Writing Curl in Mathics

Mathics is extensible; it allows creation of new functions, and new builtin functions all using the Mathics language.

So before writing this in Python or in SymPy, let us see how to do this in Mathics itself. We are only going to try the first two forms, curl in two and three dimensions.

Note: by the time you are reading this Curl has already been

added to Mathics. So we will instead define a “builtin” function MCurl (Mathics Curl) so that we don’t have function name clashes.

Two-dimensional Mathematical Definition

Curl is defined as:

\[\partial f_2 / \partial x_1 - \partial f_1 / \partial x_2\]

for two-dimensional vectors.

Two-dimensional Mathics Function

Translating the above definition into Mathics:

MCurl[{f1_, f2_}, {x1_, x2_}] :=  D[f2, x1] - D[f1, x2]

Now let’s try that inside Mathics using the two-dimensional example that can be found in the WMA reference for Curl

$ mathics

Mathics 5.0.3dev0
In[1]:= MCurl[{f1_, f2_}, {x1_, x2_}] :=  D[f2, x1] - D[f1, x2]
Out[1]= None

In[2]:= (* Test the 2D definition: *)
         MCurl[{y, -x}, {x, y}]
Out[2]= -2

In[3] := v[x_, y_] := {Cos[x] Sin[y], Cos[y] Sin[x]}
Out[3]= None

In[4] := MCurl[v[x, y], {x, y}]
Out[4]= 0

Three-dimensional Mathematical Definition

In three dimensions, things are a little more involved:

\[( \partial f_3 / \partial x_2 - \partial f_2 / \partial x_3, \ \ % \partial f_1 / \partial x_3 - \partial f_3 / \partial x_1, \ \ % \partial f_2 / \partial x_1 - \partial f_1 / \partial x_2 )\]

Three-dimensional Mathics Function

Translating the above definition into Mathics:

In[5] := MCurl[{f1_, f2_, f3_}, {x1_, x2_, x3_}] := {
         D[f3, x2] - D[f2, x3],
         D[f1, x3] - D[f3, x1],
         D[f2, x1] - D[f1, x2]
       }
Out[5]= None

In[6]:= (* An example form WMA VectorAnalysis: *)
        MCurl[{y, -x, 2 z}, {x, y, z}]
Out[6]= {0, 0, -2}

Adding Curl as an autoloaded function

The above code was done in an interactive session. Below we extract the function definitions and package this.

(* Two and Three dimensional Curl, taken from the Mathematical definitions *)

Begin["System`"] (* Add definition in System` namespace *)

(* Set Information[] or ? help *)
MCurl::usage = "returns the curl of a two-or three-dimensional vector space";

(* Curl in two dimensions *)
MCurl[{f1_, f2_}, {x1_, x2_}] :=  D[f2, x1] - D[f1, x2]

(* Curl in three dimensions *)
MCurl[{f1_, f2_, f3_}, {x1_, x2_, x3_}] := {
     D[f3, x2] - D[f2, x3],
     D[f1, x3] - D[f3, x1],
     D[f2, x1] - D[f1, x2]
     }

Protect[MCurl]  (* Make sure Function cannot easily be changed *)
End[]

Place the above code in mathics-core/mathics/autoload/rules/Curl.m, and when Mathics starts up, this code will be evaluated.

Testing autoloaded Curl function

Now let us try MCurl in a mathics session:

$  mathics

Mathics 5.0.3dev0
on CPython 3.8.12 (heads/v2.3.4.1_release:4a6b4d3504, Jun  3 2022, 15:46:12)
...

In[1]:= ?MCurl
returns the curl of a two-or three-dimensional vector space
Out[1]= Null

In[2]:= Attributes[MCurl]
Out[2]= {Protected}

In[3]:= MCurl[{y, -x}, {x, y}]
Out[3]= -2

In[4]:= v[x_, y_] := {Cos[x] Sin[y], Cos[y] Sin[x]}
Out[4]= None

In[5]:= MCurl[v[x, y], {x, y}]
Out[5]= 0

In[6]:= MCurl[{y, -x, 2 z}, {x, y, z}]
Out[6]= {0, 0, -2}

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