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<A NAME="PetscDTAltV"><H1>PetscDTAltV</H1></A>
An interface for common operations on k-forms, also known as alternating algebraic forms or alternating k-linear maps. The name of the interface comes from the notation "Alt V" for the algebra of all k-forms acting vectors in the space V, also known as the exterior algebra of V*. A recommended reference for this material is Section 2 "Exterior algebra and exterior calculus" in "Finite element
exterior calculus, homological techniques, and applications", by Arnold, Falk, & Winther (2006, doi:10.1017/S0962492906210018).
<P>
A k-form w (k is called the "form degree" of w) is an alternating k-linear map acting on tuples (v_1, ..., v_k) of
<H3><FONT COLOR="#CC3333">vectors from a vector space V and producing a real number</FONT></H3>
- alternating: swapping any two vectors in a tuple reverses the sign of the result, e.g. w(v_1, v_2, ..., v_k) = -w(v_2, v_1, ..., v_k)
- k-linear: w acts linear in each vector separately, e.g. w(a*v + b*y, v_2, ..., v_k) = a*w(v,v_2,...,v_k) + b*w(y,v_2,...,v_k)
This action is implemented as <A HREF="../DT/PetscDTAltVApply.html#PetscDTAltVApply">PetscDTAltVApply</A>.
<P>
The k-forms on a vector space form a vector space themselves, Alt^k V. The dimension of Alt^k V, if V is N dimensional, is N choose k. (This
shows that for an N dimensional space, only 0 <= k <= N are valid form degrees.)
<H3><FONT COLOR="#CC3333">The standard basis for Alt^k V, used in PetscDTAltV, has one basis k-form for each ordered subset of k coordinates of the N dimensional space</FONT></H3>
For example, if the coordinate directions of a four dimensional space are (t, x, y, z), then there are 4 choose 2 = 6 ordered subsets of two coordinates.
They are, in lexicographic order, (t, x), (t, y), (t, z), (x, y), (x, z) and (y, z). <A HREF="../DT/PetscDTAltV.html#PetscDTAltV">PetscDTAltV</A> also orders the basis of Alt^k V lexicographically
by the associated subsets.
<P>
The unit basis k-form associated with coordinates (c_1, ..., c_k) acts on a set of k vectors (v_1, ..., v_k) by creating a square matrix V where
V[i,j] = v_i[c_j] and taking the determinant of V.
<P>
If j + k <= N, then a j-form f and a k-form g can be multiplied to create a (j+k)-form using the wedge or exterior product, (f wedge g).
This is an anticommutative product, (f wedge g) = -(g wedge f). It is sufficient to describe the wedge product of two basis forms.
<H3><FONT COLOR="#CC3333">Let f be the basis j-form associated with coordinates (f_1,...,f_j) and g be the basis k-form associated with coordinates (g_1,...,g_k)</FONT></H3>
- If there is any coordinate in both sets, then (f wedge g) = 0.
- Otherwise, (f wedge g) is a multiple of the basis (j+k)-form h associated with (f_1,...,f_j,g_1,...,g_k).
- In fact it is equal to either h or -h depending on how (f_1,...,f_j,g_1,...,g_k) compares to the same list of coordinates given in ascending order: if it is an even permutation of that list, then (f wedge g) = h, otherwise (f wedge g) = -h.
The wedge product is implemented for either two inputs (f and g) in <A HREF="../DT/PetscDTAltVWedge.html#PetscDTAltVWedge">PetscDTAltVWedge</A>, or for one (just f, giving a
matrix to multiply against multiple choices of g) in <A HREF="../DT/PetscDTAltVWedgeMatrix.html#PetscDTAltVWedgeMatrix">PetscDTAltVWedgeMatrix</A>.
<P>
If k > 0, a k-form w and a vector v can combine to make a (k-1)-formm through the interior product, (w int v),
defined by (w int v)(v_1,...,v_{k-1}) = w(v,v_1,...,v_{k-1}).
<P>
The interior product is implemented for either two inputs (w and v) in <A HREF="../DT/PetscDTAltVInterior.html#PetscDTAltVInterior">PetscDTAltVInterior</A>, for one (just v, giving a
matrix to multiply against multiple choices of w) in <A HREF="../DT/PetscDTAltVInteriorMatrix.html#PetscDTAltVInteriorMatrix">PetscDTAltVInteriorMatrix</A>,
or for no inputs (giving the sparsity pattern of <A HREF="../DT/PetscDTAltVInteriorMatrix.html#PetscDTAltVInteriorMatrix">PetscDTAltVInteriorMatrix</A>) in <A HREF="../DT/PetscDTAltVInteriorPattern.html#PetscDTAltVInteriorPattern">PetscDTAltVInteriorPattern</A>.
<P>
When there is a linear map L: V -> W from an N dimensional vector space to an M dimensional vector space,
it induces the linear pullback map L^* : Alt^k W -> Alt^k V, defined by L^* w(v_1,...,v_k) = w(L v_1, ..., L v_k).
The pullback is implemented as <A HREF="../DT/PetscDTAltVPullback.html#PetscDTAltVPullback">PetscDTAltVPullback</A> (acting on a known w) and <A HREF="../DT/PetscDTAltVPullbackMatrix.html#PetscDTAltVPullbackMatrix">PetscDTAltVPullbackMatrix</A> (creating a matrix that computes the actin of L^*).
<P>
Alt^k V and Alt^(N-k) V have the same dimension, and the Hodge star operator maps between them. We note that Alt^N V is a one dimensional space, and its
basis vector is sometime called vol. The Hodge star operator has the property that (f wedge (star g)) = (f,g) vol, where (f,g) is the simple inner product
of the basis coefficients of f and g.
Powers of the Hodge star operator can be applied with <A HREF="../DT/PetscDTAltVStar.html#PetscDTAltVStar">PetscDTAltVStar</A>
<P>
<P>
<H3><FONT COLOR="#CC3333">See Also</FONT></H3>
<A HREF="../DT/PetscDTAltVApply.html#PetscDTAltVApply">PetscDTAltVApply</A>(), <A HREF="../DT/PetscDTAltVWedge.html#PetscDTAltVWedge">PetscDTAltVWedge</A>(), <A HREF="../DT/PetscDTAltVInterior.html#PetscDTAltVInterior">PetscDTAltVInterior</A>(), <A HREF="../DT/PetscDTAltVPullback.html#PetscDTAltVPullback">PetscDTAltVPullback</A>(), <A HREF="../DT/PetscDTAltVStar.html#PetscDTAltVStar">PetscDTAltVStar</A>()
<BR><P><B></B><H3><FONT COLOR="#CC3333">Level</FONT></H3>intermediate<BR>
<H3><FONT COLOR="#CC3333">Location</FONT></H3>
</B><A HREF="../../../src/dm/dt/interface/dtaltv.c.html#PetscDTAltV">src/dm/dt/interface/dtaltv.c</A>
<BR><A HREF="./index.html">Index of all DT routines</A>
<BR><A HREF="../../index.html">Table of Contents for all manual pages</A>
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