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DifferentialGeometry

 Pullback
 pullback a differential p-form by the Jacobian of a transformation

 Calling Sequence Pullback(Phi, omega)

Parameters

 Phi - a transformation from a manifold M to a manifold N omega - a differential r-form on the manifold N, where r is not greater than the dimension of M

Description

 • The pullback of omega with respect to a transformation Phi: M -> Nis an r-form theta on the manifold M and is denoted by theta = Phi^*(omega).  If p is a point of M and X_1, X_2, ..., X_r are vectors in T_pM, then:

theta(p)(X_1, X_2, ..., X_r) = omega(Phi_*(X_1), Phi_*(X_2), ..., Phi_*(X_r))  (*)

The pullback of a 0-form, that is, a real-valued function g on N, is the real-valued function f = g o Phi on M.

 • In components, let J be the Jacobian matrix of Phi computed with respect to a system of coordinates x^i on M and y^j on N and evaluated at p.  Let a be the row vector whose entries are the components of a 1-form omega at q = Phi(p), computed with respect to the coordinate basis on N. Then the matrix vector product b = a.J gives the components of theta = Phi^*(omega) with respect to the coordinate basis on M.
 • From the definition (*), it follows that Phi^* is a homomorphism from the ring of all differential forms on N to the ring of differential forms on M, that is, Phi^*(omega1 + omega2) = Phi^*(omega1) + Phi^*(omega2) and Phi^*(omega1 &w omega2) = Phi^*(omega1) &w Phi^*(omega2) (**) for all forms omega1 and omega2 on N.  Pullback uses property (*), applied to 1-forms, together with (**) to calculate the pullback of an r-form.
 • The Pullback command can be applied to a list of differential forms.
 • This command is part of the DifferentialGeometry package, and so can be used in the form Pullback(...) only after executing the command with(DifferentialGeometry).  It can always be used in the long form DifferentialGeometry:-Pullback.

Examples

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

Example 1.

Calculate the pullback of the differential form omega1 with respect to the transformation Phi1 at the point p1 = [x = 1, y = 2]. Check this result using the Jacobian of Phi1.

 > $\mathrm{DGsetup}\left(\left[x,y\right],M\right):$$\mathrm{DGsetup}\left(\left[u,v,w\right],N\right):$
 > $\mathrm{p1}≔\left[x=1,y=2\right]$
 ${\mathrm{p1}}{≔}\left[{x}{=}{1}{,}{y}{=}{2}\right]$ (1)
 > $\mathrm{Φ1}≔\mathrm{Transformation}\left(M,N,\left[u=x+2y,v=3x+4y,w=5x+6y\right]\right)$
 ${\mathrm{Φ1}}{≔}\left[{u}{=}{x}{+}{2}{}{y}{,}{v}{=}{3}{}{x}{+}{4}{}{y}{,}{w}{=}{5}{}{x}{+}{6}{}{y}\right]$ (2)
 > $\mathrm{ω1}≔v\mathrm{du}+w\mathrm{dv}+u\mathrm{dw}$
 ${\mathrm{ω1}}{≔}{\mathrm{du}}{}{v}{+}{\mathrm{dv}}{}{w}{+}{\mathrm{dw}}{}{u}$ (3)
 > $\mathrm{θ1}≔\mathrm{Pullback}\left(\mathrm{Φ1},\mathrm{ω1}\right)$
 ${\mathrm{θ1}}{≔}\left({23}{}{x}{+}{32}{}{y}\right){}{\mathrm{dx}}{+}\left({32}{}{x}{+}{44}{}{y}\right){}{\mathrm{dy}}$ (4)
 > $\mathrm{theta1_at_p1}≔\mathrm{eval}\left(\mathrm{θ1},\mathrm{p1}\right)$
 ${\mathrm{theta1_at_p1}}{≔}{87}{}{\mathrm{dx}}{+}{120}{}{\mathrm{dy}}$ (5)

We check this last result against a direct computation using the Jacobian of Phi1. First calculate the coordinates of q1 = Phi1(p1) and evaluate omega1 at this point.

