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DifferentialGeometry

 ComposeTransformations
 compose a sequence of two or more transformations

 Calling Sequence ComposeTransformation(Phi1, Phi2, Phi3, ...)

Parameters

 Phi1, Phi2, Phi3 - transformations

Description

 • ComposeTransformation(Phi1, Phi2, Phi3, ...) returns the composition of the transformations Phi1, Phi2, Phi3, ..., that is, the transformation Psi = Phi1 o Phi2 o Phi3 ....  The domain frame of Phi1 must coincide with the range frame of Phi2, the domain frame of Phi2 must coincide with the range of frame of Phi3, and so on.
 • This command is part of the DifferentialGeometry package, and so can be used in the form ComposeTransformations(...) only after executing the command with(DifferentialGeometry).  It can always be used in the long form DifferentialGeometry:-ComposeTransformations.

Examples

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

Example 1.

Define some manifolds.

 > $\mathrm{DGsetup}\left(\left[x,y\right],M\right):$$\mathrm{DGsetup}\left(\left[u,v\right],N\right):$$\mathrm{DGsetup}\left(\left[t\right],P\right):$$\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2},\mathrm{x3}\right],Q\right):$

Define transformations F: M -> N;  G: P -> M;  H: N -> Q.

 > $F≔\mathrm{Transformation}\left(M,N,\left[u=3x+2y,v=x-y\right]\right)$
 ${F}{≔}{?}$ (1)
 > $G≔\mathrm{Transformation}\left(P,M,\left[x=\mathrm{cos}\left(t\right),y=\mathrm{sin}\left(t\right)\right]\right)$
 ${G}{≔}{?}$ (2)
 > $H≔\mathrm{Transformation}\left(N,Q,\left[\mathrm{x1}=u,\mathrm{x2}=v,\mathrm{x3}=1\right]\right)$
 ${H}{≔}{?}$ (3)

Compute the compositions F o G, H o F and H o F o G.

 > $\mathrm{ComposeTransformations}\left(F,G\right)$
 ${?}$ (4)
 > $\mathrm{ComposeTransformations}\left(H,F\right)$
 ${?}$ (5)
 > $\mathrm{ComposeTransformations}\left(H,F,G\right)$
 ${?}$ (6)

Example 2.

We can express the transformation T: P -> P as the composition of 3 transformations A, B, C.

 > $T≔\mathrm{Transformation}\left(P,P,\left[t=\mathrm{sqrt}\left(\mathrm{sin}\left(t\right)+2\right)\right]\right)$
 ${T}{≔}{?}$ (7)
 > $A≔\mathrm{Transformation}\left(P,P,\left[t=\mathrm{sin}\left(t\right)\right]\right)$
 ${A}{≔}{?}$ (8)
 > $B≔\mathrm{Transformation}\left(P,P,\left[t=t+2\right]\right)$
 ${B}{≔}{?}$ (9)
 > $C≔\mathrm{Transformation}\left(P,P,\left[t=\mathrm{sqrt}\left(t\right)\right]\right)$
 ${C}{≔}{?}$ (10)
 > $S≔\mathrm{ComposeTransformations}\left(C,B,A\right)$
 ${S}{≔}{?}$ (11)
 > $\mathrm{Tools}:-\mathrm{DGequal}\left(T,S\right)$
 ${\mathrm{true}}$ (12)

Example 3.

We can check that the transformation K is the inverse of the transformation F.

 > $K≔\mathrm{Transformation}\left(N,M,\left[x=\frac{2}{5}v+\frac{1}{5}u,y=\frac{1}{5}u-\frac{3}{5}v\right]\right)$
 ${K}{≔}{?}$ (13)
 > $\mathrm{ComposeTransformations}\left(F,K\right)$
 ${?}$ (14)
 > $\mathrm{ComposeTransformations}\left(K,F\right)$
 ${?}$ (15)

Example 4.

If pi: E -> M is a fiber bundle, then a section s of E is a transformation s: M -> E such that pi o s = identity on M.

Check that the map s is a section for E.

 > $\mathrm{DGsetup}\left(\left[u,v,w\right],E\right):$$\mathrm{DGsetup}\left(\left[x,y\right],M\right):$
 > $\mathrm{pi}≔\mathrm{Transformation}\left(E,M,\left[x=uv+{w}^{2},y={u}^{2}+{w}^{2}\right]\right)$
 ${\mathrm{π}}{≔}{?}$ (16)
 > $s≔\mathrm{Transformation}\left(M,E,\left[u=\mathrm{sqrt}\left(y\right),v=\frac{x}{\mathrm{sqrt}\left(y\right)},w=0\right]\right)$
 ${s}{≔}{?}$ (17)
 > $\mathrm{ComposeTransformations}\left(\mathrm{pi},s\right)$
 ${?}$ (18)

 See Also