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Guillemin-Pollack 1.2.11:

  1. Suppose $f:X\rightarrow Y$ is a smooth map, and let $F:X\rightarrow X\times Y$ be $F(x)=(x,f(x))$. Show that $$dF_x(v)=(v,df_x(v)).$$
  2. Prove that the tangent space to the graph of $f$ at the point $(x,f(x))$ is the graph of $df_x:T_x(X)\rightarrow T_{f(x)}(Y)$.

I know how to do $(1)$ using GP $1.2.9(d)$: $d(f\times g)_{(x,y)}=df_x\times dg_y$. But I'm not sure how to approach $(2)$.

I know that $F$ is a diffeomorphism (GP 1.1.17). (And the way the book shows that the graph of a smooth function $f$ is a manifold is by asking the reader to show $F$ is a diffeomorphism; so the graph is a manifold if $X$ is.)

So let $X\subset R^N, Y\subset R^M$ be manifolds of dimension $n$ and $m$ resp. and let $\phi: U\subset R^n\rightarrow X$ be a local parametrization around $x\in X$ with $\phi(z)=x$. Then the tangent space to the graph of $f$ is the image of the map $$d(F\circ \phi)_z: R^n\rightarrow R^n\times R^m.$$

I don't know how to connect this with the graph of $df_x:T_x(X)\rightarrow T_{f(x)}(Y)$.

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    $\begingroup$ Well, what is $d(F\circ\phi)_z$? And what is $\text{im}(d\phi_z)$? $\endgroup$ – Ted Shifrin Jan 23 '18 at 1:33
  • $\begingroup$ @TedShifrin: What a great hint! I think I got it now, thank you. $\endgroup$ – user500094 Jan 23 '18 at 1:40
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    $\begingroup$ (It comes from teaching this course close to a dozen times.) :P $\endgroup$ – Ted Shifrin Jan 23 '18 at 1:46

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