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Tensor[SpinConnection] - compute the spin connection defined by a solder form

Calling Sequences

SpinConnection(${\mathbf{σ}}$)

Parameters

$\mathrm{σ}$   - a solder form

Description

 • The DifferentialGeometry Tensor package supports general computations with connections on vector bundles (Connection, Example 3; CovariantDerivative, Example 3; DirectionalCovariantDerivative, Example 3; and CurvatureTensor, Example 3).  This functionality naturally provides for covariant differentiation of spinors.
 • The command SpinConnection(${\mathbf{σ}}$) computes the connection compatible with the solder form and the epsilon spinors.
 • Given a solder form $\mathrm{σ}$, let $g$ be the associated metric. There is a unique spin connection $\nabla$ such that $\nabla \mathrm{σ}=0$ and $\nabla \mathrm{ε}=0$, where $\mathrm{ε}$ denotes either of the epsilon spinors (EpsilonSpinor). In the definition of $\nabla \mathrm{σ}$ the tensorial argument (or index) is covariantly differentiated with respect to the Christoffel connection for $g$. It is this connection $\nabla$ which is computed by the command SpinConnection(sigma).
 • Note that a generic connection for the differentiation of spinors can be constructed using the Connection command.
 • This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form SpinConnection(...) only after executing the commands with(DifferentialGeometry); with(Tensor); in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-SpinConnection.

Examples

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

Example 1.

First create a vector bundle $E\to M$ with base coordinates $\left(t,x,y,z\right)$ and fiber coordinates .

 > $\mathrm{DGsetup}\left(\left[t,x,y,z\right],\left[\mathrm{z1},\mathrm{z2},\mathrm{w1},\mathrm{w2}\right],E\right)$
 ${\mathrm{frame name: E}}$ (2.1)

Define a spacetime metric $g$ on $M$.

 E > $g≔\mathrm{evalDG}\left({x}^{4}\mathrm{dt}&t\mathrm{dt}-\mathrm{dx}&t\mathrm{dx}-\mathrm{dy}&t\mathrm{dy}-\mathrm{dz}&t\mathrm{dz}\right)$
 ${g}{≔}{{x}}^{{4}}{}{\mathrm{dt}}{}{\mathrm{dt}}{-}{\mathrm{dx}}{}{\mathrm{dx}}{-}{\mathrm{dy}}{}{\mathrm{dy}}{-}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.2)

Define an orthonormal frame on $M$ with respect to the metric $g$.

 E > $F≔\mathrm{evalDG}\left(\left[\frac{1\mathrm{D_t}}{{x}^{2}},\mathrm{D_x},\mathrm{D_y},\mathrm{D_z}\right]\right)$
 ${F}{≔}\left[\frac{{1}}{{{x}}^{{2}}}{}{\mathrm{D_t}}{,}{\mathrm{D_x}}{,}{\mathrm{D_y}}{,}{\mathrm{D_z}}\right]$ (2.3)

Calculate the solder form $\mathrm{σ}$ from the frame F.

 E > $\mathrm{σ}≔\mathrm{SolderForm}\left(F\right)$
 ${\mathrm{\sigma }}{≔}\frac{{{x}}^{{2}}{}\sqrt{{2}}}{{2}}{}{\mathrm{dt}}{}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{+}\frac{{{x}}^{{2}}{}\sqrt{{2}}}{{2}}{}{\mathrm{dt}}{}{\mathrm{D_z2}}{}{\mathrm{D_w2}}{+}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{dx}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{+}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{dx}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{-}\frac{{I}}{{2}}{}\sqrt{{2}}{}{\mathrm{dy}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{+}\frac{{I}}{{2}}{}\sqrt{{2}}{}{\mathrm{dy}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{+}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{dz}}{}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{-}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{dz}}{}{\mathrm{D_z2}}{}{\mathrm{D_w2}}$ (2.4)

Calculate the spin-connection for the solder form $\mathrm{σ}$.

