate of the tangent point on line 1 as the control point, and solve for the others in terms of it. Then substitute the parameters ino the general eqn to find the conic for this x[1]

> 

params:=solve(eqns,{y[1],x[2],y[2],x[3],y[3],x[4],y[4],A,B,C,D,E}):
con:=eval(gen,params);

-(-105664*x[1]^2-710752*x[1]-809209)*x^2/(1487929*x[1]^2-796628*x[1]-626276)-(349744*x[1]^2+2229952*x[1]+21304)*x*y/(1487929*x[1]^2-796628*x[1]-626276)-(-98256*x[1]^2-293088*x[1]-649056)*y^2/(1487929*x[1]^2-796628*x[1]-626276)+2*(272584*x[1]^2+255787*x[1]-203246)*x/(1487929*x[1]^2-796628*x[1]-626276)-8*(80083*x[1]^2-42611*x[1]+27553)*y/(1487929*x[1]^2-796628*x[1]-626276)+1 = 0

> 

display(seq(implicitplot(con,x=-4..4,y=-4..4,colour=black,gridrefine=2),x[1]=-2..2,0.3),plotpts,plotlines,axes=none,view=[-4..4,-4..4]);

> 

 


 

Download Pencil3.mw

numeric solution...

dharr 8482
November 20 2020
0 3

With some values of M and R and selected initial condition you can get a numeric solution with

dsolve(eval(sys,{M=2,R=1}) union {y(0)=2},y(r),numeric);

If you try the intial condition y(0)=1, you get the message that there is a singularity and it will work only with two initial values. This is probably why you can't get a series solution about r=0. But perhaps that's not what you want? it should be a start

poles in complex plane?...

dharr 8482
November 17 2020
0 0

You are assuming we know what field you are working in, but I don't really know what you want. A simple plot shows that the range of g is -infinity..infinity, so that is the answer to the direct question asked. But now you mention evalc, so perhaps you want the poles in the complex plane?
 

> 

g:=1/(x+x^2+x^3+2)

1/(x^3+x^2+x+2)

> 

plot(g,x=-10..10);

Poles of g in complex plane

> 

fsolve(1/g,complex);
plots:-complexplot([%],style=point,symbolsize=15,symbol=solidcircle,color=red);

HFloat(-1.3532099641993245), HFloat(0.17660498209966224)-HFloat(1.202820819285479)*I, HFloat(0.17660498209966224)+HFloat(1.202820819285479)*I

> 

 

 

Download gain.mw

pencil of conics through four points...

dharr 8482
November 13 2020
1 2

Here is how I would do it - can be refined for nicer spacing etc.

> 

restart;
> 

with(plots):local D;

Four points. We want a family of conics passing through these four points

> 

pts:=[[1,3/2],[-1/2,1],[4/3,-3/4],[-1,-1]];
p1:=plot(pts,style=point,colour=red,symbol=solidcircle,symbolsize=15):display(p1);

[[1, 3/2], [-1/2, 1], [4/3, -3/4], [-1, -1]]

General Conic, scaled so constant (F) = 1

> 

gen:=A*x^2+B*x*y+C*y^2+D*x+E*y+1=0;

A*x^2+B*x*y+C*y^2+D*x+E*y+1 = 0

Equations to be satisfied at the four points and at a general 5th point (X,Y), which we will vary

> 

eqns:=[eval(gen,[x=X,y=Y]),map(i->eval(gen,{x=i[1],y=i[2]}),pts)[]];
solve(eqns,[A,B,C,D,E]);
con:=eval(gen,%[]);

[A*X^2+B*X*Y+C*Y^2+D*X+E*Y+1 = 0, A+(3/2)*B+(9/4)*C+D+(3/2)*E+1 = 0, (1/4)*A-(1/2)*B+C-(1/2)*D+E+1 = 0, (16/9)*A-B+(9/16)*C+(4/3)*D-(3/4)*E+1 = 0, A+B+C-D-E+1 = 0]

[[A = -6*(1321*X*Y-3028*Y^2+471*X+873*Y+3051)/(18306*X^2+5401*X*Y-17332*Y^2-6054*X+12429*Y), B = (7926*X^2-2144*Y^2-3455*X+3836*Y-5401)/(18306*X^2+5401*X*Y-17332*Y^2-6054*X+12429*Y), C = -4*(4542*X^2-536*X*Y-2171*X+1844*Y-4333)/(18306*X^2+5401*X*Y-17332*Y^2-6054*X+12429*Y), D = (2826*X^2+3455*X*Y-8684*Y^2+3651*Y+6054)/(18306*X^2+5401*X*Y-17332*Y^2-6054*X+12429*Y), E = (5238*X^2-3836*X*Y+7376*Y^2-3651*X-12429)/(18306*X^2+5401*X*Y-17332*Y^2-6054*X+12429*Y)]]

-6*(1321*X*Y-3028*Y^2+471*X+873*Y+3051)*x^2/(18306*X^2+5401*X*Y-17332*Y^2-6054*X+12429*Y)+(7926*X^2-2144*Y^2-3455*X+3836*Y-5401)*x*y/(18306*X^2+5401*X*Y-17332*Y^2-6054*X+12429*Y)-4*(4542*X^2-536*X*Y-2171*X+1844*Y-4333)*y^2/(18306*X^2+5401*X*Y-17332*Y^2-6054*X+12429*Y)+(2826*X^2+3455*X*Y-8684*Y^2+3651*Y+6054)*x/(18306*X^2+5401*X*Y-17332*Y^2-6054*X+12429*Y)+(5238*X^2-3836*X*Y+7376*Y^2-3651*X-12429)*y/(18306*X^2+5401*X*Y-17332*Y^2-6054*X+12429*Y)+1 = 0

Check it out. Conic to pass through (-1/2,-3/4).

> 

con1:=eval(con,{X=-1/2,Y=-3/4});
c1:=implicitplot(con1,x=-2..2,y=-2..2,colour=black):display(c1,p1);

(66/109)*x^2+(200/327)*x*y-(512/327)*y^2-(5/109)*x+(76/109)*y+1 = 0

So we just need to systematically vary X,Y

> 

controlpts:=[seq([X,0.7*X],X=-2.1..2,0.5)];
c:=map(i->eval(con,{X=i[1],Y=i[2]}),controlpts): #list of conics

[[-2.1, -1.47], [-1.6, -1.12], [-1.1, -.77], [-.6, -.42], [-.1, -0.7e-1], [.4, .28], [.9, .63], [1.4, .98], [1.9, 1.33]]

> 

display(seq(implicitplot(c[i],x=-2..2,y=-2..2,colour=black),i=1..nops(c)),p1,axes=none);

> 

 

 

Download Pencil.mw

return as procedures...

dharr 8482
November 11 2020
1 0

I removed the with(orthopoly) since they aren't needed. I think you want to return the procedures themselves, so you need:

return psi,w; in each of functionA and functionB. Then it works.

Package.mw

packagecall.mw

 

pass Array...

dharr 8482
November 06 2020
0 0

As @acer pointed out in answer to your recent question here the way around this is to create the array outside the procedure and pass it as an argument to the procedure. I use compile quite a bit and usually end up passing many Arrays (or Vectors or Matrices); as @mmcdara says, there are lots of limitations.

convert pwlist...

dharr 8482
November 01 2020
1 0

convert(f,pwlist) puts the pieces in a list for easier manipulation.

show/hide...

dharr 8482
October 30 2020
1 8

Sounds like you are using worksheet mode rather than document mode. Try the view menu, choose show/hide contents and unclick input.

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