Skip to main content

calculus and analysis - How to collect terms with z-derivative?


my equations are very long (several pages). Here I will provide simple example:


eq = (g[x, y, z, t])^2*D[D[f[x, y, z, t], {x, 1}] {z, 1}] + \[Alpha]*

f[x, y, z, t]*D[D[g[x, y, z, t], {y, 1}], {z, 2}] +
g[x, y, z, t]*D[D[f[x, y, z, t], {t, 1}] {z, 2}]+D[f[x,y,z,t],{z,2}]

So, it has several functions, constant parameters, and consists of the sum of some terms. Each term is the product of some number of these functions and it's derivatives.


I want to collect, or sort, these terms according to z-derivative. So, first, I want to sort these terms according to the highest z-derivative of the function f[x,y,z,t]. So, in the example above, first term should be g[x, y, z, t]*D[D[f[x, y, z, t], {t, 1}] {z, 2}]+D[f[x,y,z,t],{z,2}], as long as it has second derivatives of the function f[x,y,z,t] over z. After that it should be (g[x, y, z, t])^2*D[D[f[x, y, z, t], {x, 1}] {z, 1}], and then f[x, y, z, t]*D[D[g[x, y, z, t], {y, 1}], {z, 2}].


Note, that the derivative could be taken over several arguments, like g[x, y, z, t]*D[D[f[x, y, z, t], {t, 1}] {z, 2}].


I looked through the examples of Collect, but didn't find a way to specify z-derivative, as a second argument. Also it would be good, if you point out, how to show only the terms with z derivatives.


Thanks in advance, Mikhail



Answer



I'm sure there is an easier and shorter way of doing this so consider this a starting point.



A strange thing I noticed is that when I copied and pasted your equation into a notebook it turned into a list.


eq = g[x, y, z, t]^2*D[D[f[x, y, z, t], {x, 1}] {z, 1}] + 
α*f[x, y, z, t]*D[D[g[x, y, z, t], {y, 1}], {z, 2}] +
g[x, y, z, t]*D[D[f[x, y, z, t], {t, 1}] {z, 2}] +
D[f[x, y, z, t], {z, 2}]

Mathematica graphics


I continue to be mystified by this however we need to take it into account.


Step 1 - Strip curly brackets


eq = eq[[1]]



Mathematica graphics


Step 2 - Break it into parts


eqList = eq[[#]] & /@ Range[Length[eq]]


Mathematica graphics


This effectively breaks it at the plus sign.


Step 3 - Sort the parts


This is the significant portion. We will use the value of the zth derivative to do the sorting.


sortedEqList = 
Sort[eqList,
Total@Cases[#1,

Derivative[i_Integer, j_Integer, k_Integer, l_Integer][f | g][x,
y, z, t] -> k, {0, Infinity}] >
Total@Cases[#2,
Derivative[i_Integer, j_Integer, k_Integer, l_Integer][f | g][x,
y, z, t] -> k, {0, Infinity}] &]

Mathematica graphics


Step 4 - Join the sorted parts


If you merely rejoin the parts using Plus they will be sorted back to the original order so use Inactive.


Inactive[Plus] @@ sortedEqList


Mathematica graphics


Comments

Popular posts from this blog

plotting - Filling between two spheres in SphericalPlot3D

Manipulate[ SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, Mesh -> None, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], {n, 0, 1}] I cant' seem to be able to make a filling between two spheres. I've already tried the obvious Filling -> {1 -> {2}} but Mathematica doesn't seem to like that option. Is there any easy way around this or ... Answer There is no built-in filling in SphericalPlot3D . One option is to use ParametricPlot3D to draw the surfaces between the two shells: Manipulate[ Show[SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], ParametricPlot3D[{ r {Sin[t] Cos[1.5 Pi], Sin[t] Sin[1.5 Pi], Cos[t]}, r {Sin[t] Cos[0 Pi], Sin[t] Sin[0 Pi], Cos[t]}}, {r, 1, 2 - n}, {t, 0, Pi}, PlotStyle -> Yellow, Mesh -> {2, 15}]], {n, 0, 1}]

plotting - Plot 4D data with color as 4th dimension

I have a list of 4D data (x position, y position, amplitude, wavelength). I want to plot x, y, and amplitude on a 3D plot and have the color of the points correspond to the wavelength. I have seen many examples using functions to define color but my wavelength cannot be expressed by an analytic function. Is there a simple way to do this? Answer Here a another possible way to visualize 4D data: data = Flatten[Table[{x, y, x^2 + y^2, Sin[x - y]}, {x, -Pi, Pi,Pi/10}, {y,-Pi,Pi, Pi/10}], 1]; You can use the function Point along with VertexColors . Now the points are places using the first three elements and the color is determined by the fourth. In this case I used Hue, but you can use whatever you prefer. Graphics3D[ Point[data[[All, 1 ;; 3]], VertexColors -> Hue /@ data[[All, 4]]], Axes -> True, BoxRatios -> {1, 1, 1/GoldenRatio}]

plotting - Mathematica: 3D plot based on combined 2D graphs

I have several sigmoidal fits to 3 different datasets, with mean fit predictions plus the 95% confidence limits (not symmetrical around the mean) and the actual data. I would now like to show these different 2D plots projected in 3D as in but then using proper perspective. In the link here they give some solutions to combine the plots using isometric perspective, but I would like to use proper 3 point perspective. Any thoughts? Also any way to show the mean points per time point for each series plus or minus the standard error on the mean would be cool too, either using points+vertical bars, or using spheres plus tubes. Below are some test data and the fit function I am using. Note that I am working on a logit(proportion) scale and that the final vertical scale is Log10(percentage). (* some test data *) data = Table[Null, {i, 4}]; data[[1]] = {{1, -5.8}, {2, -5.4}, {3, -0.8}, {4, -0.2}, {5, 4.6}, {1, -6.4}, {2, -5.6}, {3, -0.7}, {4, 0.04}, {5, 1.0}, {1, -6.8}, {2, -4.7}, {3, -1.