 E > $\mathrm{Γ2}≔\mathrm{SpinConnection}\left(\mathrm{σ}\right)$
 ${\mathrm{Γ2}}{≔}{x}{}{\mathrm{D_z1}}{}{\mathrm{dz2}}{}{\mathrm{dt}}{+}{x}{}{\mathrm{D_z2}}{}{\mathrm{dz1}}{}{\mathrm{dt}}{+}{x}{}{\mathrm{D_w1}}{}{\mathrm{dw2}}{}{\mathrm{dt}}{+}{x}{}{\mathrm{D_w2}}{}{\mathrm{dw1}}{}{\mathrm{dt}}$ (2.5)

Example 2.

Define a rank 1 spinor $\mathrm{φ}$. Calculate the covariant derivative of $\mathrm{φ}$. Calculate the directional derivatives of $\mathrm{φ}$.

 E > $\mathrm{φ}≔\mathrm{evalDG}\left({t}^{2}\mathrm{D_z1}-\frac{1\mathrm{D_z2}}{y}\right)$
 ${\mathrm{\phi }}{≔}{{t}}^{{2}}{}{\mathrm{D_z1}}{-}\frac{{1}}{{y}}{}{\mathrm{D_z2}}$ (2.6)
 E > $\mathrm{CovariantDerivative}\left(\mathrm{φ},\mathrm{Γ2}\right)$
 $\frac{{-}{x}{+}{2}{}{t}{}{y}}{{y}}{}{\mathrm{D_z1}}{}{\mathrm{dt}}{+}{x}{}{{t}}^{{2}}{}{\mathrm{D_z2}}{}{\mathrm{dt}}{+}\frac{{1}}{{{y}}^{{2}}}{}{\mathrm{D_z2}}{}{\mathrm{dy}}$ (2.7)
 E > $\mathrm{DirectionalCovariantDerivative}\left(\mathrm{D_x},\mathrm{φ},\mathrm{Γ2}\right)$
 ${0}{}{\mathrm{D_z1}}$ (2.8)
 E > $\mathrm{DirectionalCovariantDerivative}\left(\mathrm{D_y},\mathrm{φ},\mathrm{Γ2}\right)$
 $\frac{{1}}{{{y}}^{{2}}}{}{\mathrm{D_z2}}$ (2.9)
 E > $\mathrm{DirectionalCovariantDerivative}\left(\mathrm{D_z},\mathrm{φ},\mathrm{Γ2}\right)$
 ${0}{}{\mathrm{D_z1}}$ (2.10)
 E > $\mathrm{DirectionalCovariantDerivative}\left(y\mathrm{D_t},\mathrm{φ},\mathrm{Γ2}\right)$
 $\left({-}{x}{+}{2}{}{t}{}{y}\right){}{\mathrm{D_z1}}{+}{x}{}{y}{}{{t}}^{{2}}{}{\mathrm{D_z2}}$ (2.11)

Example 3.

Check that the covariant derivative of $\mathrm{σ}$ vanishes. Because $\mathrm{σ}$ is a spin-tensor, two connections are required. Calculate the Christoffel connection for the metric $g$.

 E > $\mathrm{Γ1}≔\mathrm{Christoffel}\left(g\right)$
 ${\mathrm{Γ1}}{≔}\frac{{2}}{{x}}{}{\mathrm{D_t}}{}{\mathrm{dt}}{}{\mathrm{dx}}{+}\frac{{2}}{{x}}{}{\mathrm{D_t}}{}{\mathrm{dx}}{}{\mathrm{dt}}{+}{2}{}{{x}}^{{3}}{}{\mathrm{D_x}}{}{\mathrm{dt}}{}{\mathrm{dt}}$ (2.12)
 E > $\mathrm{CovariantDerivative}\left(\mathrm{σ},\mathrm{Γ1},\mathrm{Γ2}\right)$
 ${0}{}{\mathrm{dt}}{}{\mathrm{D_z1}}{}{\mathrm{D_z1}}{}{\mathrm{dt}}$ (2.13)

Define an epsilon spinor and check that its covariant derivative vanishes.

