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Mathematics Equation Solving Wolfram Language

Differences between u[x,y,t] vs. u[x,y][t] vs. u[x][y][t].

Artin Hatzikioseyian

Artin Hatzikioseyian, National Technical University of Athens (NTUA)

Posted 4 years ago

Is there any difference between defining a variable by u[x,y,t] or u[x,y][t] or u[x][y][t] ? I understand that beeing used in NDSolve the appropriate way to define it is either u[x,y][t] or u[x][y][t] in order to declare the dependent variable as u[x,y] or u[x][y] and the derivative vs time with prime notation as u[x,y]'[t] or u[x][y]'[t]. For the first notation the derivative would be D[u[x,y,t] ,t ] and the dependent variable u ? I would appreciate if anybody could comment on which is the reason to select any of the three forms ?

POSTED BY: Artin Hatzikioseyian

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W. Craig Carter

W. Craig Carter, MIT

Posted 4 years ago

There is not much practical difference. There are many cases where the u[x][y][t] forms are particularly convenient, but not necessarily for the derivative application you show. For derivatives, I prefer the operator-like forms: Derivative[0, 1][ArcTan][x, y] Derivative[0, 1][ArcTan] Derivative[0, 1][ArcTan][5, y] Derivative[0, 1][ArcTan][5, 12] When I want to pass a function as an argument, the u[x][y][t] forms are particularly convenient, consider this little tool to plot tangent lines: tangentLine[f_][x_][t_] := Module[{z, derivative}, derivative = D[f[z], z] /. z -> x; Arrow[{{x, f[x]}, {x + t, f[x] + t derivative}}] ] or tangentLine[f_][x_][t_] := Module[{derivative}, derivative = Derivative[1][f][x]; Arrow[{{x, f[x]}, {x + t, f[x] + t derivative}}]] Manipulate[ Plot[f[z], {z, 0, 4}, Epilog -> {PointSize[0.05], tangentLine[f][x][t]}], {x, 0, 4}, {t, .3, 4}, {f, {Sin, Cos, Function[{u}, u/(1 + u)]}} ] Using the f[x][y][t] makes the code more tidy, I think.

There is not much practical difference. There are many cases where the u[x][y][t] forms are particularly convenient, but not necessarily for the derivative application you show.

For derivatives, I prefer the operator-like forms:

Derivative[0, 1][ArcTan][x, y]
Derivative[0, 1][ArcTan]
Derivative[0, 1][ArcTan][5, y]
Derivative[0, 1][ArcTan][5, 12]

When I want to pass a function as an argument, the u[x][y][t] forms are particularly convenient, consider this little tool to plot tangent lines:

tangentLine[f_][x_][t_] := Module[{z, derivative},
  derivative = D[f[z], z] /. z -> x;
  Arrow[{{x, f[x]}, {x + t, f[x] + t  derivative}}]
  ]

tangentLine[f_][x_][t_] := 
 Module[{derivative}, derivative = Derivative[1][f][x]; 
  Arrow[{{x, f[x]}, {x + t, f[x] + t derivative}}]]

Manipulate[
 Plot[f[z], {z, 0, 4}, 
  Epilog -> {PointSize[0.05], tangentLine[f][x][t]}],
 {x, 0, 4},
 {t, .3, 4},
 {f, {Sin, Cos, Function[{u}, u/(1 + u)]}}
 ]

Using the f[x][y][t] makes the code more tidy, I think.

POSTED BY: W. Craig Carter