Hook - Maple Help

DifferentialGeometry

 Hook
 form the interior product of a vector or a list of vectors with a differential form

 Calling Sequence Hook(X, omega) Hook(Y, T, indexlist)

Parameters

 X - a vector or a list of vectors omega - a differential p-form Y - a list of vectors or differential 1-forms T - a tensor indexlist - (optional) a list of positive integers labeling various arguments of the tensor T

Description

 • If  X is a vector and omega a differential p-form, then the interior product or hook of  X and omega is the differential (p - 1)-form theta = Hook(X, omega) defined by theta(Y1, Y2, ..., Yq) = omega(X, Y1, Y2, ..., Yq), where q = p - 1 and Y1, Y2, ...,  Yq are arbitrary vectors.
 • More generally, given a list of vectors [X1, X2, ... , Xr] and a differential p-form omega then theta = Hook([X1, X2, ..., Xr], omega) is the differential form of degree p - r defined by theta(Y1, Y2, ..., Yq) = omega(X1, X2, ..., Xr, Y1, Y2, ..., Yq), where q = p - r and Y1, Y2, ...,Yq are arbitrary vectors.
 • If  Y = [X1, X2, alpha1, ..., Xr] is a list of  vectors or differential 1-forms and T is a tensor of total rank s = r, then the second calling sequence evaluates the scalar T(X1, X2, alpha1, ... , Xr).  If s > r and  indexlist = [i_1, i_2, ..., i_r], then Hook(Y, T, indexlist) calculates the rank s - r tensor obtained by evaluating the i_k-th argument of the tensor T on the k-th element of the list Y, for k = 1, 2, ... r.
 • This command is part of the DifferentialGeometry package, and so can be used in the form Hook(...) only after executing the command with(DifferentialGeometry).  It can always be used in the long form DifferentialGeometry:-Hook.

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$
 > $\mathrm{DGsetup}\left(\left[x,y,z\right],M\right):$

Example 1.

Define vectors X1, X2, X3.

 > $\mathrm{X1}≔\mathrm{evalDG}\left(a\mathrm{D_x}+b\mathrm{D_y}+c\mathrm{D_z}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{a}\right]{,}\left[\left[{2}\right]{,}{b}\right]{,}\left[\left[{3}\right]{,}{c}\right]\right]\right]\right)$ (1)
 > $\mathrm{X2}≔\mathrm{evalDG}\left(2\mathrm{D_y}-\mathrm{D_z}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{2}\right]{,}{2}\right]{,}\left[\left[{3}\right]{,}{-}{1}\right]\right]\right]\right)$ (2)
 > $\mathrm{X3}≔\mathrm{evalDG}\left(\mathrm{D_x}+\mathrm{D_z}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]{,}\left[\left[{3}\right]{,}{1}\right]\right]\right]\right)$ (3)

Define a 1-form alpha, a 2-form beta and a 3-form omega.

 > $\mathrm{α}≔\mathrm{evalDG}\left(3\mathrm{dx}+4\mathrm{dy}-7\mathrm{dz}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{3}\right]{,}\left[\left[{2}\right]{,}{4}\right]{,}\left[\left[{3}\right]{,}{-}{7}\right]\right]\right]\right)$ (4)
 > $\mathrm{β}≔\mathrm{evalDG}\left(\mathrm{dx}&w\mathrm{dy}-3\mathrm{dx}&w\mathrm{dz}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{1}{,}{3}\right]{,}{-}{3}\right]\right]\right]\right)$ (5)
 > $\mathrm{ω}≔\mathrm{evalDG}\left(\left(\mathrm{dx}&w\mathrm{dy}\right)&w\mathrm{dz}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{3}\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{1}\right]\right]\right]\right)$ (6)

Compute various interior products.  Note that Hook(Y, Hook(X, omega) = Hook([X, Y], omega).

