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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.

Evaluate the pullback of a covector on the coordinate basis vector ${\partial }_i$. Its pushforward is $({\partial }_i{y}^{\alpha }){\partial }_{\alpha }$, soEquivalently, pulling back the coordinate one-forms gives ${\varphi }^{\ast }(d{y}^{\alpha })=d(y^{\alpha }\circ \varphi )=({\partial }_i{y}^{\alpha })d{x}^i$. The same Jacobian enters the vector pushforward and the covector pullback, with the different index contractions reflecting their duality.

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.