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MaPal93

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Function of a function: expressing deriv...

MaPal93 175 Maple
differentiation implicit
August 09 2024
2 8

Function X__2(y1,y2) is a function of X__1(y1,y2). How do I express (implicitly) d(X__2)/d(y1) and d(X__2)/d(y2) in terms of d(X__1)/d(y1) and d(X__1)/d(y2) and perhaps of X__1 and X__2 themselves?

I illustrate what I mean with an example for X__1(y1,y2), where d(X__1)/d(y1) and d(X__1)/d(y2) are written in a relatively compact form in terms of X__1 itself:
 

> 

restart;
> 

X1 := RootOf((8*y__1^14 + 32*y__1^12*y__2^2 + 48*y__1^10*y__2^4 + 32*y__1^8*y__2^6 + 8*y__1^6*y__2^8 + 16*y__1^12 + 80*y__1^10*y__2^2 + 160*y__1^8*y__2^4 + 160*y__1^6*y__2^6 + 80*y__1^4*y__2^8 + 16*y__1^2*y__2^10)*_Z^10 + (40*y__1^13*y__2^2 + 120*y__1^11*y__2^4 + 120*y__1^9*y__2^6 + 40*y__1^7*y__2^8 + 16*y__1^13 + 176*y__1^11*y__2^2 + 544*y__1^9*y__2^4 + 736*y__1^7*y__2^6 + 464*y__1^5*y__2^8 + 112*y__1^3*y__2^10)*_Z^9 + (84*y__1^12*y__2^4 + 168*y__1^10*y__2^6 + 84*y__1^8*y__2^8 - 20*y__1^14 + 28*y__1^12*y__2^2 + 552*y__1^10*y__2^4 + 1288*y__1^8*y__2^6 + 1132*y__1^6*y__2^8 + 348*y__1^4*y__2^10 - 48*y__1^12 - 192*y__1^10*y__2^2 - 272*y__1^8*y__2^4 - 128*y__1^6*y__2^6 + 48*y__1^4*y__2^8 + 64*y__1^2*y__2^10 + 16*y__2^12)*_Z^8 + (88*y__1^11*y__2^6 + 88*y__1^9*y__2^8 - 80*y__1^13*y__2^2 + 56*y__1^11*y__2^4 + 960*y__1^9*y__2^6 + 1432*y__1^7*y__2^8 + 608*y__1^5*y__2^10 - 48*y__1^13 - 416*y__1^11*y__2^2 - 944*y__1^9*y__2^4 - 736*y__1^7*y__2^6 + 16*y__1^5*y__2^8 + 256*y__1^3*y__2^10 + 80*y__1*y__2^12)*_Z^7 + (40*y__1^10*y__2^8 - 128*y__1^12*y__2^4 + 168*y__1^10*y__2^6 + 928*y__1^8*y__2^8 + 632*y__1^6*y__2^10 + 12*y__1^14 - 192*y__1^12*y__2^2 - 1084*y__1^10*y__2^4 - 1472*y__1^8*y__2^6 - 340*y__1^6*y__2^8 + 432*y__1^4*y__2^10 + 180*y__1^2*y__2^12 + 48*y__1^12 + 144*y__1^10*y__2^2 + 112*y__1^8*y__2^4 - 48*y__1^6*y__2^6 - 96*y__1^4*y__2^8 - 32*y__1^2*y__2^10)*_Z^6 + (-92*y__1^11*y__2^6 + 228*y__1^9*y__2^8 + 368*y__1^7*y__2^10 + 32*y__1^13*y__2^2 - 384*y__1^11*y__2^4 - 1272*y__1^9*y__2^6 - 704*y__1^7*y__2^8 + 376*y__1^5*y__2^10 + 224*y__1^3*y__2^12 + 48*y__1^13 + 304*y__1^11*y__2^2 + 432*y__1^9*y__2^4 + 16*y__1^7*y__2^6 - 288*y__1^5*y__2^8 - 128*y__1^3*y__2^10)*_Z^5 + (-24*y__1^10*y__2^8 + 96*y__1^8*y__2^10 + 24*y__1^12*y__2^4 - 412*y__1^10*y__2^6 - 576*y__1^8*y__2^8 + 164*y__1^6*y__2^10 + 160*y__1^4*y__2^12 + 4*y__1^14 + 172*y__1^12*y__2^2 + 532*y__1^10*y__2^4 + 252*y__1^8*y__2^6 - 328*y__1^6*y__2^8 - 216*y__1^4*y__2^10 - 16*y__1^12 - 32*y__1^10*y__2^2 + 32*y__1^6*y__2^6 + 16*y__1^4*y__2^8)*_Z^4 + (-8*y__1^11*y__2^6 - 192*y__1^9*y__2^8 + 28*y__1^7*y__2^10 + 60*y__1^5*y__2^12 + 16*y__1^13*y__2^2 + 232*y__1^11*y__2^4 + 296*y__1^9*y__2^6 - 168*y__1^7*y__2^8 - 184*y__1^5*y__2^10 - 16*y__1^13 - 64*y__1^11*y__2^2 - 32*y__1^9*y__2^4 + 64*y__1^7*y__2^6 + 48*y__1^5*y__2^8)*_Z^3 + (-15*y__1^10*y__2^8 + 9*y__1^6*y__2^12 + 24*y__1^12*y__2^4 + 116*y__1^10*y__2^6 - 36*y__1^8*y__2^8 - 80*y__1^6*y__2^10 - 4*y__1^14 - 40*y__1^12*y__2^2 - 48*y__1^10*y__2^4 + 40*y__1^8*y__2^6 + 52*y__1^6*y__2^8)*_Z^2 + (12*y__1^11*y__2^6 - 4*y__1^9*y__2^8 - 16*y__1^7*y__2^10 - 8*y__1^13*y__2^2 - 24*y__1^11*y__2^4 + 8*y__1^9*y__2^6 + 24*y__1^7*y__2^8)*_Z - y__1^10*y__2^8 - y__1^8*y__2^10 - 4*y__1^12*y__2^4 + 4*y__1^8*y__2^8):

