compute the inner product of two forms with respect to a given metric tensor - Maple Programming Help

Home : Support : Online Help : Mathematics : DifferentialGeometry : Tensor : DifferentialGeometry/Tensor/FormInnerProduct

Tensor[FormInnerProduct] - compute the inner product of two forms with respect to a given metric tensor

Calling Sequences

FormInnerProduct(g,keyword)

FormInnerProduct(g$,$ g1, ${\mathbf{α1}}$, ${\mathbf{β1}}$, keyword)

Parameters

g         - a covariant metric tensor on a manifold or on a Lie algebra with frame name, e.g., M

$\mathrm{α}$, $\mathrm{β}$        - two forms (of the same degree) on M, or lists of such

, $\mathrm{β1}$     - two forms (of the same degree) on M, or lists of such, where M is a Lie algebra with coefficients in a representation space $V$

g1                - a covariant metric tensor on the representation space $V$

keyword    - the keyword argument inversemetric = h, where h is the inverse of the metric g.

Description

 • Let   and let  be the inverse metric. If $\mathrm{α}$ and are 1-forms, then their inner product is  For monomial $p$-forms  and the inner product is given by

.

This formula is extended by bi-linearity to give the general formula for the inner product of a pair of $p-$forms.

 • In the special case of forms defined on a Lie algebra with coefficients $x$ and in a representation, the inner product formula for monomials becomes



where  and ${g}_{V}$ is the inner product on

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{Tensor}\right):$$\mathrm{with}\left(\mathrm{LieAlgebras}\right):$

First define a manifold $M$ with local coordinates $\left(x,y,z\right)$ and define a metric on $M$.

 > $\mathrm{DGsetup}\left(\left[x,y,z\right],M\right):$
 M > $g≔\mathrm{evalDG}\left(a\mathrm{dx}&t\mathrm{dx}+b\mathrm{dy}&t\mathrm{dy}+c\mathrm{dz}&t\mathrm{dz}\right)$
 ${g}{:=}{a}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}{b}{}{\mathrm{dy}}{}{\mathrm{dy}}{+}{c}{}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.1)

Example 1.

Compute the inner product of two 1-forms

 M > $\mathrm{α1}≔\mathrm{evalDG}\left(\mathrm{a1}\mathrm{dx}+\mathrm{a2}\mathrm{dy}+\mathrm{a3}\mathrm{dz}\right)$
 ${\mathrm{α1}}{:=}{\mathrm{a1}}{}{\mathrm{dx}}{+}{\mathrm{a2}}{}{\mathrm{dy}}{+}{\mathrm{a3}}{}{\mathrm{dz}}$ (2.2)
 M > $\mathrm{β1}≔\mathrm{evalDG}\left(\mathrm{b1}\mathrm{dx}+\mathrm{b2}\mathrm{dy}+\mathrm{b3}\mathrm{dz}\right)$
 ${\mathrm{β1}}{:=}{\mathrm{b1}}{}{\mathrm{dx}}{+}{\mathrm{b2}}{}{\mathrm{dy}}{+}{\mathrm{b3}}{}{\mathrm{dz}}$ (2.3)
 M > $\mathrm{FormInnerProduct}\left(g,\mathrm{α1},\mathrm{β1}\right)$
 $\frac{{\mathrm{a1}}{}{\mathrm{b1}}}{{a}}{+}\frac{{\mathrm{a2}}{}{\mathrm{b2}}}{{b}}{+}\frac{{\mathrm{a3}}{}{\mathrm{b3}}}{{c}}$ (2.4)

Example 2.

Compute the inner products of a list of monomial 2-forms.