 E > $\mathrm{Eps}≔\mathrm{EpsilonSpinor}\left("cov","spinor"\right)$
 ${\mathrm{Eps}}{≔}{\mathrm{dz1}}{}{\mathrm{dz2}}{-}{\mathrm{dz2}}{}{\mathrm{dz1}}$ (2.14)
 E > $\mathrm{CovariantDerivative}\left(\mathrm{Eps},\mathrm{Γ2}\right)$
 ${0}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dt}}$ (2.15)

Example 4.

Calculate the curvature spin-tensor for the spin-connection Gamma2.

 E > $F≔\mathrm{CurvatureTensor}\left(\mathrm{Γ2}\right)$
 ${F}{≔}{-}{\mathrm{D_z1}}{}{\mathrm{dz2}}{}{\mathrm{dt}}{}{\mathrm{dx}}{+}{\mathrm{D_z1}}{}{\mathrm{dz2}}{}{\mathrm{dx}}{}{\mathrm{dt}}{-}{\mathrm{D_z2}}{}{\mathrm{dz1}}{}{\mathrm{dt}}{}{\mathrm{dx}}{+}{\mathrm{D_z2}}{}{\mathrm{dz1}}{}{\mathrm{dx}}{}{\mathrm{dt}}{-}{\mathrm{D_w1}}{}{\mathrm{dw2}}{}{\mathrm{dt}}{}{\mathrm{dx}}{+}{\mathrm{D_w1}}{}{\mathrm{dw2}}{}{\mathrm{dx}}{}{\mathrm{dt}}{-}{\mathrm{D_w2}}{}{\mathrm{dw1}}{}{\mathrm{dt}}{}{\mathrm{dx}}{+}{\mathrm{D_w2}}{}{\mathrm{dw1}}{}{\mathrm{dx}}{}{\mathrm{dt}}$ (2.16)

The curvature tensor $R$ for the Christoffel connection can be expressed in terms of the curvature spin-tensor $F$ and the bivector solder forms by the identity

Let's check this formula for the Christoffel connection Gamma1 and the spin-connection Gamma2. First calculate the curvature tensor for Gamma1.

 E > $R≔\mathrm{CurvatureTensor}\left(\mathrm{Γ1}\right)$
 ${R}{≔}{-}\frac{{2}}{{{x}}^{{2}}}{}{\mathrm{D_t}}{}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dx}}{+}\frac{{2}}{{{x}}^{{2}}}{}{\mathrm{D_t}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dt}}{-}{2}{}{{x}}^{{2}}{}{\mathrm{D_x}}{}{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dx}}{+}{2}{}{{x}}^{{2}}{}{\mathrm{D_x}}{}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dt}}$ (2.17)

Calculate the complex conjugate of the spinor curvature F.

 E > $\mathrm{barF}≔\mathrm{ConjugateSpinor}\left(F\right)$
 ${\mathrm{barF}}{≔}{-}{\mathrm{D_z1}}{}{\mathrm{dz2}}{}{\mathrm{dt}}{}{\mathrm{dx}}{+}{\mathrm{D_z1}}{}{\mathrm{dz2}}{}{\mathrm{dx}}{}{\mathrm{dt}}{-}{\mathrm{D_z2}}{}{\mathrm{dz1}}{}{\mathrm{dt}}{}{\mathrm{dx}}{+}{\mathrm{D_z2}}{}{\mathrm{dz1}}{}{\mathrm{dx}}{}{\mathrm{dt}}{-}{\mathrm{D_w1}}{}{\mathrm{dw2}}{}{\mathrm{dt}}{}{\mathrm{dx}}{+}{\mathrm{D_w1}}{}{\mathrm{dw2}}{}{\mathrm{dx}}{}{\mathrm{dt}}{-}{\mathrm{D_w2}}{}{\mathrm{dw1}}{}{\mathrm{dt}}{}{\mathrm{dx}}{+}{\mathrm{D_w2}}{}{\mathrm{dw1}}{}{\mathrm{dx}}{}{\mathrm{dt}}$ (2.18)

Calculate the bivector soldering forms S and barS.

