In this lecture we recall the Inverse and Implicit Function Theorems for Euclidean spaces, and then prove versions valid for manifolds too. Along the way we introduce the concept of a submanifold $M$ of a larger manifold $N$. These come in two flavours: immersed submanifolds and embedded submanifolds.

Next lecture we will show that every manifold $M$ is diffeomorphic to an embedded submanifold of some Euclidean space $\mathbb{R}^k$—this is known as the Whitney Embedding Theorem.