Atlas Paclet Symbol

Dual - dual operator

Dual[expr]
allows one to calculate the dual operator on an expression (just rise and lower indexes in local coordinates)

MORE INFORMATION

expr - any tensor expression.

The operator has the following properties:

- The Dual operator is linear.

- For any vector field X in local coordinates we have: Dual(X)_j=Xⁱg_ij

- For any 1-form in local coordinates we have: Dual()ⁱ=_jg^ji

- For any tensor fields and the following rule for dual operator takes place: Dual(T)=(Dual())(Dual(T))

Declare constants:

Declare functions:

Declare p-forms:

Declare vectors:

Declare coframe:

Declare frame:

Declare metric:

Using Dual- procedure: For frame vectors:

For coframe 1-forms:

For exact 1-form:

With ToBasis procedure:

For 1-form:

For [1,1] tensor:

For metric tensor:

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Dual - dual operator

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