 E > $S≔\mathrm{BivectorSolderForm}\left(\mathrm{σ},"spinor",\mathrm{indextype}=\left["con","cov","cov","con"\right]\right)$
 ${S}{≔}\frac{{1}}{{{x}}^{{2}}}{}{\mathrm{D_t}}{}{\mathrm{dx}}{}{\mathrm{dz1}}{}{\mathrm{D_z2}}{+}\frac{{1}}{{{x}}^{{2}}}{}{\mathrm{D_t}}{}{\mathrm{dx}}{}{\mathrm{dz2}}{}{\mathrm{D_z1}}{+}\frac{{I}}{{{x}}^{{2}}}{}{\mathrm{D_t}}{}{\mathrm{dy}}{}{\mathrm{dz1}}{}{\mathrm{D_z2}}{-}\frac{{I}}{{{x}}^{{2}}}{}{\mathrm{D_t}}{}{\mathrm{dy}}{}{\mathrm{dz2}}{}{\mathrm{D_z1}}{+}\frac{{1}}{{{x}}^{{2}}}{}{\mathrm{D_t}}{}{\mathrm{dz}}{}{\mathrm{dz1}}{}{\mathrm{D_z1}}{-}\frac{{1}}{{{x}}^{{2}}}{}{\mathrm{D_t}}{}{\mathrm{dz}}{}{\mathrm{dz2}}{}{\mathrm{D_z2}}{+}{{x}}^{{2}}{}{\mathrm{D_x}}{}{\mathrm{dt}}{}{\mathrm{dz1}}{}{\mathrm{D_z2}}{+}{{x}}^{{2}}{}{\mathrm{D_x}}{}{\mathrm{dt}}{}{\mathrm{dz2}}{}{\mathrm{D_z1}}{-}{I}{}{\mathrm{D_x}}{}{\mathrm{dy}}{}{\mathrm{dz1}}{}{\mathrm{D_z1}}{+}{I}{}{\mathrm{D_x}}{}{\mathrm{dy}}{}{\mathrm{dz2}}{}{\mathrm{D_z2}}{-}{\mathrm{D_x}}{}{\mathrm{dz}}{}{\mathrm{dz1}}{}{\mathrm{D_z2}}{+}{\mathrm{D_x}}{}{\mathrm{dz}}{}{\mathrm{dz2}}{}{\mathrm{D_z1}}{+}{I}{}{{x}}^{{2}}{}{\mathrm{D_y}}{}{\mathrm{dt}}{}{\mathrm{dz1}}{}{\mathrm{D_z2}}{-}{I}{}{{x}}^{{2}}{}{\mathrm{D_y}}{}{\mathrm{dt}}{}{\mathrm{dz2}}{}{\mathrm{D_z1}}{+}{I}{}{\mathrm{D_y}}{}{\mathrm{dx}}{}{\mathrm{dz1}}{}{\mathrm{D_z1}}{-}{I}{}{\mathrm{D_y}}{}{\mathrm{dx}}{}{\mathrm{dz2}}{}{\mathrm{D_z2}}{-}{I}{}{\mathrm{D_y}}{}{\mathrm{dz}}{}{\mathrm{dz1}}{}{\mathrm{D_z2}}{-}{I}{}{\mathrm{D_y}}{}{\mathrm{dz}}{}{\mathrm{dz2}}{}{\mathrm{D_z1}}{+}{{x}}^{{2}}{}{\mathrm{D_z}}{}{\mathrm{dt}}{}{\mathrm{dz1}}{}{\mathrm{D_z1}}{-}{{x}}^{{2}}{}{\mathrm{D_z}}{}{\mathrm{dt}}{}{\mathrm{dz2}}{}{\mathrm{D_z2}}{+}{\mathrm{D_z}}{}{\mathrm{dx}}{}{\mathrm{dz1}}{}{\mathrm{D_z2}}{-}{\mathrm{D_z}}{}{\mathrm{dx}}{}{\mathrm{dz2}}{}{\mathrm{D_z1}}{+}{I}{}{\mathrm{D_z}}{}{\mathrm{dy}}{}{\mathrm{dz1}}{}{\mathrm{D_z2}}{+}{I}{}{\mathrm{D_z}}{}{\mathrm{dy}}{}{\mathrm{dz2}}{}{\mathrm{D_z1}}$ (2.19)
