1\begin{patch}{DeRhamComplexXmpPagePatch1}
2\begin{paste}{DeRhamComplexXmpPageFull1}{DeRhamComplexXmpPageEmpty1}
3\pastebutton{DeRhamComplexXmpPageFull1}{\hidepaste}
4\tab{5}\spadcommand{coefRing := Integer\bound{coefRing }}
5\indentrel{3}\begin{verbatim}
6   (1)  Integer
7                                             Type: Type
8\end{verbatim}
9\indentrel{-3}\end{paste}\end{patch}
10
11\begin{patch}{DeRhamComplexXmpPageEmpty1}
12\begin{paste}{DeRhamComplexXmpPageEmpty1}{DeRhamComplexXmpPagePatch1}
13\pastebutton{DeRhamComplexXmpPageEmpty1}{\showpaste}
14\tab{5}\spadcommand{coefRing := Integer\bound{coefRing }}
15\end{paste}\end{patch}
16
17\begin{patch}{DeRhamComplexXmpPagePatch2}
18\begin{paste}{DeRhamComplexXmpPageFull2}{DeRhamComplexXmpPageEmpty2}
19\pastebutton{DeRhamComplexXmpPageFull2}{\hidepaste}
20\tab{5}\spadcommand{lv : List Symbol := [x,y,z]\bound{lv }}
21\indentrel{3}\begin{verbatim}
22   (2)  [x, y, z]
23                                     Type: List(Symbol)
24\end{verbatim}
25\indentrel{-3}\end{paste}\end{patch}
26
27\begin{patch}{DeRhamComplexXmpPageEmpty2}
28\begin{paste}{DeRhamComplexXmpPageEmpty2}{DeRhamComplexXmpPagePatch2}
29\pastebutton{DeRhamComplexXmpPageEmpty2}{\showpaste}
30\tab{5}\spadcommand{lv : List Symbol := [x,y,z]\bound{lv }}
31\end{paste}\end{patch}
32
33\begin{patch}{DeRhamComplexXmpPagePatch3}
34\begin{paste}{DeRhamComplexXmpPageFull3}{DeRhamComplexXmpPageEmpty3}
35\pastebutton{DeRhamComplexXmpPageFull3}{\hidepaste}
36\tab{5}\spadcommand{der := DERHAM(coefRing,lv)\free{coefRing }\free{lv }\bound{der }}
37\indentrel{3}\begin{verbatim}
38   (3)  DeRhamComplex(Integer,[x,y,z])
39                                             Type: Type
40\end{verbatim}
41\indentrel{-3}\end{paste}\end{patch}
42
43\begin{patch}{DeRhamComplexXmpPageEmpty3}
44\begin{paste}{DeRhamComplexXmpPageEmpty3}{DeRhamComplexXmpPagePatch3}
45\pastebutton{DeRhamComplexXmpPageEmpty3}{\showpaste}
46\tab{5}\spadcommand{der := DERHAM(coefRing,lv)\free{coefRing }\free{lv }\bound{der }}
47\end{paste}\end{patch}
48
49\begin{patch}{DeRhamComplexXmpPagePatch4}
50\begin{paste}{DeRhamComplexXmpPageFull4}{DeRhamComplexXmpPageEmpty4}
51\pastebutton{DeRhamComplexXmpPageFull4}{\hidepaste}
52\tab{5}\spadcommand{R := Expression coefRing\free{coefRing }\bound{R }}
53\indentrel{3}\begin{verbatim}
54   (4)  Expression(Integer)
55                                             Type: Type
56\end{verbatim}
57\indentrel{-3}\end{paste}\end{patch}
58
59\begin{patch}{DeRhamComplexXmpPageEmpty4}
60\begin{paste}{DeRhamComplexXmpPageEmpty4}{DeRhamComplexXmpPagePatch4}
61\pastebutton{DeRhamComplexXmpPageEmpty4}{\showpaste}
62\tab{5}\spadcommand{R := Expression coefRing\free{coefRing }\bound{R }}
63\end{paste}\end{patch}
64
65\begin{patch}{DeRhamComplexXmpPagePatch5}
66\begin{paste}{DeRhamComplexXmpPageFull5}{DeRhamComplexXmpPageEmpty5}
67\pastebutton{DeRhamComplexXmpPageFull5}{\hidepaste}
68\tab{5}\spadcommand{f : R := x^2*y*z-5*x^3*y^2*z^5\free{R }\bound{f }}
69\indentrel{3}\begin{verbatim}
70             3 2 5    2
71   (5)  - 5 x y z  + x y z
72                              Type: Expression(Integer)
73\end{verbatim}
