and an order- tensor may likewise be defined as an element of a tensor product of different vector spaces. A type tensor, in the sense defined previously, is also a tensor of order in this more general sense. The concept of tensor product can be extended to arbitrary modules over a ring. Tensors in infinite dimensions The notion of a tensor can be generalized in a variety of ways to infinite dimensions. One, for instance, is via the tensor product of Hilbert spaces. Another way of generalizing the idea of tensor, common in nonlinear analysis, is via the multilinear maps definition where instead of using finite-dimensional vector spaces and their algebraic duals, one uses infinite-dimensional Banach spaces and their continuous dual. Tensors thus live naturally on Banach manifolds and Fréchet manifolds. Tensor densities Suppose that a homogeneous medium fills , so that the density of the medium is described by a single scalar value in . The mass, in kg, of a region is obtained by multiplying by the volume of the region , or equivalently integrating the constant over the region: where the Cartesian coordinates , , are measured in . If the units of length are changed into , then the numerical values of the coordinate functions must be rescaled by a factor of 100: The numerical value of the density must then also transform by to compensate, so that the numerical value of the mass in kg is still given by integral of . Thus (in units of ). More generally, if the Cartesian coordinates , , undergo a linear transformation, then the numerical value of the density must change by a factor of the reciprocal of the absolute value of the determinant of the coordinate transformation, so that the integral remains invariant, by the change of variables formula for integration. Such a quantity that scales by the reciprocal of the absolute value of the determinant of the coordinate transition map is called a scalar density. To model a non-constant density, is a function of the variables , , (a scalar field), and under a curvilinear change of coordinates, it transforms by the reciprocal of the Jacobian of the coordinate change. For more on the intrinsic meaning, see Density on a manifold. A tensor density transforms like a tensor under a coordinate change, except that it in addition picks up a factor of the absolute value of