 M > $\mathrm{g2}≔\mathrm{evalDG}\left(a\mathrm{dx}&t\mathrm{dx}+b\mathrm{dy}&t\mathrm{dy}+c\mathrm{dz}&t\mathrm{dz}\right)$
 ${\mathrm{g2}}{:=}{a}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}{b}{}{\mathrm{dy}}{}{\mathrm{dy}}{+}{c}{}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.5)
 M > $\mathrm{Ω}≔\mathrm{evalDG}\left(\left[\mathrm{dx}&w\mathrm{dy},\mathrm{dx}&w\mathrm{dz},\mathrm{dy}&w\mathrm{dz}\right]\right)$
 ${\mathrm{Ω}}{:=}\left[{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dy}}{,}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dz}}{,}{\mathrm{dy}}{}{\bigwedge }{}{\mathrm{dz}}\right]$ (2.6)
 M > $\mathrm{FormInnerProduct}\left(\mathrm{g2},\mathrm{Ω},\mathrm{Ω}\right)$
 $\left[\begin{array}{ccc}\frac{{1}}{{a}{}{b}}& {0}& {0}\\ {0}& \frac{{1}}{{a}{}{c}}& {0}\\ {0}& {0}& \frac{{1}}{{b}{}{c}}\end{array}\right]$ (2.7)

Compute the inner product of a pair of 2-forms.

 M > $\mathrm{α2}≔\mathrm{evalDG}\left(2\mathrm{dx}&w\mathrm{dy}+\mathrm{dy}&w\mathrm{dz}\right)$
 ${\mathrm{α2}}{:=}{2}{}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dy}}{+}{\mathrm{dy}}{}{\bigwedge }{}{\mathrm{dz}}$ (2.8)
 M > $\mathrm{β2}≔\mathrm{evalDG}\left(3\mathrm{dx}&w\mathrm{dz}+4\mathrm{dy}&w\mathrm{dz}\right)$
 ${\mathrm{β2}}{:=}{3}{}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dz}}{+}{4}{}{\mathrm{dy}}{}{\bigwedge }{}{\mathrm{dz}}$ (2.9)
 M > $\mathrm{FormInnerProduct}\left(\mathrm{g2},\mathrm{α2},\mathrm{α2}\right)$
 $\frac{{4}}{{a}{}{b}}{+}\frac{{1}}{{b}{}{c}}$ (2.10)

Example 3.

In this example we compute the inner products of forms defined on a Lie algebra with coefficients in a representation.

 M > $\mathrm{LD}≔\mathrm{SimpleLieAlgebraData}\left("so\left(4\right)",\mathrm{so4}\right)$
 ${\mathrm{LD}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e4}}\right]$ (2.11)
 M > $\mathrm{DGsetup}\left(\mathrm{LD}\right)$
 ${\mathrm{Lie algebra: so4}}$ (2.12)
 so4 > $\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4}\right],V\right)$
 ${\mathrm{frame name: V}}$ (2.13)
 so4 > $\mathrm{ρ}≔\mathrm{StandardRepresentation}\left(\mathrm{so4},\mathrm{representationspace}=V\right)$
 ${\mathrm{ρ}}{:=}\left[\left[{\mathrm{e1}}{,}\left[\begin{array}{cccc}{0}& {-1}& {0}& {0}\\ {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e2}}{,}\left[\begin{array}{cccc}{0}& {0}& {-1}& {0}\\ {0}& {0}& {0}& {0}\\ {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e3}}{,}\left[\begin{array}{cccc}{0}& {0}& {0}& {-1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {1}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e4}}{,}\left[\begin{array}{cccc}{0}& {0}& {0}& {0}\\ {0}& {0}& {-1}& {0}\\ {0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e5}}{,}\left[\begin{array}{cccc}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-1}\\ {0}& {0}& {0}& {0}\\ {0}& {1}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e6}}{,}\left[\begin{array}{cccc}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-1}\\ {0}& {0}& {1}& {0}\end{array}\right]\right]\right]$ (2.14)
 V > $\mathrm{DGsetup}\left(\mathrm{ρ},\mathrm{so4V},\left[\mathrm{O}\right],\left[o\right]\right)$
 ${\mathrm{Lie algebra with coefficients: so4V}}$ (2.15)
 so4V > $g≔\mathrm{KillingForm}\left(\mathrm{so4V}\right)$
 ${g}{:=}{-}{4}{}{\mathrm{o1}}{}{\mathrm{o1}}{-}{4}{}{\mathrm{o2}}{}{\mathrm{o2}}{-}{4}{}{\mathrm{o3}}{}{\mathrm{o3}}{-}{4}{}{\mathrm{o4}}{}{\mathrm{o4}}{-}{4}{}{\mathrm{o5}}{}{\mathrm{o5}}{-}{4}{}{\mathrm{o6}}{}{\mathrm{o6}}$ (2.16)
 so4V > $h≔\mathrm{InverseMetric}\left(g\right)$
 ${h}{:=}{-}\frac{{1}}{{4}}{}{\mathrm{O1}}{}{\mathrm{O1}}{-}\frac{{1}}{{4}}{}{\mathrm{O2}}{}{\mathrm{O2}}{-}\frac{{1}}{{4}}{}{\mathrm{O3}}{}{\mathrm{O3}}{-}\frac{{1}}{{4}}{}{\mathrm{O4}}{}{\mathrm{O4}}{-}\frac{{1}}{{4}}{}{\mathrm{O5}}{}{\mathrm{O5}}{-}\frac{{1}}{{4}}{}{\mathrm{O6}}{}{\mathrm{O6}}$ (2.17)
 so4V > $\mathrm{gV}≔\mathrm{evalDG}\left(\mathrm{dx1}&t\mathrm{dx1}+\mathrm{dx2}&t\mathrm{dx2}+\mathrm{dx3}&t\mathrm{dx3}+\mathrm{dx4}&t\mathrm{dx4}\right)$
 ${\mathrm{gV}}{:=}{\mathrm{dx1}}{}{\mathrm{dx1}}{+}{\mathrm{dx2}}{}{\mathrm{dx2}}{+}{\mathrm{dx3}}{}{\mathrm{dx3}}{+}{\mathrm{dx4}}{}{\mathrm{dx4}}$ (2.18)

