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The pushforward of a contravariant tensor applies $d{\varphi }_p$ to each of its $r$ vector slots. Choose a basis $e_i$ of $T_p\mathcal{M}$ and expandIts pushforward is $T^{i_1\cdots i_r}(d\varphi e_{i_1})\otimes \cdots \otimes (d\varphi e_{i_r})$. Each factor is tangent to the hypersurface and fixed by $P$. ConsequentlyThe basis expansion proves the result for arbitrary tensors, rather than only a single decomposable tensor product.

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.