The Reynolds transport theorem gives a generic conservation law for a conserved quantity carried by fluid flow through a continuous control volume, providing the foundation for all conservation laws of fluid mechanics. We present new spatial and parametric multivariate generalisations of this theorem, each of which provides a continuous mapping of the contents of a domain within a coordinate space. The most general form involves mappings on a manifold, presented in a multivariate extension of

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... terior calculus. These theorems are used to generate tables of new conservation laws in integral form in different spaces, including Eulerian velocity space and velocity-volume phase space, based on fluid densities defined in these spaces.