d - exterior differentiation

d[expr]
allows one to calculate the exterior derivative on an expression that is p-form.
  • expr - any expression.
  • The exterior derivative is the operator d : p -> p+1 where p is p-form and p+1 is (p+1)-form. The operator has the following properties.
  • For any 0-form f=f(x1, x2, ..xn) we have: d[f]=
  • For any p-forms , and constants , we have: d[ + ]=d[]+d[]
  • If 1 is p-form and 2 is q-form then exterior derivative of their exterior product is as follows: d[[1][2]]=[d[1]][2]+(-1)p[1][d[2]]
  • For any p-form Poincare's lemma takes place: d[d[]] = 0
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Declare constants:
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Declare functions:
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Declare p-forms:
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Using d- procedure:
f - declared as a function with indefinite number of variables:
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F - declared as function on three variables:
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G - declared as function G=G[x,z] and z=z[phi] thus:
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As declared as constant thus:
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Exterior derivative is linear operation:
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There were not any declarations about x and y so they are 0-forms by defaults:
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Verify that double-d gives 0 (Poincare's lemma):
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As y is 0-form (by defaults) and has been declared as P-form then:
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1 and 2 declared as p- and q-forms respectively:
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