Where have I been?

Given smooth manifolds $M$ and $N$, $T(M\times N)\cong TM\times TN$.

Proof. Let $p : T(M\times N)\to TM\times TN$ given by $p(x,X) =
((\pi_M(x),X_M),(\pi_N(x),X_N))$, where $\pi_M$ is the projection
$M\times N\to M$, $X_m(f)=X(f\circ \pi_M)$ for $f:M\to \R$, and
$\pi_N, X_N$ are defined analogously. $p^{-1}$ is given as
follows: Choose coordinates $x_1,\ldots,x_m$ on $M$, and
$y_1,\ldots,y_n$ on $N$; note $x_1,\ldots,x_m,y_1,\ldots,y_n$ are
coordinates on $M\times N$. Then $p^{-1}$ takes $((x,X),(y,Y))$ to
\[\left((x,y),\sum_{i=1}^m X(x_i) \frac{\partial}{\partial x_i} +
\sum_{j=1}^n Y(y_j) \frac{\partial}{\partial y_j}\right)\] by
Lemma 2.3. Since $p$ and $p^{-1}$ are both clearly continuous, $p$
is the desired diffeomorphism.\qed

(Plus 5 other problems like this, and 120 pages of John Locke to read.)