Compute the inner product of a pair of zero forms.

 V > $\mathrm{FormInnerProduct}\left(g,\mathrm{gV},a\mathrm{x1}+b\mathrm{x2},c\mathrm{x1}+d\mathrm{x2}\right)$
 ${a}{}{c}{+}{b}{}{d}$ (2.19)

Compute the inner product of a pair of 1-forms.

 V > $\mathrm{FormInnerProduct}\left(g,\mathrm{gV},\mathrm{x1}\mathrm{o1},\mathrm{x1}\mathrm{o3}\right)$
 ${0}$ (2.20)
 so4V > $\mathrm{FormInnerProduct}\left(g,\mathrm{gV},\mathrm{x2}\mathrm{o1},\mathrm{x1}\mathrm{o1}\right)$
 ${0}$ (2.21)
 so4V > $\mathrm{FormInnerProduct}\left(g,\mathrm{gV},\mathrm{x2}\mathrm{o2},\mathrm{x2}\mathrm{o2}\right)$
 ${-}\frac{{1}}{{4}}$ (2.22)
 V > $\mathrm{FormInnerProduct}\left(g,\mathrm{gV},\mathrm{x2}\mathrm{o1}&w\mathrm{o2},\mathrm{x2}\mathrm{o1}&w\mathrm{o2}\right)$
 $\frac{{1}}{{16}}$ (2.23)

Compute the the length of a 2-form.

 V > $\mathrm{α3}≔\mathrm{evalDG}\left(a\mathrm{x2}\mathrm{o1}&w\mathrm{o2}+b\mathrm{x4}\mathrm{o1}&w\mathrm{o3}+cx\mathrm{o2}&w\mathrm{o5}\right)$
 ${\mathrm{α3}}{:=}{a}{}{\mathrm{x2}}{}{\mathrm{o1}}{}{\bigwedge }{}{\mathrm{o2}}{+}{b}{}{\mathrm{x4}}{}{\mathrm{o1}}{}{\bigwedge }{}{\mathrm{o3}}{+}{c}{}{x}{}{\mathrm{o2}}{}{\bigwedge }{}{\mathrm{o5}}$ (2.24)
 so4V > $\sqrt{\mathrm{FormInnerProduct}\left(g,\mathrm{gV},\mathrm{α3},\mathrm{α3}\right)}$
 $\frac{{1}}{{4}}{}\sqrt{{{a}}^{{2}}{+}{{b}}^{{2}}}$ (2.25)