 E > $\mathrm{barS}≔\mathrm{BivectorSolderForm}\left(\mathrm{σ},"barspinor",\mathrm{indextype}=\left["con","cov","cov","con"\right]\right)$
 ${\mathrm{barS}}{≔}\frac{{1}}{{{x}}^{{2}}}{}{\mathrm{D_t}}{}{\mathrm{dx}}{}{\mathrm{dw1}}{}{\mathrm{D_w2}}{+}\frac{{1}}{{{x}}^{{2}}}{}{\mathrm{D_t}}{}{\mathrm{dx}}{}{\mathrm{dw2}}{}{\mathrm{D_w1}}{-}\frac{{I}}{{{x}}^{{2}}}{}{\mathrm{D_t}}{}{\mathrm{dy}}{}{\mathrm{dw1}}{}{\mathrm{D_w2}}{+}\frac{{I}}{{{x}}^{{2}}}{}{\mathrm{D_t}}{}{\mathrm{dy}}{}{\mathrm{dw2}}{}{\mathrm{D_w1}}{+}\frac{{1}}{{{x}}^{{2}}}{}{\mathrm{D_t}}{}{\mathrm{dz}}{}{\mathrm{dw1}}{}{\mathrm{D_w1}}{-}\frac{{1}}{{{x}}^{{2}}}{}{\mathrm{D_t}}{}{\mathrm{dz}}{}{\mathrm{dw2}}{}{\mathrm{D_w2}}{+}{{x}}^{{2}}{}{\mathrm{D_x}}{}{\mathrm{dt}}{}{\mathrm{dw1}}{}{\mathrm{D_w2}}{+}{{x}}^{{2}}{}{\mathrm{D_x}}{}{\mathrm{dt}}{}{\mathrm{dw2}}{}{\mathrm{D_w1}}{+}{I}{}{\mathrm{D_x}}{}{\mathrm{dy}}{}{\mathrm{dw1}}{}{\mathrm{D_w1}}{-}{I}{}{\mathrm{D_x}}{}{\mathrm{dy}}{}{\mathrm{dw2}}{}{\mathrm{D_w2}}{-}{\mathrm{D_x}}{}{\mathrm{dz}}{}{\mathrm{dw1}}{}{\mathrm{D_w2}}{+}{\mathrm{D_x}}{}{\mathrm{dz}}{}{\mathrm{dw2}}{}{\mathrm{D_w1}}{-}{I}{}{{x}}^{{2}}{}{\mathrm{D_y}}{}{\mathrm{dt}}{}{\mathrm{dw1}}{}{\mathrm{D_w2}}{+}{I}{}{{x}}^{{2}}{}{\mathrm{D_y}}{}{\mathrm{dt}}{}{\mathrm{dw2}}{}{\mathrm{D_w1}}{-}{I}{}{\mathrm{D_y}}{}{\mathrm{dx}}{}{\mathrm{dw1}}{}{\mathrm{D_w1}}{+}{I}{}{\mathrm{D_y}}{}{\mathrm{dx}}{}{\mathrm{dw2}}{}{\mathrm{D_w2}}{+}{I}{}{\mathrm{D_y}}{}{\mathrm{dz}}{}{\mathrm{dw1}}{}{\mathrm{D_w2}}{+}{I}{}{\mathrm{D_y}}{}{\mathrm{dz}}{}{\mathrm{dw2}}{}{\mathrm{D_w1}}{+}{{x}}^{{2}}{}{\mathrm{D_z}}{}{\mathrm{dt}}{}{\mathrm{dw1}}{}{\mathrm{D_w1}}{-}{{x}}^{{2}}{}{\mathrm{D_z}}{}{\mathrm{dt}}{}{\mathrm{dw2}}{}{\mathrm{D_w2}}{+}{\mathrm{D_z}}{}{\mathrm{dx}}{}{\mathrm{dw1}}{}{\mathrm{D_w2}}{-}{\mathrm{D_z}}{}{\mathrm{dx}}{}{\mathrm{dw2}}{}{\mathrm{D_w1}}{-}{I}{}{\mathrm{D_z}}{}{\mathrm{dy}}{}{\mathrm{dw1}}{}{\mathrm{D_w2}}{-}{I}{}{\mathrm{D_z}}{}{\mathrm{dy}}{}{\mathrm{dw2}}{}{\mathrm{D_w1}}$ (2.20)

