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Tuesday, November 1, 2016

Explicit pushforwards and pullbacks

 Preliminary exam prep

Here we consider a map f:MN between manifolds of dimension m and n, respectively, and the maps that it induces. Let pM with x1,,xm a local chart for Up and y1,,yn a local chart for Vf(p). Induced from f are the differential (or pushforward) df and the pullback df, which are duals of each other:
dfp : TpMTf(p)Ndf : TMTNα(βα(βf))dfp : Tf(p)NTpMdf : TNTMωωfkTNkTMω dy1dyk(ωf) d(y1f)d(ykf)

These maps may be described by the diagram below.
Example: For example, consider the map f:R3R3 given by f(x,y,z)=(xy,3z2,xz+yz), with the image having coordinates (u,v,w). With elements
2xx5zyTM,2uv+w5C(N),cos(uv)TN,

we have
dfp(2xx5zy)(2uv+w5)=(2xx5zy)(6(xy)z2+xz+yz5)(p),dfp(cos(uv))=cos((xy)3z2),(2dfp)(cos(uv)dudw)=cos((xy)3z2)d(3z2)d(xz+yz)=cos((xy)3z2)(6z2 dxdz6z2 dydz).

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