Pushforward (differential)

"If a map, φ, carries every point on manifold M to manifold N then the pushforward of φ carries vectors in the tangent space at every point in M to a tangent space at every point in N." — If a map, φ, carries every point on manifold M to manifold N then the pushforward of φ carries vectors in the tangent space at every point in M to a tangent space at every point in N.

In differential geometry, pushforward is a linear approximation of smooth maps (formulating manifold) on tangent spaces. Suppose that $\varphi \colon M\to N$ is a smooth map between smooth manifolds; then the differential of $\varphi$ at a point $x$ , denoted $\mathrm {d} \varphi _{x}$ , is, in some sense, the best linear approximation of $\varphi$ near $x$ . It can be viewed as a generalization of the total derivative of ordinary calculus. Explicitly, the differential is a linear map from the tangent space of $M$ at $x$ to the tangent space of $N$ at $\varphi (x)$ , $\mathrm {d} \varphi _{x}\colon T_{x}M\to T_{\varphi (x)}N$ . Hence it can be used to push tangent vectors on $M$ forward to tangent vectors on $N$ . The differential of a map $\varphi$ is also called, by various authors, the derivative or total derivative of $\varphi$ .