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Divergence- L q and divergence-measure tensor fields are introduced and their properties are analyzed. These measure-theoretic tools are then used to study the gradient flow of linear growth functionals of maps into the unit sphere. A BV-solution concept is introduced and the existence of such a solution is established for the flow using the energy method together with a regularization and a penalization technique. Based on the analytical results, practical fully discrete finite element methods are then proposed for computing weak solutions of the gradient flow, and the convergence of the proposed numerical method is also proved. [ABSTRACT FROM AUTHOR]