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

Let ${\displaystyle \varphi :U\to V}$ be a smooth map from an open subset ${\displaystyle U}$ of ${\displaystyle \mathbb {R} ^{m}}$ to an open subset ${\displaystyle V}$ of ${\displaystyle \mathbb {R} ^{n}}$. For any point ${\displaystyle x}$ in ${\displaystyle U}$, the Jacobian of ${\displaystyle \varphi }$ at ${\displaystyle x}$ (with respect to the standard coordinates) is the matrix representation of the total derivative of ${\displaystyle \varphi }$ at ${\displaystyle x}$, which is a linear map

between their tangent spaces. Note the tangent spaces ${\displaystyle T_{x}\mathbb {R} ^{m},T_{\varphi (x)}\mathbb {R} ^{n}}$ are isomorphic to ${\displaystyle \mathbb {R} ^{m}}$ and ${\displaystyle \mathbb {R} ^{n}}$, respectively. The pushforward generalizes this construction to the case that ${\displaystyle \varphi }$ is a smooth function between any smooth manifolds ${\displaystyle M}$ and ${\displaystyle N}$.

Let ${\displaystyle \varphi \colon M\to N}$ be a smooth map of smooth manifolds. Given ${\displaystyle x\in M,}$ the differential of ${\displaystyle \varphi }$ at ${\displaystyle x}$ is a linear map

from the tangent space of ${\displaystyle M}$ at ${\displaystyle x}$ to the tangent space of ${\displaystyle N}$ at ${\displaystyle \varphi (x).}$ The image ${\displaystyle d\varphi _{x}X}$ of a tangent vector ${\displaystyle X\in T_{x}M}$ under ${\displaystyle d\varphi _{x}}$ is sometimes called the pushforward of ${\displaystyle X}$ by ${\displaystyle \varphi .}$ The exact definition of this pushforward depends on the definition one uses for tangent vectors (for the various definitions see tangent space).

If tangent vectors are defined as equivalence classes of the curves ${\displaystyle \gamma }$ for which ${\displaystyle \gamma (0)=x,}$ then the differential is given by

Here, ${\displaystyle \gamma }$ is a curve in ${\displaystyle M}$ with ${\displaystyle \gamma (0)=x,}$ and ${\displaystyle \gamma '(0)}$ is tangent vector to the curve ${\displaystyle \gamma }$ at ${\displaystyle 0.}$ In other words, the pushforward of the tangent vector to the curve ${\displaystyle \gamma }$ at ${\displaystyle 0}$ is the tangent vector to the curve ${\displaystyle \varphi \circ \gamma }$ at ${\displaystyle 0.}$

Alternatively, if tangent vectors are defined as derivations acting on smooth real-valued functions, then the differential is given by