Need help to interpret unusual Maxima notation for a derivative i.e. (d^{y+1}/dx^{y+1}) rho

Your notation of the mixed derivative is incorrect.  For example, in 
calculating %sigma[12,2] you should say

(diff(%rho, x, 1, y, 1)

(instead of (diff(%rho, x, y))

-- Martin








(%i22) ? diff;
  -- Function: diff (<expr>, <x_1>, <n_1>, ..., <x_m>, <n_m>)
  -- Function: diff (<expr>, <x>, <n>)
  -- Function: diff (<expr>, <x>)
  -- Function: diff (<expr>)
      Returns the derivative or differential of <expr> with respect to
      some or all variables in <expr>.
      `diff (<expr>, <x>, <n>)' returns the <n>'th derivative of <expr>
      with respect to <x>.
      `diff (<expr>, <x_1>, <n_1>, ..., <x_m>, <n_m>)' returns the mixed
      partial derivative of <expr> with respect to <x_1>, ..., <x_m>.
      It is equivalent to `diff (... (diff (<expr>, <x_m>, <n_m>) ...),
      <x_1>, <n_1>)'.
      `diff (<expr>, <x>)' returns the first derivative of <expr> with
      respect to the variable <x>.
      `diff (<expr>)' returns the total differential of <expr>, that is,
      the sum of the derivatives of <expr> with respect to each its
      variables times the differential `del' of each variable.  No
      further simplification of `del' is offered.
      The noun form of `diff' is required in some contexts, such as
      stating a differential equation.  In these cases, `diff' may be
      quoted (as `'diff') to yield the noun form instead of carrying out
      the differentiation.
      When `derivabbrev' is `true', derivatives are displayed as
      subscripts.  Otherwise, derivatives are displayed in the Leibniz
      notation, `dy/dx'.
      Examples:
           (%i1) diff (exp (f(x)), x, 2);
                                2
                         f(x)  d               f(x)  d         2
           (%o1)       %e     (--- (f(x))) + %e     (-- (f(x)))
                                 2                   dx
                               dx
           (%i2) derivabbrev: true$
           (%i3) 'integrate (f(x, y), y, g(x), h(x));
                                    h(x)
                                   /
                                   [
           (%o3)                   I     f(x, y) dy
                                   ]
                                   /
                                    g(x)
           (%i4) diff (%, x);
                  h(x)
                 /
                 [
           (%o4) I     f(x, y)  dy + f(x, h(x)) h(x)  - f(x, g(x)) g(x)
                 ]            x                     x                  x
                 /
                  g(x)
      For the tensor package, the following modifications have been
      incorporated:
      (1) The derivatives of any indexed objects in <expr> will have the
      variables <x_i> appended as additional arguments.  Then all the
      derivative indices will be sorted.
      (2) The <x_i> may be integers from 1 up to the value of the
      variable `dimension' [default value: 4].  This will cause the
      differentiation to be carried out with respect to the <x_i>'th
      member of the list `coordinates' which should be set to a list of
      the names of the coordinates, e.g., `[x, y, z, t]'. If
      `coordinates' is bound to an atomic variable, then that variable
      subscripted by <x_i> will be used for the variable of
      differentiation.  This permits an array of coordinate names or
      subscripted names like `X[1]', `X[2]', ... to be used.  If
      `coordinates' has not been assigned a value, then the variables
      will be treated as in (1) above.
   There are also some inexact matches for `diff'.
   Try `?? diff' to see them.
(%o22) true


