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What is a codifferential?

What is a codifferential? Here are some definitions.

Noun
  1. (mathematics) The projected differential of an extensor field.
  2. (differential geometry) the formal adjoint of the exterior derivative; a differential-geometric version of the divergence operator; the exterior derivative sandwiched between two Hodge star operators with some additional factor(s) that take(s) care of the sign; the Hermitian conjugate of the exterior derivative under the inner product for k-form fields over some manifold M:
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