alias(X__1=X1);

X__2 := -y__1*y__2^2*X__1*(8*X__1^7*y__1^10 + 24*X__1^7*y__1^8*y__2^2 + 24*X__1^7*y__1^6*y__2^4 + 8*X__1^7*y__1^4*y__2^6 + 24*X__1^6*y__1^9*y__2^2 + 48*X__1^6*y__1^7*y__2^4 + 24*X__1^6*y__1^5*y__2^6 + 26*X__1^5*y__1^8*y__2^4 + 26*X__1^5*y__1^6*y__2^6 + 8*X__1^4*y__1^7*y__2^6 + 32*X__1^7*y__1^8 + 128*X__1^7*y__1^6*y__2^2 + 192*X__1^7*y__1^4*y__2^4 + 128*X__1^7*y__1^2*y__2^6 + 32*X__1^7*y__2^8 + 16*X__1^6*y__1^9 + 192*X__1^6*y__1^7*y__2^2 + 480*X__1^6*y__1^5*y__2^4 + 448*X__1^6*y__1^3*y__2^6 + 144*X__1^6*y__1*y__2^8 - 16*X__1^5*y__1^10 + 32*X__1^5*y__1^8*y__2^2 + 392*X__1^5*y__1^6*y__2^4 + 624*X__1^5*y__1^4*y__2^6 + 280*X__1^5*y__1^2*y__2^8 - 36*X__1^4*y__1^9*y__2^2 + 72*X__1^4*y__1^7*y__2^4 + 396*X__1^4*y__1^5*y__2^6 + 288*X__1^4*y__1^3*y__2^8 - 28*X__1^3*y__1^8*y__2^4 + 84*X__1^3*y__1^6*y__2^6 + 156*X__1^3*y__1^4*y__2^8 - 7*X__1^2*y__1^7*y__2^6 + 33*X__1^2*y__1^5*y__2^8 - 64*X__1^5*y__1^8 - 192*X__1^5*y__1^6*y__2^2 - 192*X__1^5*y__1^4*y__2^4 - 64*X__1^5*y__1^2*y__2^6 - 32*X__1^4*y__1^9 - 288*X__1^4*y__1^7*y__2^2 - 480*X__1^4*y__1^5*y__2^4 - 224*X__1^4*y__1^3*y__2^6 + 8*X__1^3*y__1^10 - 88*X__1^3*y__1^8*y__2^2 - 408*X__1^3*y__1^6*y__2^4 - 312*X__1^3*y__1^4*y__2^6 + 12*X__1^2*y__1^9*y__2^2 - 108*X__1^2*y__1^7*y__2^4 - 196*X__1^2*y__1^5*y__2^6 + 4*X__1^2*y__1^3*y__2^8 + 2*X__1*y__1^8*y__2^4 - 46*X__1*y__1^6*y__2^6 + 4*X__1*y__1^4*y__2^8 - y__1^7*y__2^6 + y__1^5*y__2^8 + 32*X__1^3*y__1^8 + 64*X__1^3*y__1^6*y__2^2 + 32*X__1^3*y__1^4*y__2^4 + 16*X__1^2*y__1^9 + 96*X__1^2*y__1^7*y__2^2 + 80*X__1^2*y__1^5*y__2^4 + 32*X__1*y__1^8*y__2^2 + 64*X__1*y__1^6*y__2^4 + 16*y__1^7*y__2^4)/(-8*X__1^7*y__1^11*y__2^2 - 24*X__1^7*y__1^9*y__2^4 - 24*X__1^7*y__1^7*y__2^6 - 8*X__1^7*y__1^5*y__2^8 - 32*X__1^6*y__1^10*y__2^4 - 64*X__1^6*y__1^8*y__2^6 - 32*X__1^6*y__1^6*y__2^8 - 50*X__1^5*y__1^9*y__2^6 - 50*X__1^5*y__1^7*y__2^8 - 28*X__1^4*y__1^8*y__2^8 + 