74\indentrel{-3}\end{paste}\end{patch}
75
76\begin{patch}{DeRhamComplexXmpPageEmpty5}
77\begin{paste}{DeRhamComplexXmpPageEmpty5}{DeRhamComplexXmpPagePatch5}
78\pastebutton{DeRhamComplexXmpPageEmpty5}{\showpaste}
79\tab{5}\spadcommand{f : R := x^2*y*z-5*x^3*y^2*z^5\free{R }\bound{f }}
80\end{paste}\end{patch}
81
82\begin{patch}{DeRhamComplexXmpPagePatch6}
83\begin{paste}{DeRhamComplexXmpPageFull6}{DeRhamComplexXmpPageEmpty6}
84\pastebutton{DeRhamComplexXmpPageFull6}{\hidepaste}
85\tab{5}\spadcommand{g : R := z^2*y*cos(z)-7*sin(x^3*y^2)*z^2\free{R }\bound{g }}
86\indentrel{3}\begin{verbatim}
87             2     3 2       2
88   (6)  - 7 z sin(x y ) + y z cos(z)
89                              Type: Expression(Integer)
90\end{verbatim}
91\indentrel{-3}\end{paste}\end{patch}
92
93\begin{patch}{DeRhamComplexXmpPageEmpty6}
94\begin{paste}{DeRhamComplexXmpPageEmpty6}{DeRhamComplexXmpPagePatch6}
95\pastebutton{DeRhamComplexXmpPageEmpty6}{\showpaste}
96\tab{5}\spadcommand{g : R := z^2*y*cos(z)-7*sin(x^3*y^2)*z^2\free{R }\bound{g }}
97\end{paste}\end{patch}
98
99\begin{patch}{DeRhamComplexXmpPagePatch7}
100\begin{paste}{DeRhamComplexXmpPageFull7}{DeRhamComplexXmpPageEmpty7}
101\pastebutton{DeRhamComplexXmpPageFull7}{\hidepaste}
102\tab{5}\spadcommand{h : R :=x*y*z-2*x^3*y*z^2\free{R }\bound{h }}
103\indentrel{3}\begin{verbatim}
104             3   2
105   (7)  - 2 x y z  + x y z
106                              Type: Expression(Integer)
107\end{verbatim}
108\indentrel{-3}\end{paste}\end{patch}
109
110\begin{patch}{DeRhamComplexXmpPageEmpty7}
111\begin{paste}{DeRhamComplexXmpPageEmpty7}{DeRhamComplexXmpPagePatch7}
112\pastebutton{DeRhamComplexXmpPageEmpty7}{\showpaste}
113\tab{5}\spadcommand{h : R :=x*y*z-2*x^3*y*z^2\free{R }\bound{h }}
114\end{paste}\end{patch}
115
116\begin{patch}{DeRhamComplexXmpPagePatch8}
117\begin{paste}{DeRhamComplexXmpPageFull8}{DeRhamComplexXmpPageEmpty8}
118\pastebutton{DeRhamComplexXmpPageFull8}{\hidepaste}
119\tab{5}\spadcommand{dx : der := generator(1)\free{der }\bound{dx }}
120\indentrel{3}\begin{verbatim}
121   (8)  dx
122                   Type: DeRhamComplex(Integer,[x,y,z])
123\end{verbatim}
124\indentrel{-3}\end{paste}\end{patch}
125
126\begin{patch}{DeRhamComplexXmpPageEmpty8}
127\begin{paste}{DeRhamComplexXmpPageEmpty8}{DeRhamComplexXmpPagePatch8}
128\pastebutton{DeRhamComplexXmpPageEmpty8}{\showpaste}
129\tab{5}\spadcommand{dx : der := generator(1)\free{der }\bound{dx }}
130\end{paste}\end{patch}
131
132\begin{patch}{DeRhamComplexXmpPagePatch9}
133\begin{paste}{DeRhamComplexXmpPageFull9}{DeRhamComplexXmpPageEmpty9}
134\pastebutton{DeRhamComplexXmpPageFull9}{\hidepaste}
135\tab{5}\spadcommand{dy : der := generator(2)\free{der }\bound{dy }}
136\indentrel{3}\begin{verbatim}
137   (9)  dy
138                   Type: DeRhamComplex(Integer,[x,y,z])
139\end{verbatim}
140\indentrel{-3}\end{paste}\end{patch}
141
142\begin{patch}{DeRhamComplexXmpPageEmpty9}
143\begin{paste}{DeRhamComplexXmpPageEmpty9}{DeRhamComplexXmpPagePatch9}
144\pastebutton{DeRhamComplexXmpPageEmpty9}{\showpaste}
145\tab{5}\spadcommand{dy : der := generator(2)\free{der }\bound{dy }}
146\end{paste}\end{patch}
147