 > $\mathrm{q1}≔\mathrm{ApplyTransformation}\left(\mathrm{Φ1},\mathrm{p1}\right)$
 ${\mathrm{q1}}{≔}\left[{u}{=}{5}{,}{v}{=}{11}{,}{w}{=}{17}\right]$ (6)
 > $\mathrm{omega1_at_q1}≔\mathrm{eval}\left(\mathrm{ω1},\mathrm{q1}\right)$
 ${\mathrm{omega1_at_q1}}{≔}{11}{}{\mathrm{du}}{+}{17}{}{\mathrm{dv}}{+}{5}{}{\mathrm{dw}}$ (7)
 > $a≔\mathrm{Vector}[\mathrm{row}]\left(\left[11,17,5\right]\right)$
 ${a}{≔}\left[\begin{array}{ccc}{11}& {17}& {5}\end{array}\right]$ (8)
 > $J≔\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{Φ1},"JacobianMatrix"\right)$
 ${J}{≔}\left[\begin{array}{rr}{1}& {2}\\ {3}& {4}\\ {5}& {6}\end{array}\right]$ (9)
 > $b≔a\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}J$
 ${b}{≔}\left[\begin{array}{cc}{87}& {120}\end{array}\right]$ (10)

The entries of b coincide with the components of theta1_at_p1.

Example 2.

The Pullback command can be applied to a list of differential forms.

 > $\mathrm{Pullback}\left(\mathrm{Φ1},\left[\mathrm{du},\mathrm{dv}\right]\right)$
 $\left[{\mathrm{dx}}{+}{2}{}{\mathrm{dy}}{,}{3}{}{\mathrm{dx}}{+}{4}{}{\mathrm{dy}}\right]$ (11)

Example 3.

Express the function f and the 2-form omega2 in spherical coordinates.

 > $\mathrm{DGsetup}\left(\left[x,y,z\right],\mathrm{E3}\right):$$\mathrm{DGsetup}\left(\left[\mathrm{ρ},\mathrm{θ},\mathrm{φ}\right],\mathrm{Sp}\right):$
 > $\mathrm{f2}≔{x}^{2}+{y}^{2}+{z}^{2}$
 ${\mathrm{f2}}{≔}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}$ (12)
 > $\mathrm{ω2}≔\mathrm{evalDG}\left(z\mathrm{dx}&w\mathrm{dy}-y\mathrm{dx}&w\mathrm{dz}+x\mathrm{dy}&w\mathrm{dz}\right)$
 ${\mathrm{ω2}}{≔}\left({z}{}{\mathrm{dx}}\right){}{\bigwedge }{}{\mathrm{dy}}{-}\left(\left({y}{}{\mathrm{dx}}\right){}{\bigwedge }{}{\mathrm{dz}}\right){+}\left({x}{}{\mathrm{dy}}\right){}{\bigwedge }{}{\mathrm{dz}}$ (13)
 > $\mathrm{Φ2}≔\mathrm{Transformation}\left(\mathrm{Sp},\mathrm{E3},\left[x=\mathrm{ρ}\mathrm{cos}\left(\mathrm{θ}\right)\mathrm{sin}\left(\mathrm{φ}\right),y=\mathrm{ρ}\mathrm{sin}\left(\mathrm{θ}\right)\mathrm{sin}\left(\mathrm{φ}\right),z=\mathrm{ρ}\mathrm{cos}\left(\mathrm{φ}\right)\right]\right)$
 ${\mathrm{Φ2}}{≔}\left[{x}{=}{\mathrm{\rho }}{}{\mathrm{cos}}{}\left({\mathrm{\theta }}\right){}{\mathrm{sin}}{}\left({\mathrm{\phi }}\right){,}{y}{=}{\mathrm{\rho }}{}{\mathrm{sin}}{}\left({\mathrm{\theta }}\right){}{\mathrm{sin}}{}\left({\mathrm{\phi }}\right){,}{z}{=}{\mathrm{\rho }}{}{\mathrm{cos}}{}\left({\mathrm{\phi }}\right)\right]$ (14)
 > $\mathrm{simplify}\left(\mathrm{Pullback}\left(\mathrm{Φ2},\mathrm{f2}\right)\right)$
 ${{\mathrm{ρ}}}^{{2}}$ (15)
 > $\mathrm{simplify}\left(\mathrm{Pullback}\left(\mathrm{Φ2},\mathrm{ω2}\right)\right)$
 ${-}\left(\left({\mathrm{sin}}{}\left({\mathrm{\phi }}\right){}{{\mathrm{\rho }}}^{{3}}{}{\mathrm{dtheta}}\right){}{\bigwedge }{}{\mathrm{dphi}}\right)$ (16)
 Sp >