16. Separation of variables in higher dimensions

Corrections

At 16:00, \( Y_n(x) \) should obviously be \( Y_n(y) \).

At 18:01, \( \lambda_5 \) should be \( 1^2 + 3^2 = 10 \) rather than \( 2^2 + 3^2 = 13 \), and similarly \( \lambda_6 = 3^2 + 1^2 = 10 \). And then, at 18:23, they should of course be labelled \( \lambda_{13} \) and \( \lambda_{31} \). (The sequence continues with \( \lambda_{7} = \lambda_{8} = 2^2 + 3^2 = 13 \), \( \lambda_{9} = \lambda_{10} = 1^2 + 4^2 = 17 \), \( \lambda_{11} = 3^2 + 3^2 = 18 \), \( \lambda_{12} = \lambda_{13} = 2^2 + 4^2 = 20 \), etc.)

At 27:00, when I said that \( R(0) \) should be infinite, of course I meant to say finite (as it is written on screen).

At 43:20, "[free to choose] \( a_1 \)" should be \( a_0 \) (obviously).

At 58:00 and 58:34, \( u(x,y) \) should be \( u(x,y,t) \).

At 1:06:28, \( \lambda^2 = (am/\pi)^2 + (bn/\pi)^2 \) should be \( \lambda = (\pi m/a)^2 + (\pi n/b)^2 \). (This is corrected a little later in the video, but maybe not in a perfectly clear way.)

Mathematica code

The plot at 55:25 is produced by the following Mathematica code:

With[{n = 4, m = 2}, With[{z = BesselJZero[n, m]},
  ParametricPlot3D[{r Cos[v], r Sin[v], Cos[n v] BesselJ[n, r z]}, {r, 0, 1}, {v, 0, 2 Pi},
    PlotPoints -> 50]]]

And the first row of plots at 1:11:40 is produced as follows:

region = Polygon[{{0, 0}, {1, 1}, {2, 0}, {1, -1}, {2, -2}, {2, -4}, {1, -3}, {0, -4}}];

{eigenvalues, eigenfunctions} =
  NDEigensystem[{-Laplacian[u[x, y], {x, y}], DirichletCondition[u[x, y] == 0, True]},
   u[x, y], {x, y} \[Element] region, 4];
  
Table[Plot3D[eigenfunctions[[i]] // N // Evaluate, {x, y} \[Element] region,
  PlotRange -> All, BoxRatios -> Automatic, PlotLabel -> eigenvalues[[i]],
  PlotTheme -> "Web", ImageSize -> Medium, PlotPoints -> 50], {i, Length[eigenvalues]}]

For the second row, do the same but with a another region:

region = Polygon[{{0, 0}, {2, 0}, {3, -1}, {1, -3}, {2, -4}, {0, -4}, {0, -2}, {1, -1}}];

Sidansvarig: Hans Lundmark
Senast uppdaterad: 2026-05-04