148\begin{patch}{DeRhamComplexXmpPagePatch10}
149\begin{paste}{DeRhamComplexXmpPageFull10}{DeRhamComplexXmpPageEmpty10}
150\pastebutton{DeRhamComplexXmpPageFull10}{\hidepaste}
151\tab{5}\spadcommand{dz : der := generator(3)\free{der }\bound{dz }}
152\indentrel{3}\begin{verbatim}
153   (10)  dz
154                   Type: DeRhamComplex(Integer,[x,y,z])
155\end{verbatim}
156\indentrel{-3}\end{paste}\end{patch}
157
158\begin{patch}{DeRhamComplexXmpPageEmpty10}
159\begin{paste}{DeRhamComplexXmpPageEmpty10}{DeRhamComplexXmpPagePatch10}
160\pastebutton{DeRhamComplexXmpPageEmpty10}{\showpaste}
161\tab{5}\spadcommand{dz : der := generator(3)\free{der }\bound{dz }}
162\end{paste}\end{patch}
163
164\begin{patch}{DeRhamComplexXmpPagePatch11}
165\begin{paste}{DeRhamComplexXmpPageFull11}{DeRhamComplexXmpPageEmpty11}
166\pastebutton{DeRhamComplexXmpPageFull11}{\hidepaste}
167\tab{5}\spadcommand{[dx,dy,dz] := [generator(i)$der for i in 1..3]\free{der }\bound{dxyz }}
168\indentrel{3}\begin{verbatim}
169   (11)  [dx, dy, dz]
170             Type: List(DeRhamComplex(Integer,[x,y,z]))
171\end{verbatim}
172\indentrel{-3}\end{paste}\end{patch}
173
174\begin{patch}{DeRhamComplexXmpPageEmpty11}
175\begin{paste}{DeRhamComplexXmpPageEmpty11}{DeRhamComplexXmpPagePatch11}
176\pastebutton{DeRhamComplexXmpPageEmpty11}{\showpaste}
177\tab{5}\spadcommand{[dx,dy,dz] := [generator(i)$der for i in 1..3]\free{der }\bound{dxyz }}
178\end{paste}\end{patch}
179
180\begin{patch}{DeRhamComplexXmpPagePatch12}
181\begin{paste}{DeRhamComplexXmpPageFull12}{DeRhamComplexXmpPageEmpty12}
182\pastebutton{DeRhamComplexXmpPageFull12}{\hidepaste}
183\tab{5}\spadcommand{alpha : der := f*dx + g*dy + h*dz\bound{alpha }\free{der f g h dxyz }}
184\indentrel{3}\begin{verbatim}
185   (12)
186           3   2
187     (- 2 x y z  + x y z)dz
188   +
189           2     3 2       2
190     (- 7 z sin(x y ) + y z cos(z))dy
191   +
192           3 2 5    2
193     (- 5 x y z  + x y z)dx
194                   Type: DeRhamComplex(Integer,[x,y,z])
195\end{verbatim}
196\indentrel{-3}\end{paste}\end{patch}
197
198\begin{patch}{DeRhamComplexXmpPageEmpty12}
199\begin{paste}{DeRhamComplexXmpPageEmpty12}{DeRhamComplexXmpPagePatch12}
200\pastebutton{DeRhamComplexXmpPageEmpty12}{\showpaste}
201\tab{5}\spadcommand{alpha : der := f*dx + g*dy + h*dz\bound{alpha }\free{der f g h dxyz }}
202\end{paste}\end{patch}
203
204\begin{patch}{DeRhamComplexXmpPagePatch13}
205\begin{paste}{DeRhamComplexXmpPageFull13}{DeRhamComplexXmpPageEmpty13}
206\pastebutton{DeRhamComplexXmpPageFull13}{\hidepaste}
207\tab{5}\spadcommand{beta : der := cos(tan(x*y*z)+x*y*z)*dx + x*dy\bound{beta }\free{der f g h dxyz }}
208\indentrel{3}\begin{verbatim}
209   (13)  x dy + cos(tan(x y z) + x y z)dx
210                   Type: DeRhamComplex(Integer,[x,y,z])
211\end{verbatim}
212\indentrel{-3}\end{paste}\end{patch}
213
214\begin{patch}{DeRhamComplexXmpPageEmpty13}
215\begin{paste}{DeRhamComplexXmpPageEmpty13}{DeRhamComplexXmpPagePatch13}
216\pastebutton{DeRhamComplexXmpPageEmpty13}{\showpaste}
217\tab{5}\spadcommand{beta : der := cos(tan(x*y*z)+x*y*z)*dx + x*dy\bound{beta }\free{der f g h dxyz }}
