petsc-3.15.0 2021-04-05
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# PetscDTAltVPullback

Compute the pullback of a k-form under a linear transformation of the coordinate space

### Synopsis

```#include "petscdt.h"
PetscErrorCode PetscDTAltVPullback(PetscInt N, PetscInt M, const PetscReal *L, PetscInt k, const PetscReal *w, PetscReal *Lstarw)
```

### Input Arguments

 N - the dimension of the origin vector space of the linear transformation, M >= 0 M - the dimension of the image vector space of the linear transformation, N >= 0 L - a linear transformation, an [M x N] matrix in row-major format k - the *signed* degree k of the |k|-form w, -(min(M,N)) <= k <= min(M,N). A negative form degree indicates that the pullback should be conjugated by the Hodge star operator (see note). w - a |k|-form in the image space, size [M choose |k|]

### Output Arguments

 Lstarw - the pullback of w to a |k|-form in the origin space, size [N choose |k|]: (Lstarw)(v_1,...v_k) = w(L*v_1,...,L*v_k).

Note: negative form degrees accomodate, e.g., H-div conforming vector fields. An H-div conforming vector field stores its degrees of freedom as (dx, dy, dz), like a 1-form, but its normal trace is integrated on faces, like a 2-form. The correct pullback then is to apply the Hodge star transformation from (M-2)-form to 2-form, pullback as a 2-form, then the inverse Hodge star transformation.