Mathematica: plot and evaluate function of 2 variables

[hasidim]

Hi all,

I am trying to evaluate and plot a function of two variable in mathematica.

Function:

R(t,\theta) = a(t) + b(t)*Y_n(\theta)

Where a(t) and b(t) are functions (that I have evaluated already), and Y is the spherical harmonic for a given n, and R is the radius.

In mathematica I am unsure how to evaluate and plot a function of two variables in spherical coordinates. I attempted to use "SphericalPlot3D" to plot R as a function of \theta and \phi, and "Manipulate" to add a slider to adjust the time but am having no luck.

Any help would be APPRECIATED!

 

[Bill Simpson]

Is this something like what you are trying to do?

a[t_] := 2t; b[t_] := t^2; m = 1;(* < --- Substitute your functions here *)
R[t_, n_, m_, theta_, phi_] := a[t] + b[t]*SphericalHarmonicY[n, m, theta, phi]
Manipulate[SphericalPlot3D[R[t, n, m, theta, phi], {theta, 0, 2Pi}, {phi, 0, Pi}], {t, 0, 1}, {n, 0, 4, 1}]

Can you perhaps adapt this for what you need?

 

[hasidim]

Is this something like what you are trying to do?

a[t_] := 2t; b[t_] := t^2; m = 1;(* < --- Substitute your functions here *)
R[t_, n_, m_, theta_, phi_] := a[t] + b[t]*SphericalHarmonicY[n, m, theta, phi]
Manipulate[SphericalPlot3D[R[t, n, m, theta, phi], {theta, 0, 2Pi}, {phi, 0, Pi}], {t, 0, 1}, {n, 0, 4, 1}]

Can you perhaps adapt this for what you need?

That is pretty similar to what I have tried... but, I'll try copying that and modifying it.

As usual, I suspect that I am missing just one small detail.

 

[Bill Simpson]

If your modification doesn't work then divide the problem into simpler tasks.
Confirm that
SphericalPlot3D[SphericalHarmonicY[1, 2, theta, phi]], {theta, 0, 2Pi}, {phi, 0, Pi}]
plot matches http://mathworld.wolfram.com/SphericalHarmonic.html
and substitute a few other values for the 1 and 2.
Then try a simpler Manipulate example using a Real and Integer value from the Help and confirm that works. That might uncover some of your missing details.

 

[blue_raver22]

Hello,

Thank you for reaching out for assistance with plotting and evaluating a function of two variables in Mathematica. I am happy to provide some guidance and tips for achieving your goal.

Firstly, in order to plot a function of two variables, you will need to use a 3D plot function such as "Plot3D" or "SphericalPlot3D". These functions take in three arguments: the function itself, the ranges of the two variables, and any additional options for the plot. In your case, the function is R(t,\theta) and the two variables are t and \theta. The ranges for t and \theta can be specified using the "Range" option in the plot function.

Next, you will need to define your function R(t,\theta) in Mathematica. You can do this by using the "DefineFunction" command. For example, you can write:

R[t_, \theta_] := a[t] + b[t] * SphericalHarmonicY[n, \theta, \phi]

Note that the variables t and \theta are defined as arguments in the function, and the SphericalHarmonicY function is used to represent Y_n(\theta) in your original function.

Once you have defined your function, you can then use it in the plot function. For example, you could write:

SphericalPlot3D[R[t, \theta], {t, tmin, tmax}, {\theta, \theta_min, \theta_max}, PlotRange -> All]

where tmin, tmax, \theta_min, and \theta_max are the desired ranges for the variables t and \theta.

Additionally, if you want to add a slider to adjust the time (t) in your plot, you can use the "Manipulate" function. This function allows you to create interactive plots by varying the values of certain parameters. For example, you could write:

Manipulate[SphericalPlot3D[R[t, \theta], {t, tmin, tmax}, {\theta, \theta_min, \theta_max}, PlotRange -> All], {t, tmin, tmax}]

This will create a slider for the variable t, allowing you to adjust the time in your plot.

I hope this helps you with your plot and evaluation of a function of two variables in Mathematica. Please let me know if you have any further questions or need additional assistance. Best of luck with your work!