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For a smooth map between manifolds $\varphi :\mathcal{M}\to \mathcal{N}$, the four constructions go in the directions dictated by composition and the differential of a smooth map.

The pullback of a smooth function $f\in C^{\infty }(\mathcal{N})$ is ${\varphi }^{\ast }f=f\circ \varphi$, a function on $\mathcal{M}$. The pushforward of a curve $\lambda :I\to \mathcal{M}$ is ${\varphi }_{\ast }\lambda =\varphi \circ \lambda$, a curve in $\mathcal{N}$ with the same parameter interval.

For a tangent vector $V\in T_p\mathcal{M}$, its pushforward is ${\varphi }_{\ast }V=d{\varphi }_p(V)\in T_{\varphi (p)}\mathcal{N}$. Intrinsically, treating a tangent vector as a derivation on functions,For a covector $\eta \in T_{\varphi (p)}^{\ast }\mathcal{N}$, the pullback of a covector is the dual linear map:No inverse map is required. A pushed-forward field for a general map is a field along that map; it need not assign a unique vector to an image point with several preimages.

Treat $V$ as a derivation. For an arbitrary smooth function $f$ on $\mathcal{N}$, the definition of the differential of a smooth map and the chain rule giveComparison with $({\varphi }_{\ast }V{)}^{\alpha }{\partial }_{\alpha }f$ for all $f$ provesThe Jacobian is an $n\times m$ matrix; it need not be square or invertible.

At $p\in \mathcal{Q}$, the differential of a smooth map defines the pushforward of a vector field byThis is a tangent vector at $\psi (p)$, and varies smoothly as a section of the pullback tangent bundle. In local coordinates its components are $X^i(p){\partial }_i{\psi }^a(p)$.

For a general smooth map between manifolds, this is a vector field along a map, rather than an intrinsic vector field on all of $N$. A vector field $Y$ on the image can be defined only if these vectors agree whenever two points have the same image, and the resulting field must also be smooth; a projectable vector field must meet that requirement. For a diffeomorphism, there is no ambiguity and $Y_q=d{\psi }_{{\psi }^{-1}(q)}{X}_{{\psi }^{-1}(q)}$. For example, $\psi (x)=x^2$ on $\mathbb{R}$ and $X={\partial }_x$ give $2x{\partial }_y$, with opposite nonzero values at the two preimages of $y>0$. Thus the source's notation must be read with this qualification.

The pullback of a covariant tensor composes each of its vector arguments with the differential of a smooth map. It takes a type $(0,s)$ tensor on the target to one on the source. Antisymmetry is not required; the pullback of a differential form is the alternating special case. If the target vectors are projected onto the image tangent space first, evaluating the pullback is unchanged.

The pullback of a covector is the dual linear map to the differential of a smooth map. In coordinates its components are $({\varphi }^{\ast }\eta {)}_i=({\partial }_i{y}^{\alpha }){\eta }_{\alpha }$. The same rectangular Jacobian defines vector pushforward, with its other index contracted. No invertibility is needed.

A pullback of a smooth function composes a function on the target with a smooth map between manifolds, producing a function on the source. It needs no inverse map. Acting on the pulled-back function defines the differential of a smooth map intrinsically as a map of derivations.

The pushforward of a contravariant tensor applies the differential of a smooth map to each vector factor of a type $(r,0)$ tensor. It is defined pointwise at a specified source point. A field generally gives a field along the map, not necessarily a unique tensor field on the image when several points have the same image. A diffeomorphism allows the usual transport of arbitrary mixed tensors through its inverse as well.

A pushforward of a curve composes a parametrized curve in the source with a smooth map between manifolds. Its tangent is the image of the original tangent under the differential of a smooth map. A constant image curve has zero tangent, so a general smooth map need not preserve regularity of curves.