218\end{paste}\end{patch}
219
220\begin{patch}{DeRhamComplexXmpPagePatch14}
221\begin{paste}{DeRhamComplexXmpPageFull14}{DeRhamComplexXmpPageEmpty14}
222\pastebutton{DeRhamComplexXmpPageFull14}{\hidepaste}
223\tab{5}\spadcommand{exteriorDifferential alpha;\free{alpha }\bound{ed }}
224\indentrel{3}\begin{verbatim}
225                   Type: DeRhamComplex(Integer,[x,y,z])
226\end{verbatim}
227\indentrel{-3}\end{paste}\end{patch}
228
229\begin{patch}{DeRhamComplexXmpPageEmpty14}
230\begin{paste}{DeRhamComplexXmpPageEmpty14}{DeRhamComplexXmpPagePatch14}
231\pastebutton{DeRhamComplexXmpPageEmpty14}{\showpaste}
232\tab{5}\spadcommand{exteriorDifferential alpha;\free{alpha }\bound{ed }}
233\end{paste}\end{patch}
234
235\begin{patch}{DeRhamComplexXmpPagePatch15}
236\begin{paste}{DeRhamComplexXmpPageFull15}{DeRhamComplexXmpPageEmpty15}
237\pastebutton{DeRhamComplexXmpPageFull15}{\hidepaste}
238\tab{5}\spadcommand{exteriorDifferential \%\free{ed }}
239\indentrel{3}\begin{verbatim}
240   (15)  0
241                   Type: DeRhamComplex(Integer,[x,y,z])
242\end{verbatim}
243\indentrel{-3}\end{paste}\end{patch}
244
245\begin{patch}{DeRhamComplexXmpPageEmpty15}
246\begin{paste}{DeRhamComplexXmpPageEmpty15}{DeRhamComplexXmpPagePatch15}
247\pastebutton{DeRhamComplexXmpPageEmpty15}{\showpaste}
248\tab{5}\spadcommand{exteriorDifferential \%\free{ed }}
249\end{paste}\end{patch}
250
251\begin{patch}{DeRhamComplexXmpPagePatch16}
252\begin{paste}{DeRhamComplexXmpPageFull16}{DeRhamComplexXmpPageEmpty16}
253\pastebutton{DeRhamComplexXmpPageFull16}{\hidepaste}
254\tab{5}\spadcommand{gamma := alpha * beta\bound{gamma }\free{alpha }\free{beta }}
255\indentrel{3}\begin{verbatim}
256   (16)
257         4   2    2
258     (2 x y z  - x y z)dy dz
259   +
260         3   2
261     (2 x y z  - x y z)cos(tan(x y z) + x y z)dx dz
262   +
263               2     3 2       2
264           (7 z sin(x y ) - y z cos(z))
265        *
266           cos(tan(x y z) + x y z)
267       +
268              4 2 5    3
269         - 5 x y z  + x y z
270    *
271       dx dy
272                   Type: DeRhamComplex(Integer,[x,y,z])
273\end{verbatim}
274\indentrel{-3}\end{paste}\end{patch}
275
276\begin{patch}{DeRhamComplexXmpPageEmpty16}
277\begin{paste}{DeRhamComplexXmpPageEmpty16}{DeRhamComplexXmpPagePatch16}
278\pastebutton{DeRhamComplexXmpPageEmpty16}{\showpaste}
279\tab{5}\spadcommand{gamma := alpha * beta\bound{gamma }\free{alpha }\free{beta }}
280\end{paste}\end{patch}
281
282\begin{patch}{DeRhamComplexXmpPagePatch17}
283\begin{paste}{DeRhamComplexXmpPageFull17}{DeRhamComplexXmpPageEmpty17}
284\pastebutton{DeRhamComplexXmpPageFull17}{\hidepaste}
285\tab{5}\spadcommand{exteriorDifferential(gamma) - (exteriorDifferential(alpha)*beta - alpha * exteriorDifferential(beta))\free{alpha beta gamma }}
286\indentrel{3}\begin{verbatim}
287   (17)  0
288                   Type: DeRhamComplex(Integer,[x,y,z])
289\end{verbatim}
290\indentrel{-3}\end{paste}\end{patch}
291
292\begin{patch}{DeRhamComplexXmpPageEmpty17}
293\begin{paste}{DeRhamComplexXmpPageEmpty17}{DeRhamComplexXmpPagePatch17}
294\pastebutton{DeRhamComplexXmpPageEmpty17}{\showpaste}
