In differential geometry, the Jacobian of a transformation Phi: M -> N is considered to be a linear transformation between tangent spaces Phi_*: T_pM -> T_qN, where p is a point of M and q = Phi(p).  Let X be a tangent vector in T_pM.  Then Y = Phi_*(X) may be defined in one of two ways.  If X is viewed as a derivation on the smooth functions defined in a neighborhood of p in M, then Y = Phi_*(X) is the derivation on the smooth functions g defined in a neighborhood q of N by Y(g) = X(Phi o g).  If X is viewed as the tangent vector to a curve alpha(t) at t = 0, then Y is the tangent vector to the curve beta = (Phi o alpha)(t) at t = 0.

The vector Y = Phi_*(X) is called the pushforward of X by Phi.  The vector Y is computed by Pushforward(Phi, X, pt), where pt = [a1, a2, a3, ...] or pt = [x1 = a1, x2 = a2, x3 = a3, ...] are the coordinates of the point p.

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 column vector whose entries are the components of X computed with respect to the coordinate basis on M. Then the matrix vector product b = J.a gives the components of Y = Phi_*(X) with respect to the coordinate basis on N.

The command Pushforward(Phi, X) returns a vector on N whose coefficients are functions of the coordinates on M.  The result defines the pushforward of the vector field X at an arbitrary point of M.

If Phi is an invertible transformation with inverse Psi, then a vector field  Y on N can be constructed from a vector field X on M by setting Y(q) = Phi_*(X(p)), where q is an arbitrary point of N and p = Psi(q).  The command Pushforward(Phi,  Psi, X) returns the vector field Y.

The Pushforward command can be applied to a list of vectors.

This command is part of the DifferentialGeometry package, and so can be used in the form Pushforward(...) only after executing the command with(DifferentialGeometry).  It can always be used in the long form DifferentialGeometry:-Pushforward.

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

$\mathrm{DGsetup}\left(\left[x\,y\right]\,M\right)\:$$\mathrm{DGsetup}\left(\left[u\,v\right]\,N\right)\:$

$\mathrm{\Φ1}≔\mathrm{Transformation}\left(M\,N\,\left[u=\ln \left(x^2+y^2\right)\,v=\frac{x}{y}\right]\right)$

$\mathrm{\Φ1}≔\left[u=\ln \left(x^2+y^2\right)\,v=\frac{x}{y}\right]$

$\mathrm{X1}≔\mathrm{evalDG}\left(a\mathrm{D\_x}+b\mathrm{D\_y}\right)$

$\mathrm{X1}≔a\mathrm{D\_x}+b\mathrm{D\_y}$

$\mathrm{Y1}≔\mathrm{Pushforward}\left(\mathrm{\Φ1}\,\mathrm{X1}\,\left[x=1\,y=3\right]\right)$

$\mathrm{Y1}≔\left(\frac{a}{5}+\frac{3b}{5}\right)\mathrm{D\_u}+\left(\frac{a}{3}-\frac{b}{9}\right)\mathrm{D\_v}$

$J≔\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{\Φ1}\,''JacobianMatrix''\right)$

$J≔\left[\begin{array}{cc}\frac{2x}{x^2+y^2}& \frac{2y}{x^2+y^2}\\ \frac{1}{y}& -\frac{x}{y^2}\end{array}\right]$

$\mathrm{J1}≔\mathrm{eval}\left(J\,\left[x=1\,y=3\right]\right)$

$\mathrm{J1}≔\left[\begin{array}{cc}\frac{1}{5}& \frac{3}{5}\\ \frac{1}{3}& -\frac{1}{9}\end{array}\right]$

$C≔\mathrm{J1}·\mathrm{Vector}\left(\left[a\,b\right]\right)$

$C≔\left[\begin{array}{c}\frac{a}{5}+\frac{3b}{5}\\ \frac{a}{3}-\frac{b}{9}\end{array}\right]$

$\mathrm{\Φ2}≔\mathrm{Transformation}\left(M\,N\,\left[u=\ln \left(x^2+y^2\right)\,v=\frac{x}{y}\right]\right)$

