L13_A

       1. Review: The Tangent Plane
       2. Proof: The Tangent Plane to f^{-1}(a) Is the Kernel of df_p
       3. Example: Sphere

L13_B

       1. Example: Sphere (cont.)
       2. Proposition: Differential of a Map

L13_C

       1. Example: Rotation Map on Unit Sphere
       2. Definition: Local Diffeomorphism
       3. Proposition: The Differential Is a Local Isomorphism
           Implies a Local Diffeomorphism