295\tab{5}\spadcommand{exteriorDifferential(gamma) - (exteriorDifferential(alpha)*beta - alpha * exteriorDifferential(beta))\free{alpha beta gamma }}
296\end{paste}\end{patch}
297
298\begin{patch}{DeRhamComplexXmpPagePatch18}
299\begin{paste}{DeRhamComplexXmpPageFull18}{DeRhamComplexXmpPageEmpty18}
300\pastebutton{DeRhamComplexXmpPageFull18}{\hidepaste}
301\tab{5}\spadcommand{a : BOP := operator('a)\bound{ao }}
302\indentrel{3}\begin{verbatim}
303   (18)  a
304                                    Type: BasicOperator
305\end{verbatim}
306\indentrel{-3}\end{paste}\end{patch}
307
308\begin{patch}{DeRhamComplexXmpPageEmpty18}
309\begin{paste}{DeRhamComplexXmpPageEmpty18}{DeRhamComplexXmpPagePatch18}
310\pastebutton{DeRhamComplexXmpPageEmpty18}{\showpaste}
311\tab{5}\spadcommand{a : BOP := operator('a)\bound{ao }}
312\end{paste}\end{patch}
313
314\begin{patch}{DeRhamComplexXmpPagePatch19}
315\begin{paste}{DeRhamComplexXmpPageFull19}{DeRhamComplexXmpPageEmpty19}
316\pastebutton{DeRhamComplexXmpPageFull19}{\hidepaste}
317\tab{5}\spadcommand{b : BOP := operator('b)\bound{bo }}
318\indentrel{3}\begin{verbatim}
319   (19)  b
320                                    Type: BasicOperator
321\end{verbatim}
322\indentrel{-3}\end{paste}\end{patch}
323
324\begin{patch}{DeRhamComplexXmpPageEmpty19}
325\begin{paste}{DeRhamComplexXmpPageEmpty19}{DeRhamComplexXmpPagePatch19}
326\pastebutton{DeRhamComplexXmpPageEmpty19}{\showpaste}
327\tab{5}\spadcommand{b : BOP := operator('b)\bound{bo }}
328\end{paste}\end{patch}
329
330\begin{patch}{DeRhamComplexXmpPagePatch20}
331\begin{paste}{DeRhamComplexXmpPageFull20}{DeRhamComplexXmpPageEmpty20}
332\pastebutton{DeRhamComplexXmpPageFull20}{\hidepaste}
333\tab{5}\spadcommand{c : BOP := operator('c)\bound{co }}
334\indentrel{3}\begin{verbatim}
335   (20)  c
336                                    Type: BasicOperator
337\end{verbatim}
338\indentrel{-3}\end{paste}\end{patch}
339
340\begin{patch}{DeRhamComplexXmpPageEmpty20}
341\begin{paste}{DeRhamComplexXmpPageEmpty20}{DeRhamComplexXmpPagePatch20}
342\pastebutton{DeRhamComplexXmpPageEmpty20}{\showpaste}
343\tab{5}\spadcommand{c : BOP := operator('c)\bound{co }}
344\end{paste}\end{patch}
345
346\begin{patch}{DeRhamComplexXmpPagePatch21}
347\begin{paste}{DeRhamComplexXmpPageFull21}{DeRhamComplexXmpPageEmpty21}
348\pastebutton{DeRhamComplexXmpPageFull21}{\hidepaste}
349\tab{5}\spadcommand{sigma := a(x,y,z) * dx + b(x,y,z) * dy + c(x,y,z) * dz\bound{sigma }\free{ao bo co dxyz }}
350\indentrel{3}\begin{verbatim}
351   (21)  c(x,y,z)dz + b(x,y,z)dy + a(x,y,z)dx
352                   Type: DeRhamComplex(Integer,[x,y,z])
353\end{verbatim}
354\indentrel{-3}\end{paste}\end{patch}
355
356\begin{patch}{DeRhamComplexXmpPageEmpty21}
357\begin{paste}{DeRhamComplexXmpPageEmpty21}{DeRhamComplexXmpPagePatch21}
358\pastebutton{DeRhamComplexXmpPageEmpty21}{\showpaste}
359\tab{5}\spadcommand{sigma := a(x,y,z) * dx + b(x,y,z) * dy + c(x,y,z) * dz\bound{sigma }\free{ao bo co dxyz }}
360\end{paste}\end{patch}
361
362\begin{patch}{DeRhamComplexXmpPagePatch22}
363\begin{paste}{DeRhamComplexXmpPageFull22}{DeRhamComplexXmpPageEmpty22}
364\pastebutton{DeRhamComplexXmpPageFull22}{\hidepaste}
