Differential Geometry¶

Chacana supports type-checked differential geometry operators. The type checker validates argument types when a context is provided.

Exterior derivative¶

chacana.parse("d(omega)")            # ExteriorDerivative
chacana.parse("d(A ^ B)")           # Derivative of a wedge product
chacana.parse("d(omega){_a _b}")    # With explicit indices on result

The exterior derivative satisfies the Leibniz rule $d(A \wedge B) = dA \wedge B + (-1)^p A \wedge dB$. This identity is a processor concern, not enforced by the type checker.

Hodge star¶

chacana.parse("star(omega)", context=ctx)
chacana.parse("hodge(omega)", context=ctx)  # Synonym

Requires active metric

The Hodge star operator requires active_metric in the TOML context. Without it, the type checker raises ChacanaTypeError.

Interior product¶

chacana.parse("i(X, omega)", context=ctx)

The type checker validates:

First argument must be a vector field (rank 1, contravariant)
Second argument must be a p-form with p >= 1 (interior product is undefined for 0-forms)

Lie derivative¶

chacana.parse("L(X, g{_a _b})", context=ctx)

The first argument must be a vector field (rank 1, contravariant).

Trace¶

chacana.parse("Tr(T{^a _b})", context=ctx)

The argument must have rank >= 2 (needs at least two indices to contract).

Determinant¶

chacana.parse("det(g{_a _b})", context=ctx)

The argument must be a rank-2 tensor.

Inverse¶

chacana.parse("inv(M)", context=ctx)

The argument must be a rank-2 tensor. Non-rank-2 tensors raise ChacanaTypeError.

Operator head mapping¶

The visitor maps functional operator names to canonical AST heads:

Syntax	AST head
`d(...)`	`ExteriorDerivative`
`L(...)`	`LieDerivative`
`Tr(...)`	`Trace`
`det(...)`	`Determinant`
`inv(...)`	`Inverse`
`star(...)` / `hodge(...)`	`HodgeStar`
`i(...)`	`InteriorProduct`
`foo(...)`	`foo` (passthrough)