0x032 Differential Geometry
Topological Manifold
Differential Manifold
***Definition (differential, pushforward)** Let \(\varphi: M \to N\) be a smooth map between smooth manifolds. Given \(x \in M\), the differential of \(\varphi\) at \(x\) is a linear map:
\[d\varphi(x): T_x M \to T_{\varphi(x)}N\]
from tangent space of \(M\) at \(x\) to the tangent space of \(N\) at \(\varphi(x)\)
