Covariant field families and variations

Let $M$ be a smooth four-manifold and let $\pi :E\to M$ be a smooth fibre bundle. A field configuration is a smooth section $\varphi :M\to E$, so $\pi \circ \varphi ={\mathrm{id}}_M$. For a finite-dimensional smooth manifold $P$, a smooth field family is a smooth map $\mathrm{\Phi }:P\times M\to E$ with $\pi (\mathrm{\Phi }(p,x))=x$ for all $(p,x)$; it gives configurations ${\varphi }_p(x)=\mathrm{\Phi }(p,x)$. For $u\in T_{p}P$, the variation ${\delta }_u{\varphi }_p$ at $x$ is the tangent vector $d(r\mapsto \mathrm{\Phi }(c(r),x)){\mathrm{\mid }}_{r=0}$, where $c(0)=p$ and $c^{\mathrm{\prime }}(0)=u$. In a local fibre chart this is the ordinary derivative of the field components with respect to the parameter. For commuting parameter coordinates $s_1,s_2$, ${\delta }_i\varphi ={\mathrm{\partial }}_{s_i}\varphi$ and ${\delta }_1{\delta }_2\varphi ={\delta }_2{\delta }_1\varphi$.

Smooth fibre bundles and local trivializations

Smooth maps between manifolds

Smooth fibre bundles and local trivializations

$C^r$ and smooth maps between smooth manifolds

The parameter manifold is finite-dimensional. This definition does not assume that the full space of field configurations or its solution subset has a global infinite-dimensional manifold structure.