365\tab{5}\spadcommand{theta := a(x,y,z) * dx * dy + b(x,y,z) * dx * dz + c(x,y,z) * dy * dz\bound{theta }\free{ao bo co dxyz }}
366\indentrel{3}\begin{verbatim}
367   (22)  c(x,y,z)dy dz + b(x,y,z)dx dz + a(x,y,z)dx dy
368                   Type: DeRhamComplex(Integer,[x,y,z])
369\end{verbatim}
370\indentrel{-3}\end{paste}\end{patch}
371
372\begin{patch}{DeRhamComplexXmpPageEmpty22}
373\begin{paste}{DeRhamComplexXmpPageEmpty22}{DeRhamComplexXmpPagePatch22}
374\pastebutton{DeRhamComplexXmpPageEmpty22}{\showpaste}
375\tab{5}\spadcommand{theta := a(x,y,z) * dx * dy + b(x,y,z) * dx * dz + c(x,y,z) * dy * dz\bound{theta }\free{ao bo co dxyz }}
376\end{paste}\end{patch}
377
378\begin{patch}{DeRhamComplexXmpPagePatch23}
379\begin{paste}{DeRhamComplexXmpPageFull23}{DeRhamComplexXmpPageEmpty23}
380\pastebutton{DeRhamComplexXmpPageFull23}{\hidepaste}
381\tab{5}\spadcommand{totalDifferential(a(x,y,z))$der\free{ao der }}
382\indentrel{3}\begin{verbatim}
383   (23)  a  (x,y,z)dz + a  (x,y,z)dy + a  (x,y,z)dx
384          ,3             ,2             ,1
385                   Type: DeRhamComplex(Integer,[x,y,z])
386\end{verbatim}
387\indentrel{-3}\end{paste}\end{patch}
388
389\begin{patch}{DeRhamComplexXmpPageEmpty23}
390\begin{paste}{DeRhamComplexXmpPageEmpty23}{DeRhamComplexXmpPagePatch23}
391\pastebutton{DeRhamComplexXmpPageEmpty23}{\showpaste}
392\tab{5}\spadcommand{totalDifferential(a(x,y,z))$der\free{ao der }}
393\end{paste}\end{patch}
394
395\begin{patch}{DeRhamComplexXmpPagePatch24}
396\begin{paste}{DeRhamComplexXmpPageFull24}{DeRhamComplexXmpPageEmpty24}
397\pastebutton{DeRhamComplexXmpPageFull24}{\hidepaste}
398\tab{5}\spadcommand{exteriorDifferential sigma\free{sigma }}
399\indentrel{3}\begin{verbatim}
400   (24)
401     (c  (x,y,z) - b  (x,y,z))dy dz
402       ,2           ,3
403   +
404     (c  (x,y,z) - a  (x,y,z))dx dz
405       ,1           ,3
406   +
407     (b  (x,y,z) - a  (x,y,z))dx dy
408       ,1           ,2
409                   Type: DeRhamComplex(Integer,[x,y,z])
410\end{verbatim}
411\indentrel{-3}\end{paste}\end{patch}
412
413\begin{patch}{DeRhamComplexXmpPageEmpty24}
414\begin{paste}{DeRhamComplexXmpPageEmpty24}{DeRhamComplexXmpPagePatch24}
415\pastebutton{DeRhamComplexXmpPageEmpty24}{\showpaste}
416\tab{5}\spadcommand{exteriorDifferential sigma\free{sigma }}
417\end{paste}\end{patch}
418
419\begin{patch}{DeRhamComplexXmpPagePatch25}
420\begin{paste}{DeRhamComplexXmpPageFull25}{DeRhamComplexXmpPageEmpty25}
421\pastebutton{DeRhamComplexXmpPageFull25}{\hidepaste}
422\tab{5}\spadcommand{exteriorDifferential theta\free{theta }}
423\indentrel{3}\begin{verbatim}
424   (25)  (c  (x,y,z) - b  (x,y,z) + a  (x,y,z))dx dy dz
425           ,1           ,2           ,3
426                   Type: DeRhamComplex(Integer,[x,y,z])
427\end{verbatim}
428\indentrel{-3}\end{paste}\end{patch}
429
430\begin{patch}{DeRhamComplexXmpPageEmpty25}
431\begin{paste}{DeRhamComplexXmpPageEmpty25}{DeRhamComplexXmpPagePatch25}
432\pastebutton{DeRhamComplexXmpPageEmpty25}{\showpaste}
433\tab{5}\spadcommand{exteriorDifferential theta\free{theta }}
434\end{paste}\end{patch}
435