The first term on the right-hand side of (*) is

 E > $\mathrm{R1}≔\mathrm{ContractIndices}\left(S,F,\left[\left[3,1\right],\left[4,2\right]\right]\right)$
 ${\mathrm{R1}}{≔}{-}\frac{{2}}{{{x}}^{{2}}}{}{\mathrm{D_t}}{}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dx}}{+}\frac{{2}}{{{x}}^{{2}}}{}{\mathrm{D_t}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dt}}{-}{2}{}{{x}}^{{2}}{}{\mathrm{D_x}}{}{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dx}}{+}{2}{}{{x}}^{{2}}{}{\mathrm{D_x}}{}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dt}}{+}{2}{}{I}{}{\mathrm{D_y}}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dx}}{-}{2}{}{I}{}{\mathrm{D_y}}{}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dt}}{-}{2}{}{I}{}{\mathrm{D_z}}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dx}}{+}{2}{}{I}{}{\mathrm{D_z}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dt}}$ (2.21)

The second term on the right-hand side of (*) is

 E > $\mathrm{R2}≔\mathrm{ContractIndices}\left(\mathrm{barS},\mathrm{barF},\left[\left[3,1\right],\left[4,2\right]\right]\right)$
 ${\mathrm{R2}}{≔}{-}\frac{{2}}{{{x}}^{{2}}}{}{\mathrm{D_t}}{}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dx}}{+}\frac{{2}}{{{x}}^{{2}}}{}{\mathrm{D_t}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dt}}{-}{2}{}{{x}}^{{2}}{}{\mathrm{D_x}}{}{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dx}}{+}{2}{}{{x}}^{{2}}{}{\mathrm{D_x}}{}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dt}}{-}{2}{}{I}{}{\mathrm{D_y}}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dx}}{+}{2}{}{I}{}{\mathrm{D_y}}{}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dt}}{+}{2}{}{I}{}{\mathrm{D_z}}{}{\mathrm{dy}}{}{\mathrm{dt}}{}{\mathrm{dx}}{-}{2}{}{I}{}{\mathrm{D_z}}{}{\mathrm{dy}}{}{\mathrm{dx}}{}{\mathrm{dt}}$ (2.22)
 E > $\mathrm{LHS}≔2&multR$
 ${\mathrm{LHS}}{≔}{-}\frac{{4}}{{{x}}^{{2}}}{}{\mathrm{D_t}}{}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dx}}{+}\frac{{4}}{{{x}}^{{2}}}{}{\mathrm{D_t}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dt}}{-}{4}{}{{x}}^{{2}}{}{\mathrm{D_x}}{}{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dx}}{+}{4}{}{{x}}^{{2}}{}{\mathrm{D_x}}{}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dt}}$ (2.23)
 E > $\mathrm{RHS}≔\mathrm{R1}&plus\mathrm{R2}$
 ${\mathrm{RHS}}{≔}{-}\frac{{4}}{{{x}}^{{2}}}{}{\mathrm{D_t}}{}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dx}}{+}\frac{{4}}{{{x}}^{{2}}}{}{\mathrm{D_t}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dt}}{-}{4}{}{{x}}^{{2}}{}{\mathrm{D_x}}{}{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dx}}{+}{4}{}{{x}}^{{2}}{}{\mathrm{D_x}}{}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dt}}$ (2.24)
 E > $\mathrm{LHS}&minus\mathrm{RHS}$
 ${0}{}{\mathrm{D_t}}{}{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dt}}$ (2.25)