32*X__1^8*y__1^10 + 160*X__1^8*y__1^8*y__2^2 + 320*X__1^8*y__1^6*y__2^4 + 320*X__1^8*y__1^4*y__2^6 + 160*X__1^8*y__1^2*y__2^8 + 32*X__1^8*y__2^10 + 160*X__1^7*y__1^9*y__2^2 + 640*X__1^7*y__1^7*y__2^4 + 960*X__1^7*y__1^5*y__2^6 + 640*X__1^7*y__1^3*y__2^8 + 160*X__1^7*y__1*y__2^10 - 8*X__1^6*y__1^12 - 24*X__1^6*y__1^10*y__2^2 + 328*X__1^6*y__1^8*y__2^4 + 1048*X__1^6*y__1^6*y__2^6 + 1056*X__1^6*y__1^4*y__2^8 + 352*X__1^6*y__1^2*y__2^10 - 24*X__1^5*y__1^11*y__2^2 - 48*X__1^5*y__1^9*y__2^4 + 400*X__1^5*y__1^7*y__2^6 + 848*X__1^5*y__1^5*y__2^8 + 424*X__1^5*y__1^3*y__2^10 - 38*X__1^4*y__1^10*y__2^4 - 54*X__1^4*y__1^8*y__2^6 + 274*X__1^4*y__1^6*y__2^8 + 290*X__1^4*y__1^4*y__2^10 - 44*X__1^3*y__1^9*y__2^6 - 32*X__1^3*y__1^7*y__2^8 + 104*X__1^3*y__1^5*y__2^10 - 27*X__1^2*y__1^8*y__2^8 + 15*X__1^2*y__1^6*y__2^10 - 64*X__1^6*y__1^10 - 256*X__1^6*y__1^8*y__2^2 - 384*X__1^6*y__1^6*y__2^4 - 256*X__1^6*y__1^4*y__2^6 - 64*X__1^6*y__1^2*y__2^8 - 256*X__1^5*y__1^9*y__2^2 - 768*X__1^5*y__1^7*y__2^4 - 768*X__1^5*y__1^5*y__2^6 - 256*X__1^5*y__1^3*y__2^8 + 16*X__1^4*y__1^12 + 32*X__1^4*y__1^10*y__2^2 - 408*X__1^4*y__1^8*y__2^4 - 848*X__1^4*y__1^6*y__2^6 - 424*X__1^4*y__1^4*y__2^8 + 48*X__1^3*y__1^11*y__2^2 + 48*X__1^3*y__1^9*y__2^4 - 352*X__1^3*y__1^7*y__2^6 - 352*X__1^3*y__1^5*y__2^8 + 54*X__1^2*y__1^10*y__2^4 - 150*X__1^2*y__1^6*y__2^8 + 22*X__1*y__1^9*y__2^6 - 30*X__1*y__1^7*y__2^8 - 2*y__1^8*y__2^8 + 32*X__1^4*y__1^10 + 96*X__1^4*y__1^8*y__2^2 + 96*X__1^4*y__1^6*y__2^4 + 32*X__1^4*y__1^4*y__2^6 + 96*X__1^3*y__1^9*y__2^2 + 192*X__1^3*y__1^7*y__2^4 + 96*X__1^3*y__1^5*y__2^6 - 8*X__1^2*y__1^12 - 8*X__1^2*y__1^10*y__2^2 + 104*X__1^2*y__1^8*y__2^4 + 104*X__1^2*y__1^6*y__2^6 - 16*X__1*y__1^11*y__2^2 + 48*X__1*y__1^7*y__2^6 - 8*y__1^10*y__2^4 + 8*y__1^8*y__2^6):


Synthetic representation of derivatives


derivatives for X__1 okay: written in terms of y__1, y__2, and X__1 itself.