436\begin{patch}{DeRhamComplexXmpPagePatch26}
437\begin{paste}{DeRhamComplexXmpPageFull26}{DeRhamComplexXmpPageEmpty26}
438\pastebutton{DeRhamComplexXmpPageFull26}{\hidepaste}
439\tab{5}\spadcommand{one : der := 1\bound{one }\free{der }}
440\indentrel{3}\begin{verbatim}
441   (26)  1
442                   Type: DeRhamComplex(Integer,[x,y,z])
443\end{verbatim}
444\indentrel{-3}\end{paste}\end{patch}
445
446\begin{patch}{DeRhamComplexXmpPageEmpty26}
447\begin{paste}{DeRhamComplexXmpPageEmpty26}{DeRhamComplexXmpPagePatch26}
448\pastebutton{DeRhamComplexXmpPageEmpty26}{\showpaste}
449\tab{5}\spadcommand{one : der := 1\bound{one }\free{der }}
450\end{paste}\end{patch}
451
452\begin{patch}{DeRhamComplexXmpPagePatch27}
453\begin{paste}{DeRhamComplexXmpPageFull27}{DeRhamComplexXmpPageEmpty27}
454\pastebutton{DeRhamComplexXmpPageFull27}{\hidepaste}
455\tab{5}\spadcommand{g1 : der := a([x,t,y,u,v,z,e]) * one\free{der one ao }\bound{g1 }}
456\indentrel{3}\begin{verbatim}
457   (27)  a(x,t,y,u,v,z,e)
458                   Type: DeRhamComplex(Integer,[x,y,z])
459\end{verbatim}
460\indentrel{-3}\end{paste}\end{patch}
461
462\begin{patch}{DeRhamComplexXmpPageEmpty27}
463\begin{paste}{DeRhamComplexXmpPageEmpty27}{DeRhamComplexXmpPagePatch27}
464\pastebutton{DeRhamComplexXmpPageEmpty27}{\showpaste}
465\tab{5}\spadcommand{g1 : der := a([x,t,y,u,v,z,e]) * one\free{der one ao }\bound{g1 }}
466\end{paste}\end{patch}
467
468\begin{patch}{DeRhamComplexXmpPagePatch28}
469\begin{paste}{DeRhamComplexXmpPageFull28}{DeRhamComplexXmpPageEmpty28}
470\pastebutton{DeRhamComplexXmpPageFull28}{\hidepaste}
471\tab{5}\spadcommand{h1 : der := a([x,y,x,t,x,z,y,r,u,x]) * one\free{der one ao }\bound{h1 }}
472\indentrel{3}\begin{verbatim}
473   (28)  a(x,y,x,t,x,z,y,r,u,x)
474                   Type: DeRhamComplex(Integer,[x,y,z])
475\end{verbatim}
476\indentrel{-3}\end{paste}\end{patch}
477
478\begin{patch}{DeRhamComplexXmpPageEmpty28}
479\begin{paste}{DeRhamComplexXmpPageEmpty28}{DeRhamComplexXmpPagePatch28}
480\pastebutton{DeRhamComplexXmpPageEmpty28}{\showpaste}
481\tab{5}\spadcommand{h1 : der := a([x,y,x,t,x,z,y,r,u,x]) * one\free{der one ao }\bound{h1 }}
482\end{paste}\end{patch}
483
484\begin{patch}{DeRhamComplexXmpPagePatch29}
485\begin{paste}{DeRhamComplexXmpPageFull29}{DeRhamComplexXmpPageEmpty29}
486\pastebutton{DeRhamComplexXmpPageFull29}{\hidepaste}
487\tab{5}\spadcommand{exteriorDifferential g1\free{g1 }}
488\indentrel{3}\begin{verbatim}
489   (29)
490     a  (x,t,y,u,v,z,e)dz + a  (x,t,y,u,v,z,e)dy
491      ,6                     ,3
492   +
493     a  (x,t,y,u,v,z,e)dx
494      ,1
495                   Type: DeRhamComplex(Integer,[x,y,z])
496\end{verbatim}
497\indentrel{-3}\end{paste}\end{patch}
498
499\begin{patch}{DeRhamComplexXmpPageEmpty29}
500\begin{paste}{DeRhamComplexXmpPageEmpty29}{DeRhamComplexXmpPagePatch29}
501\pastebutton{DeRhamComplexXmpPageEmpty29}{\showpaste}
502\tab{5}\spadcommand{exteriorDifferential g1\free{g1 }}
503\end{paste}\end{patch}
504
505\begin{patch}{DeRhamComplexXmpPagePatch30}
506\begin{paste}{DeRhamComplexXmpPageFull30}{DeRhamComplexXmpPageEmpty30}
507\pastebutton{DeRhamComplexXmpPageFull30}{\hidepaste}