Vishal Ramnath schrieb:
> Hello,
>  
> I need some help in understanding what this term in a PDE that I set up 
> in wxMaxima 0.8.2 Windows XP actually is viz.
>  
>    y+1
> d      
> -------- rho
>    y+1
> dx
>  
> (LaTeX code is $\frac{d^{y+1}}{dx^{y+1}} \rho$)
>  
> It doesn't quite make sense if interpreted as
>  
> (d/dy) (d/dy) (d/dx) rho
>  
> i.e. (y+1)-th derivative of the (d/dx) derivative of rho, for example in 
> Maxima (d/dy)(d/dx rho) would be written as (d^y rho)/(dx^y) which is 
> straightforward and understandable.
>  
> I attach the code below.
>  
> Thank you for any assistance.
>  
> Regards,
>  
> Vishal
>  
> ps. I use the Greek font symbols
>  
> "Burnett Equations - 8 June 2009"$
> "This program sets up the Burnett equations (mass, momentum, energy) as 
> explicit PDEs"$
>  
> "set \rho(x,y) u(x,y) v(x,y)"$
> depends([%rho, u, v], [x, y])$
>  
> "first order components for stress tensor"$
> %sigma[11, 1]: -%mu*((4/3)*diff(u, x) - (2/3)*diff(v, y));
> %sigma[22, 1]: -%mu*((4/3)*diff(v, y) - (2/3)*diff(u, x));
>  
> %sigma[12, 1]: -%mu*(diff(u, y) + diff(v, x));
> %sigma[21, 1]: -%mu*(diff(u, y) + diff(v, x));
>  
> "heat transfer components to first order"$
> q[1, 1]: -%kappa*diff(T, x);
> q[2, 1]: -%kappa*diff(T, y);
>  
> "set up Navier-Stokes equations i.e. Burnett equation truncated to first 
> order"$
> eq[1, 1]: diff(%rho*u, x) + diff(%rho*v, y);
> eq[2, 1]: diff(%rho*u^2 + %sigma[11, 1], x) + diff(%rho*u*v + %sigma[21, 
> 1], y);
> eq[3, 1]: diff(%rho*u*v + %sigma[12, 1], x) + diff(%rho*v^2 + %sigma[22, 
> 1], y);
>  
> "ideal gas equation of state"$
> "Comment: I want to solve for u(x,y,t), v(x,y,t), \rho(x,y,t)"$
> p: %rho*R*T;
>  
> %sigma[11, 2]: %sigma[11, 1]
>     + (%mu^2/p)*(%alpha[1]*(diff(u, x))^2
>               + %alpha[2]*(diff(u, y))^2
>               + %alpha[3]*(diff(v, x))^2
>               + %alpha[4]*(diff(v, y))^2
>               + %alpha[5]*(diff(u, x))*(diff(v, y))
>               + %alpha[6]*(diff(u, y))*(diff(v, x))
>         + %alpha[7]*R*(diff(T, x, 2))
>         + %alpha[8]*R*(diff(T, y, 2))
>         + %alpha[9]*(R*T/%rho)*(diff(%rho, x, 2))
>         + %alpha[10]*(R*T/%rho)*(diff(%rho, y, 2))
>         + %alpha[11]*(R*T/%rho^2)*((diff(%rho, x))^2)
>         + %alpha[12]*(R*T/%rho^2)*((diff(%rho, y))^2)
>         + %alpha[13]*(R/T)*((diff(T, x))^2)
>         + %alpha[14]*(R/T)*((diff(T, y))^2)
>         + %alpha[15]*(R/%rho)*(diff(T, x))*(diff(%rho, x))
>         + %alpha[16]*(R/%rho)*(diff(T, y))*(diff(%rho, y))
>         );
>      
> %sigma[22, 2]: %sigma[22, 1]
>     + (%mu^2/p)*(%alpha[1]*(diff(v, y))^2
>         + %alpha[2]*(diff(v, x))^2
>         + %alpha[3]*(diff(u, y))^2
>         + %alpha[4]*(diff(u, x))^2
>         + %alpha[5]*(diff(u, x))*(diff(v, y))
>         + %alpha[6]*(diff(u, y))*(diff(v, x))
>         + %alpha[7]*R*(diff(T, y, 2))
>         + %alpha[8]*R*(diff(T, x, 2))
>         + %alpha[9]*(R*T/%rho)*(diff(%rho, y, 2))
>         + %alpha[10]*(R*T/%rho)*(diff(%rho, x, 2))
>         + %alpha[11]*(R*T/%rho^2)*((diff(%rho, y))^2)
>         + %alpha[12]*(R*T/%rho^2)*((diff(%rho, x))^2)
>         + %alpha[13]*(R/T)*((diff(T, y))^2)
>         + %alpha[14]*(R/T)*((diff(T, x))^2)
>         + %alpha[15]*(R/%rho)*(diff(T, y))*(diff(%rho, y))
>         + %alpha[16]*(R/%rho)*(diff(T, x))*(diff(%rho, x))
>         );
>         
> %sigma[12, 2]: %sigma[12, 1] + (%mu^2/p)*(??ta[1]*(diff(u, x))*(diff(u, y))
>            + ??ta[2]*(diff(v, x))*(diff(v, y))
>            + ??ta[3]*(diff(u, x))*(diff(v, x))
>            + ??ta[4]*(diff(u, y))*(diff(v, y))
>            + ??ta[5]*R*(diff(T,x,y))
>            + ??ta[6]*(R*T/%rho)*(diff(%rho, x, y))
>            + ??ta[7]*(R/T)*(diff(T, x))*(diff(T, y))
>            + ??ta[8]*(R*T/%rho^2)*(diff(%rho, x))*(diff(%rho, y))
>            + ??ta[9]*(R/%rho)*(diff(%rho, x))*(diff(T, y))
>            + ??ta[10]*(R/%rho)*(diff(T, x))*(diff(%rho, y))
>            );
> q[1, 2]: q[1, 1] + (%mu^2/%rho)*(%gamma[1]*(1/T)*(diff(T, x))*(diff(u, x))
>          + %gamma[2]*(1/T)*(diff(T, x))*(diff(v, y))
>          + %gamma[3]*(1/T)*(diff(T, y))*(diff(v, x))
>          + %gamma[4]*(1/T)*(diff(T, y))*(diff(u, y))
>          + %gamma[5]*(diff(u, x, 2))
>          + %gamma[6]*(diff(u, y, 2))
>          + %gamma[7]*(diff(v, x, y))
>          + %gamma[8]*(1/%rho)*(diff(%rho, x))*(diff(u, x))
>          + %gamma[9]*(1/%rho)*(diff(%rho, x))*(diff(v, y))
>          + %gamma[10]*(1/%rho)*(diff(%rho, y))*(diff(v, x))
>          + %gamma[11]*(1/%rho)*(diff(%rho, y))*(diff(u, y))
>          );
>            
>            
> eq[1, 2]: diff(%rho*u, x) + diff(%rho*v, y);
> eq[2, 2]: diff(%rho*u^2 + %sigma[11, 2], x) + diff(%rho*u*v + %sigma[21, 
> 2], y);
> eq[3, 2]: diff(%rho*u*v + %sigma[12, 2], x) + diff(%rho*v^2 + %sigma[22, 
> 2], y);
>  
> "Equations are built up now!"$
>  
> "Discretize first order equations using finite differences..."$
>  
>  
> 
> 
> ------------------------------------------------------------------------
> 
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