> 

derX1_y1 := diff(X1, y__1):
derX1_y2 := diff(X1, y__2):

Diff('X__1(y__1,y__2)', y__1) = collect~(normal(eval(derX1_y1, X1 = 'X__1(y__1,y__2)')), 'X__1(y__1,y__2)');
Diff('X__1(y__1,y__2)', y__2) = collect~(normal(eval(derX1_y2, X1 = 'X__1(y__1,y__2)')), 'X__1(y__1,y__2)');


derivatives for X__2 not okay (X__2 is itself a function of X__1): how do I write them in terms of y__1, y__2, X__1 AND the derivatives for X__1 just found? What's the most compact way to express these two derivatives below?

> 

derX2_y1 := diff(X__2, y__1):
derX2_y2 := diff(X__2, y__2):

Diff('X__2(y__1,y__2)', y__1) = collect~(normal(eval(eval(derX2_y1, derX1_y1='Diff('X__1(y__1,y__2)', y__1)'), X__1 = 'X__1(y__1,y__2)')), 'X__1(y__1,y__2)');
Diff('X__2(y__1,y__2)', y__2) = collect~(normal(eval(derX2_y2, X__1 = 'X__1(y__1,y__2)')), 'X__1(y__1,y__2)');
> 

``

 

Download compact_derivatives.mw


Thank you for the help.

Combining standard font and Greek symbol...

MaPal93 175 Maple
subscript superscript legend
August 05 2024
1 3

I can't combine standard font and Greek symbols in a plot legend. I want to type:

"Limit of X_{i}^{\rho=0} (\alpha\mapsto+\infty)>0"

where I used LaTeX notation to emphasise that I want (1) superscript and subscript of X displayed as aligned exactly on top of each other, (2) rho, alpha, mapsto-style arrow, and infinity typed in their respective symbolic forms.

thanks!

How to display an expression as I typed ...

MaPal93 175 Maple
typesetting textplot plotting
June 06 2024
2 2

I want to display W__LJ as I typed it, without Maple running the calculations resulting in the output here below:

> 

restart;
> 

'W__LJ = 0.75 + 0.98*((1.18/Gamma)^1.9 - (1.15/Gamma)^0.98)';

W__LJ = .75+1.342152577*(1/Gamma)^1.9-1.123854165*(1/Gamma)^.98

 (1)
 

NULL

Download display_formula.mw

This is important as I will place such expression in a plot like this:
textplot([2,0.9,typeset('W__LJ=0.75+0.98((1.18/Gamma)^1.9-(1.15/Gamma)^0.98)')],'font'=["helvetica","roman",35])
and I need it in the original functional form, which is easier to interpret.

For which x- and y- values these two bou...

MaPal93 175 Maple 2023
solve fsolve plot3d
June 06 2024
-1 11

I have two surfaces crossing the z=0 plane for some ranges of x and y values.

For the first surface, x=Gamma is bounded between 0 and 10 and y=rho between -1 and +1. For the second surface, x=Gamma_1 is bounded between 0 and 10 and y=Gamma_2 between 0 and 10 as well. I want to clearly identify (parametric):

  1. For which Gamma and rho ranges of values the first surface is positive (and for which negative)
  2. For which Gamma_1 and Gamma_2 ranges of values the second surface is positive (and for which negative)

Worksheet: sign_regions.mw (highlighted in yellow my two failed attempts)

How to apply the zgradient of colorschem...

MaPal93 175 Maple
plot3d view plotting
June 05 2024
1 2

I have a positive surface that I plot3d for bounded x- and y- value ranges. It reaches 1000 but it's mostly flat except almost at the edges of the x- and y-axes. Therefore, I inlcude view=[default,default,0..10] among the options of plot3d so that I can focus on the features of interest.

For coloring I use 'colorscheme'=["zgradient",["LightGray", "Gray", "Green"]]. However, this applies to the whole surface (up to 1000) and NOT exclusively to the viewed part as I would like to. 

Question: how to scale the zgradient so that it only applies to my "view" portion?

Note that I don't want to do this by using the 'markers' options to readjust the color splits manually, as I don't know the exact proportion and I don't want to randomly play around with different triplets until I achieve a visually satisfying result...

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