508\tab{5}\spadcommand{exteriorDifferential h1\free{h1 }}
509\indentrel{3}\begin{verbatim}
510   (30)
511     a  (x,y,x,t,x,z,y,r,u,x)dz
512      ,6
513   +
514           a  (x,y,x,t,x,z,y,r,u,x)
515            ,7
516         +
517           a  (x,y,x,t,x,z,y,r,u,x)
518            ,2
519    *
520       dy
521   +
522         a   (x,y,x,t,x,z,y,r,u,x)
523          ,10
524       +
525         a  (x,y,x,t,x,z,y,r,u,x)
526          ,5
527       +
528       a  (x,y,x,t,x,z,y,r,u,x) + a  (x,y,x,t,x,z,y,r,u,x)
529        ,3                         ,1
530    *
531       dx
532                   Type: DeRhamComplex(Integer,[x,y,z])
533\end{verbatim}
534\indentrel{-3}\end{paste}\end{patch}
535
536\begin{patch}{DeRhamComplexXmpPageEmpty30}
537\begin{paste}{DeRhamComplexXmpPageEmpty30}{DeRhamComplexXmpPagePatch30}
538\pastebutton{DeRhamComplexXmpPageEmpty30}{\showpaste}
539\tab{5}\spadcommand{exteriorDifferential h1\free{h1 }}
540\end{paste}\end{patch}
541
542\begin{patch}{DeRhamComplexXmpPagePatch31}
543\begin{paste}{DeRhamComplexXmpPageFull31}{DeRhamComplexXmpPageEmpty31}
544\pastebutton{DeRhamComplexXmpPageFull31}{\hidepaste}
545\tab{5}\spadcommand{coefficient(gamma, dx*dy)\free{gamma dxyz }}
546\indentrel{3}\begin{verbatim}
547   (31)
548         2     3 2       2
549     (7 z sin(x y ) - y z cos(z))cos(tan(x y z) + x y z)
550   +
551          4 2 5    3
552     - 5 x y z  + x y z
553                              Type: Expression(Integer)
554\end{verbatim}
555\indentrel{-3}\end{paste}\end{patch}
556
557\begin{patch}{DeRhamComplexXmpPageEmpty31}
558\begin{paste}{DeRhamComplexXmpPageEmpty31}{DeRhamComplexXmpPagePatch31}
559\pastebutton{DeRhamComplexXmpPageEmpty31}{\showpaste}
560\tab{5}\spadcommand{coefficient(gamma, dx*dy)\free{gamma dxyz }}
561\end{paste}\end{patch}
562
563\begin{patch}{DeRhamComplexXmpPagePatch32}
564\begin{paste}{DeRhamComplexXmpPageFull32}{DeRhamComplexXmpPageEmpty32}
565\pastebutton{DeRhamComplexXmpPageFull32}{\hidepaste}
566\tab{5}\spadcommand{coefficient(gamma, one)\free{gamma one }}
567\indentrel{3}\begin{verbatim}
568   (32)  0
569                              Type: Expression(Integer)
570\end{verbatim}
571\indentrel{-3}\end{paste}\end{patch}
572
573\begin{patch}{DeRhamComplexXmpPageEmpty32}
574\begin{paste}{DeRhamComplexXmpPageEmpty32}{DeRhamComplexXmpPagePatch32}
575\pastebutton{DeRhamComplexXmpPageEmpty32}{\showpaste}
576\tab{5}\spadcommand{coefficient(gamma, one)\free{gamma one }}
577\end{paste}\end{patch}
578
579\begin{patch}{DeRhamComplexXmpPagePatch33}
580\begin{paste}{DeRhamComplexXmpPageFull33}{DeRhamComplexXmpPageEmpty33}
581\pastebutton{DeRhamComplexXmpPageFull33}{\hidepaste}
582\tab{5}\spadcommand{coefficient(g1,one)\free{g1 one }}
583\indentrel{3}\begin{verbatim}
584   (33)  a(x,t,y,u,v,z,e)
585                              Type: Expression(Integer)
586\end{verbatim}
587\indentrel{-3}\end{paste}\end{patch}
588
589\begin{patch}{DeRhamComplexXmpPageEmpty33}
590\begin{paste}{DeRhamComplexXmpPageEmpty33}{DeRhamComplexXmpPagePatch33}
591\pastebutton{DeRhamComplexXmpPageEmpty33}{\showpaste}
592\tab{5}\spadcommand{coefficient(g1,one)\free{g1 one }}
593\end{paste}\end{patch}
594
595