A development of generalized differentiation with respect to time is given within the structure of the functional approach to generalized function theory. The analysis is in parallel with techniques used in an earlier publication by F. Farassat and the author. The results are used to derive a generalized three dimensional Leibnitz' theorem that leads to a generalized Reynolds' transport theorem. This is then applied to derive the governing equations of continuum mechanics. It is shown that the process produces both the governing field equations and their associated discontinuity conditions in a single operation.
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