Скачать или смотреть 370 Video 17: pushforward

In this video we explain how maps f:\R^n\to\R^m “push forward” vectors from T_p\R^n to T_p\R^m.

In linear algebra, you learned the effect of coordinate change on vectors: a vector v, when viewed in a new coordinate system, has to be written in a new way. The concept of “pushforward” is how to extend these ideas to manifolds.

When f:\R^n\to\R^m, we know it has a derivative, its Jacobian Df(p):\R^n\to\R^m. But Df varies from point to point, so is there a better way to understand what it is?

Write f(p)=(f_1(p),...,f_m(p)), where f_i:\R^n\to\R. Then the i^th row of Df(p) is simply df_i(p), and we plug vectors from T_p\R^n, not from \R^n, into df_i(p). So to really make sense of Df(p), we should be plugging in vectors from T_p\R^n into Df(p), and it should spit out vectors in T_{f(p)}\R^m.

For example, let’s consider f:\R^2_{r\theta}\to\R^2_{xy} given by f(r,\theta)=(r\cos\theta,r\sin\theta). (If you want a linear example, check out S6.1-6.2).
Draw the curves of constant r and constant \theta on the xy-plane (images under f).
Compute Df(r,\theta) and draw the images of several vectors [1 0]_{(r,\theta)} and [0 1]_{(r,\theta)}.

This story generalizes to all manifolds. If f:M\to N, then Df(p):T_pM\to T_pN. We also denote Df(p) by T_pf or f_* (“f-pushforward”).
def: f:M\to N. The pushforward of f is the map f_*:TM \to TN given by f_*(v_p)=Df(p)(v_p).

As an example, consider f:S^2\to\R given by f(x,y,z)=z. Draw vectors which are sent to zero under Df and those which are not.