Is the Pushforward the Same as the Differential?

For a smooth map between manifolds, the differential at a point is the pushforward of tangent vectors. These are two names for the same basic tangent-space map. The distinction that often causes confusion is that the differential of a real-valued function is a covector, while the pushforward of a tangent vector is a tangent vector in the target.

The differential as a map of tangent spaces

Let $f:M\to N$ be smooth and let $p\in M$. The differential of $f$ at $p$ is a linear map

$$df_p:T_pM\longrightarrow T_{f(p)}N.$$

It takes a tangent vector based at $p$ and produces a tangent vector based at the image point $f(p)$. The same map is commonly denoted by $T_pf$ or $f_*|_p$. In this setting, pushforward means applying this map to a tangent vector:

$$f_*v=df_p(v),\qquad v\in T_pM.$$

A tangent vector can be understood as the velocity of a curve. If $c(t)$ is a curve in $M$ with $c(0)=p$ and $c'(0)=v$, then

$$df_p(v)=(f\circ c)'(0).$$

This curve description also shows why the map is called a derivative: it records how the map $f$ changes to first order in the direction $v$.

Why the differential of a function can look different

In calculus, “the differential of a function” often means a real-valued function $g:M\to\mathbb{R}$. Since $T_{g(p)}\mathbb{R}$ is naturally identified with $\mathbb{R}$, its differential has the form

$$dg_p:T_pM\longrightarrow\mathbb{R}.$$

A linear map from a tangent space to the real numbers is a covector, an element of the cotangent space $T_p^*M$. Thus $dg_p$ is not a tangent vector; it is a linear functional that evaluates tangent vectors. For $v\in T_pM$, its value $dg_p(v)$ is the directional derivative of $g$ in direction $v$.

For a general map $f:M\to N$, the differential still has the same role, but its output lies in $T_{f(p)}N$, not necessarily in $\mathbb{R}$. The scalar-valued case is simply the special case $N=\mathbb{R}$.

A worked coordinate example

Consider

$$f:\mathbb{R}^2\to\mathbb{R}^2,\qquad f(x,y)=(x^2+y,xy).$$

In the standard coordinates, the differential is represented by the Jacobian matrix

$$Df(x,y)=\begin{pmatrix}2x&1\\y&x\end{pmatrix}.$$

At $p=(1,2)$, the image is $f(p)=(3,2)$, and

$$df_p=\begin{pmatrix}2&1\\2&1\end{pmatrix}.$$

Take the tangent vector $v=(3,-1)\in T_p\mathbb{R}^2$. Its pushforward is

$$df_p(v)=\begin{pmatrix}2&1\\2&1\end{pmatrix}\begin{pmatrix}3\\-1\end{pmatrix}=\begin{pmatrix}5\\5\end{pmatrix}.$$

This resulting vector belongs to $T_{(3,2)}\mathbb{R}^2$, the tangent space at the image point. In Euclidean space all these tangent spaces can be identified with $\mathbb{R}^2$, but the base point still matters on a general manifold.

For comparison, let $h(x,y)=x^2+y$, a scalar-valued function, and use the same point and vector. Then

$$dh_p(3,-1)=2(1)(3)+1(-1)=5.$$

Here the output is a number, because $dh_p$ is a covector.

Pushforward of a vector field

There is a related terminology issue. A vector field $X$ on $M$ assigns a vector $X_p$ to each point $p$. Applying $df_p$ gives a vector based at $f(p)$, so across all points this defines a vector field along the map $f$. It does not automatically define a single vector field on $N$: different points of $M$ can map to the same point of $N$ and produce different vectors there.

If $f$ is a diffeomorphism, there is no such ambiguity, and the pushforward vector field on $N$ is defined by

$$(f_*X)_{f(p)}=df_p(X_p).$$

For more general maps, extra conditions are needed to obtain a vector field on the target.

Common mistakes

• Treating $df_p$ as a vector. It is a linear map. For a real-valued function, it is specifically a covector.
• Ignoring the base point. The map goes from $T_pM$ to $T_{f(p)}N$, not generally from one fixed vector space to itself.
• Assuming every pushed-forward field is a field on $N$. Pointwise tangent vectors always push forward, but a whole field may not descend unambiguously to the target.
• Confusing pushforward with pullback. Differentials push tangent vectors forward. Covectors and differential forms are pulled back in the opposite direction using the differential; for example, $f^*\alpha$ is a covector on $M$ when $\alpha$ is a covector on $N$.

So, when the object being pushed forward is a tangent vector, “pushforward” is the pointwise differential of the map. The surrounding context determines whether the phrase refers to that tangent-space map or to another induced operation.