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("the anticommutative algebra's operations on indexed objects")$
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("the operations are allowed only on the covariant tensors")$
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("The exterior product is denoted by &. Take it on two 1-forms
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("The exterior product of the three 1-forms
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a([i])&b([j])&c([k])")$
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show(a([i])&b([j])&c([k]))$
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/* Copyright (C) 2003 Valerij Pipin <pip@iszf.irk.ru>
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* This program is free software; you can redistribute it and/or
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* modify it under the terms of the GNU General Public License as
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* published by the Free Software Foundation; either version 2 of
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* the License, or (at your option) any later version.
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* This program is distributed in the hope that it will be
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* useful, but WITHOUT ANY WARRANTY; without even the implied
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* warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR
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* PURPOSE. See the GNU General Public License for more details.
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* Demonstrating exterior algebra operations
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if get('itensor,'version)=false then load(itensor);
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("Exterior algebra for fully antisymmetric covariant tensors")$
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("The exterior product is denoted by ~. Apply it to two 1-forms")$
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("The exterior product of the three 1-forms")$
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ishow(a([i])~b([j])~c([k]))$
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("Take a sum of two 3 forms")$
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show( factor(a([i])&(bk*b([j]))&c([k]) +(ak*a([i]))&c([j])&b([k])))$
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ishow( factor(a([i])~(bk*b([j]))~c([k]) +(ak*a([i]))~c([j])~b([k])))$
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("declare 2-form (don't forget to put allsym:false) and take & with 1-form")$
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("declare a 2-form and compute its exterior product with a 1-form")$
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decsym(p,2,0,[anti(all)],[]);
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show(q([i])&p([j,k]))$
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("the exterior (anticommutative) derivative
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is denoted by @ind, where ind denotes the component")$
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("So actually d_k & form means in our notation form @ k, e.g.")$
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show((a([j])&b([k]))@i)$
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show((a([j])&b([k]))@k)$
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("the interior product is denoted by |_, say form |_ vector, where
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form is indexed tensor and vector is vector's name")$
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show((a([i])&b([j]))|_a)$
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("to avoid the mistake please use the literally sorted indices
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when you apply the |_ ")$
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show(factor((a([i2])&b([i1]))|_a+ (a([i1])&b([i2]))|_a))$
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("the Lie derivative is denoted by @L[vector,ind], (say form @L[vector,ind])
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where vector is vector name and ind is the index of the component")$
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("it is currently not applicable to functions.
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Also you have to use the literally sorted indices")$
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show(a([i1])@L[v,i2])$
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("tests the consequence of the
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Cartan identity for 1-forms. The Lie derivative
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has to commute with the exterior one ")$
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show((a([i1])@i2)@L[v,i3]-(a([i1])@L[v,i2])@i3)$
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("where the (a([i1])@i2)@L[v,i3] is")$
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show((a([i1])@i2)@L[v,i3])$
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("the verifications of this consequence of the Cartan identity
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for the higher order forms are in car_iden.dem")$
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ishow(q([i])~p([j,k]))$
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("the exterior derivative by index k is denoted by extdiff(x,k)")$
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ishow(extdiff(p([j,k]),i))$
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("In other words, instead of 'd_k~f' we write extdiff(f,k)")$
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ishow(extdiff((a([j])~b([k])),i))$
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ishow(extdiff((a([j])~b([k])),k))$
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("Contraction (interior product) with a vector v is denoted by f|v")$
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ishow((a([i])~b([j]))|a)$
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("This operator always uses the first index in literal order")$
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ishow(factor((a([i2])~b([i1]))|a+(a([i1])~b([i2]))|a))$
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("The Lie derivative is liediff(v,x); v is a vector name, x a tensor")$
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ishow(liediff(v,a([i1])))$
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("The Lie-derivative works on all tensors (but not on functions)")$
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ishow(liediff(v,a([i,j],[k,l],m)))$
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("To see all these at work, we verify the Cartan identity for 1-forms")$
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ishow(canform(liediff(v,extdiff(a([i1]),i2))-extdiff(liediff(v,a([i1])),i2)))$
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("Where liediff(v,extdiff(a([i1]),i2)) evaluates to:")$
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ishow(liediff(v,extdiff(a([i1]),i2)))$
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("Cartan identities for higher-order forms are shown in car_iden.dem")$
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/* End of demo -- comment line needed by MAXIMA to resume demo menu */
added
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share/tensor/ex_calc.dem
