iota - interior product operator

iota[v1, v2, ..., vn, expr]
iotav1, v2, ..., vn[expr]
allows one to calculate the interior product of given expression and vector fields.
  • expr - any expression (on which interior product operator is defined).
  • v1, v2, ..., vn - vector fields.
  • The main syntax is X[] where X is a vector and is a p-form.
  • Let X be a vector and be n-form in some k-dimensional manifold then under definition: [X()]i1, i2, ..in-1=Xjj, i1, i2, ..in-1
  • Multiple iota operator defined as follows: X1, X2, ..Xj()=X1(X2(..Xj()))
  • It is well known that iota operator is anti-differentiation for p-forms. Thus if 1 is p-form then: X((1)(2))=(X(1)) (2)+(-1)p(1)(X(1))
  • iota can be entered as \[Iota] or Esc i Esc.
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Declare constants:
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Declare functions:
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Declare p-forms:
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Declare vectors:
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Using - procedure:
Just definition for "long" iota operator:
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As 1 is p-form and 2 is q-form then under main rule for interior product we have:
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It is obvious that (-1)(-1+p)q(2)(X(1))=(X(1))(2) (see Wedge).
Interior product on any 0-form equals zero:
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Interior product is linear with respect to any argument:
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And:
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Iota operator reduces covariance:
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Calculate on 2-form:
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Calculate on p-form:
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Calculate triple interior product on p-form:
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Verify the main rule for interior product:
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But for tensor product we have:
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And then:
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