bothcases:false;
("the anticommutative algebra's operations on indexed objects")$
("the operations are allowed only on the covariant tensors")$
load("ex_calc.mc");

("The exterior product is denoted by &. Take it on two 1-forms
  a([i])&b([j])")$

 show(a([i])&b([j]))$

("The exterior product of the three  1-forms
  a([i])&b([j])&c([k])")$

show(a([i])&b([j])&c([k]))$

("Take a sum of two 3 forms")$
show( factor(a([i])&(bk*b([j]))&c([k]) +(ak*a([i]))&c([j])&b([k])))$

("declare 2-form (don't forget to put allsym:false) and take & with 1-form")$
allsym:false;
decsym(p,2,0,[anti(all)],[]);

show(q([i])&p([j,k]))$
("the exterior (anticommutative) derivative
  is denoted by @ind, where ind denotes the component")$

("So actually d_k & form means in our notation form @ k, e.g.")$

show(p([j,k])@i)$
show((a([j])&b([k]))@i)$
show((a([j])&b([k]))@k)$

("the interior product is denoted by |_, say form |_ vector, where
  form is indexed tensor and vector is vector's name")$
show(a([i])|_a)$

show((a([i])&b([j]))|_a)$
("to avoid the mistake please use the literaly sorted indices
  when you apply the |_ ")$

show(factor((a([i2])&b([i1]))|_a+ (a([i1])&b([i2]))|_a))$

("the Lie derivative is denoted by @L[vector,ind],( say form @L[vector,ind])
  where  vector is vector name and ind is the index of component")$
("it is currently not applicable to functions.
  Also you have to use the literaly sorted indices")$

show(a([i1])@L[v,i2])$
("tests the consequence of the
  Cartan identity for 1-forms.The Lie derivativies
  has to commute  with the exterior one ")$

show((a([i1])@i2)@L[v,i3]-(a([i1])@L[v,i2])@i3)$

("where the (a([i1])@i2)@L[v,i3] is")$
show((a([i1])@i2)@L[v,i3])$

("the verifications of this consequense of the Cartan identity
  for the higher order forms are in car_iden.mc")$