# 5. The Tangent Bundle

In this lecture we define the *tangent bundle* $T M$ as the disjoint union of all the tangent spaces:

$$ T M := \bigsqcup_{p \in M} T_pM.$$

and prove this is naturally a smooth manifold of twice the dimension of $M$. We then play a similar game with the *cotangent bundle*, which is the disjoint union of all the *cotangent spaces. *Next, we explain how the maps $D \varphi(p)$ for $p \in M$ combine to give a single smooth map $ D \varphi \colon T M \to TN$.

Finally, we recall the Inverse and Implicit Function Theorems. We state the Euclidean version of the Inverse Function Theorem, and use it to prove the manifold version of the Inverse Function Theorem, and the Euclidean version of the Implicit Function Theorem. Next lecture we will take this one step further and prove a version of the Implicit Function Theorem for manifolds.

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