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