$\mathrm{\Φ2}≔\left[u=\ln \left(x^2+y^2\right)\,v=\frac{x}{y}\right]$

$\mathrm{p1}≔\left[x=1\,y=1\right]$

$\mathrm{p1}≔\left[x=1\,y=1\right]$

$\mathrm{p2}≔\left[x=-1\,y=-1\right]$

$\mathrm{p2}≔\left[x=\mathrm{-1}\,y=\mathrm{-1}\right]$

$\mathrm{ApplyTransformation}\left(\mathrm{\Φ2}\,\mathrm{p1}\right),\mathrm{ApplyTransformation}\left(\mathrm{\Φ2}\,\mathrm{p2}\right)$

$\left[u=\ln \left(2\right)\,v=1\right],\left[u=\ln \left(2\right)\,v=1\right]$

$\mathrm{Pushforward}\left(\mathrm{\Φ2}\,\mathrm{D\_x}\,\mathrm{p1}\right),\mathrm{Pushforward}\left(\mathrm{\Φ2}\,\mathrm{D\_x}\,\mathrm{p2}\right)$

$\mathrm{D\_u}+\mathrm{D\_v},-\mathrm{D\_u}-\mathrm{D\_v}$

$\mathrm{\Φ3}≔\mathrm{Transformation}\left(M\,N\,\left[u=\ln \left(x^2+y^2\right)\,v=\frac{x}{y}\right]\right)$

$\mathrm{\Φ3}≔\left[u=\ln \left(x^2+y^2\right)\,v=\frac{x}{y}\right]$

$\mathrm{X3}≔\mathrm{evalDG}\left(x^2\mathrm{D\_x}+x^2\mathrm{D\_y}\right)$

$\mathrm{X3}≔x^2\mathrm{D\_x}+x^2\mathrm{D\_y}$

$\mathrm{Pushforward}\left(\mathrm{\Φ3}\,\mathrm{X3}\right)$

$\frac{2x^2\left(x+y\right)\mathrm{D\_u}}{x^2+y^2}-\frac{x^2\left(x-y\right)\mathrm{D\_v}}{y^2}$

$\mathrm{DGsetup}\left(\left[r\,\mathrm{\theta }\right]\,P\right)\:$

$\mathrm{X4}≔\mathrm{evalDG}\left(\frac{y}{x}\mathrm{D\_x}-2\mathrm{D\_y}\right)$

$\mathrm{X4}≔\frac{y\mathrm{D\_x}}{x}-2\mathrm{D\_y}$

$\mathrm{\Φ4}≔\mathrm{Transformation}\left(P\,M\,\left[x=r\cos \left(\mathrm{\theta }\right)\,y=r\sin \left(\mathrm{\theta }\right)\right]\right)$

$\mathrm{\Φ4}≔\left[x=r\cos \left(\mathrm{\theta }\right)\,y=r\sin \left(\mathrm{\theta }\right)\right]$

$\mathrm{\_EnvExplicit}≔\mathrm{true}\:$

$\mathrm{invPhi4}≔\mathrm{InverseTransformation}\left(\mathrm{\Φ4}\right)$

$\mathrm{invPhi4}≔\left[r=\sqrt{x^2+y^2}\,\mathrm{\theta }=\arctan \left(\frac{y}{\sqrt{x^2+y^2}}\,\frac{x}{\sqrt{x^2+y^2}}\right)\right]$

$\mathrm{Pushforward}\left(\mathrm{invPhi4}\&comma;\mathrm{\&Phi;4}\&comma;\mathrm{X4}\right)\mathbf{assuming}0<r$

$-\sin \left(\mathrm{\theta }\right)\mathrm{D\_r}+\frac{\left(-\cos \left(\mathrm{\theta }\right)-\sec \left(\mathrm{\theta }\right)\right)\mathrm{D\_\&theta;}}{r}$

$\mathrm{DGsetup}\left(\left[t\right]\&comma;R\right)\&colon;$$\mathrm{DGsetup}\left(\left[x\&comma;y\&comma;z\right]\&comma;\mathrm{E3}\right)\&colon;$

$C≔\mathrm{Transformation}\left(R\&comma;\mathrm{E3}\&comma;\left[x=t^2\&comma;y=t^3\&comma;z=t^3\right]\right)$