 > $f≔\mathrm{Hook}\left(\mathrm{X1},\mathrm{α}\right)$
 ${f}{≔}{-}{7}{}{c}{+}{4}{}{b}{+}{3}{}{a}$ (7)
 > $\mathrm{θ}≔\mathrm{Hook}\left(\mathrm{X1},\mathrm{β}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{3}{}{c}{-}{b}\right]{,}\left[\left[{2}\right]{,}{a}\right]{,}\left[\left[{3}\right]{,}{-}{3}{}{a}\right]\right]\right]\right)$ (8)
 > $\mathrm{Hook}\left(\mathrm{X2},\mathrm{θ}\right)$
 ${5}{}{a}$ (9)
 > $\mathrm{Hook}\left(\left[\mathrm{X1},\mathrm{X2}\right],\mathrm{β}\right)$
 ${5}{}{a}$ (10)
 > $\mathrm{Hook}\left(\left[\mathrm{X2},\mathrm{X1}\right],\mathrm{β}\right)$
 ${-}{5}{}{a}$ (11)
 > $\mathrm{Hook}\left(\left[\mathrm{X1},\mathrm{X2},\mathrm{X3}\right],\mathrm{ω}\right)$
 ${2}{}{a}{-}{b}{-}{2}{}{c}$ (12)

Example 2.

Evaluate a  type (1,3) tensor T on various vectors and differential forms.

 > $T≔\mathrm{evalDG}\left(2\left(\left(\mathrm{dx}&t\mathrm{dy}\right)&t\mathrm{dz}\right)&t\mathrm{D_y}-3\left(\left(\mathrm{dy}&t\mathrm{dx}\right)&t\mathrm{dz}\right)&t\mathrm{D_z}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}{,}{"con_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}{,}{2}\right]{,}{2}\right]{,}\left[\left[{2}{,}{1}{,}{3}{,}{3}\right]{,}{-}{3}\right]\right]\right]\right)$ (13)
 > $\mathrm{Hook}\left(\left[\mathrm{D_x},\mathrm{D_y},\mathrm{D_z},\mathrm{dy}\right],T\right)$
 ${2}$ (14)
 > $\mathrm{Hook}\left(\left[\mathrm{X1},\mathrm{X3}\right],T\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"con_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{3}{,}{3}\right]{,}{-}{3}{}{b}\right]\right]\right]\right)$ (15)
 > $\mathrm{Hook}\left(\left[\mathrm{X1},\mathrm{X2},\mathrm{X3},\mathrm{α}\right],T\right)$
 ${16}{}{a}$ (16)
 > $\mathrm{Hook}\left(\left[\mathrm{D_y},\mathrm{dz}\right],T,\left[1,4\right]\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{-}{3}\right]\right]\right]\right)$ (17)
 > $\mathrm{Hook}\left(\left[\mathrm{X1},\mathrm{X2}\right],T,\left[1,3\right]\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"con_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{3}{}{b}\right]{,}\left[\left[{2}{,}{2}\right]{,}{-}{2}{}{a}\right]\right]\right]\right)$ (18)

Example 3.

The interior product can be calculated for abstract forms.

 > $\mathrm{DGsetup}\left(\left[\left[\mathrm{ω1},\mathrm{ω2},\mathrm{ω3}\right],\mathrm{β1}=\mathrm{dgform}\left(2\right)\right],\left[d\left(\mathrm{ω1}\right)=\mathrm{ω2}&w\mathrm{ω3},d\left(\mathrm{ω2}\right)=\mathrm{β1}\right],\mathrm{M3}\right)$
 ${\mathrm{frame name: M3}}$ (19)
 > $\mathrm{Hook}\left(\mathrm{D_omega1},\mathrm{ω1}\right);$$\mathrm{Hook}\left(\mathrm{D_omega1},\mathrm{β1}\right)$
 ${1}$
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{M3}}{,}{1}\right]{,}\left[\left[\left[{5}\right]{,}{1}\right]\right]\right]\right)$ (20)

Structure equations for interior products can be specified.

 > $\mathrm{DGsetup}\left(\mathrm{M3},\left[\right],\left[\mathrm{hook}\left(\mathrm{D_omega1},\mathrm{β1}\right)=\mathrm{ω3}\right]\right)$
 ${\mathrm{updated frame: M3}}$ (21)
 > $\mathrm{Hook}\left(\mathrm{D_omega1},\mathrm{β1}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{M3}}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right)$ (22)