$C≔\left[x=t^2\&comma;y=t^3\&comma;z=t^3\right]$

$\mathrm{Pushforward}\left(C\&comma;\mathrm{D\_t}\right)$

$2t\mathrm{D\_x}+3t^2\mathrm{D\_y}+3t^2\mathrm{D\_z}$

$\mathrm{DGsetup}\left(\left[x\&comma;y\right]\&comma;\mathrm{E2}\right)\&colon;$$\mathrm{DGsetup}\left(\left[x\&comma;y\&comma;z\right]\&comma;\mathrm{E3}\right)\&colon;$

$S≔\mathrm{Transformation}\left(\mathrm{E2}\&comma;\mathrm{E3}\&comma;\left[x=x\&comma;y=y\&comma;z=x^3-3x{y}^2\right]\right)$

$S≔\left[x=x\&comma;y=y\&comma;z=x^3-3y^{2}x\right]$

$\mathrm{Pushforward}\left(S\&comma;\left[\mathrm{D\_x}\&comma;\mathrm{D\_y}\right]\right)$

$\left[\mathrm{D\_x}+\left(3x^2-3y^2\right)\mathrm{D\_z}\&comma;\mathrm{D\_y}-6xy\mathrm{D\_z}\right]$

$\mathrm{DGsetup}\left(\left[u\&comma;v\right]\&comma;\mathrm{E2}\right)\&colon;$$\mathrm{DGsetup}\left(\left[x\&comma;y\&comma;z\right]\&comma;\mathrm{E3}\right)\&colon;$

$\mathrm{\&Phi;5}≔\mathrm{Transformation}\left(\mathrm{E3}\&comma;N\&comma;\left[u=\frac{x}{z}\&comma;v=\frac{y}{z}\right]\right)$

$\mathrm{\&Phi;5}≔\left[u=\frac{x}{z}\&comma;v=\frac{y}{z}\right]$

$\mathrm{X5}≔\mathrm{evalDG}\left(\frac{1}{x^2+y^2+z^2}\left(x^{2}y\mathrm{D\_x}+y{z}^2\mathrm{D\_y}-xyz\mathrm{D\_z}\right)\right)$

$\mathrm{X5}≔\frac{y{x}^2\mathrm{D\_x}}{x^2+y^2+z^2}+\frac{z^{2}y\mathrm{D\_y}}{x^2+y^2+z^2}-\frac{zxy\mathrm{D\_z}}{x^2+y^2+z^2}$

$\mathrm{Pushforward}\left(\mathrm{\&Phi;5}\&comma;\mathrm{X5}\right)$

$\frac{2y{x}^2\mathrm{D\_u}}{z\left(x^2+y^2+z^2\right)}+\frac{y\left(xy+z^2\right)\mathrm{D\_v}}{z\left(x^2+y^2+z^2\right)}$

$\mathrm{\Sigma }≔\mathrm{Transformation}\left(N\&comma;\mathrm{E3}\&comma;\left[x=u\&comma;y=v\&comma;z=1\right]\right)$

$\mathrm{\Sigma }≔\left[x=u\&comma;y=v\&comma;z=1\right]$

$\mathrm{ComposeTransformations}\left(\mathrm{\&Phi;5}\&comma;\mathrm{\Sigma }\right)$

$\left[u=u\&comma;v=v\right]$

$\mathrm{Pushforward}\left(\mathrm{\&Phi;5}\&comma;\mathrm{\Sigma }\&comma;\mathrm{X5}\right)$

$\frac{2v{u}^2\mathrm{D\_u}}{u^2+v^2+1}+\frac{v\left(uv+1\right)\mathrm{D\_v}}{u^2+v^2+1}$

$\mathrm{ChangeFrame}\left(M\right)$

$N$

$\mathrm{Pushforward}\left(\mathrm{\&Phi;1}\&comma;\left[\mathrm{D\_x}\&comma;\mathrm{D\_y}\right]\&comma;\left[x=1\&comma;y=3\right]\right)$

$\left[\frac{\mathrm{D\_u}}{5}+\frac{\mathrm{D\_v}}{3}\&comma;\frac{3\mathrm{D\_u}}{5}-\frac{\mathrm{D\_v}}{9}\right]$